Element (category theory)

In category theory, the concept of an element, or a point, generalizes the more usual set theoretic concept of an element of a set to an object of any category. This idea often allows restating of definitions or properties of morphisms (such as monomorphism or product) given by a universal property in more familiar terms, by stating their relation to elements. Some very general theorems, such as Yoneda's lemma and the Mitchell embedding theorem, are of great utility for this, by allowing one to work in a context where these translations are valid. This approach to category theory, in particular the use of the Yoneda lemma in this way, is due to Grothendieck, and is often called the method of the functor of points.
Definition.
Suppose C is any category and "A", "T" are two objects of C. A "T"-valued point of "A" is simply an arrow formula_1. The set of all "T"-valued points of "A" varies functorially with "T", giving rise to the "functor of points" of "A"; according to the Yoneda lemma, this completely determines "A" as an object of C.
Properties of morphisms.
Many properties of morphisms can be restated in terms of points. For example, a map formula_2 is said to be a monomorphism if
Suppose formula_7 and formula_8 in "C". Then "g" and "h" are "A"-valued points of "B", and therefore monomorphism is equivalent to the more familiar statement
Some care is necessary. "f" is an epimorphism if the dual condition holds:
In set theory, the term "epimorphism" is synonymous with "surjection", i.e.
This is clearly not the translation of the first statement into the language of points, and in fact these statements are "not" equivalent in general. However, in some contexts, such as abelian categories, "monomorphism" and "epimorphism" are backed by sufficiently strong conditions that in fact they do allow such a reinterpretation on points.
Similarly, categorical constructions such as the product have pointed analogues. Recall that if "A", "B" are two objects of C, their product "A"×"B" is an object such that
In this definition, "f" and "g" are "T"-valued points of "A" and "B", respectively, while "h" is a "T"-valued point of "A"×"B". An alternative definition of the product is therefore:
This is the more familiar definition of the product of two sets.
Geometric origin.
The terminology is geometric in origin; in algebraic geometry, Grothendieck introduced the notion of a scheme in order to unify the subject with arithmetic geometry, which dealt with the same idea of studying solutions to polynomial equations (i.e. algebraic varieties) but where the solutions are not complex numbers but rational numbers, integers, or even elements of some finite field. A scheme is then just that: a scheme for collecting together all the manifestations of a variety defined by the same equations but with solutions taken in different number sets. One scheme gives a complex variety, whose points are its formula_19-valued points, as well as the set of formula_20-valued points (rational solutions to the equations), and even formula_21-valued points (solutions modulo "p").
One feature of the language of points is evident from this example: it is, in general, not enough to consider just points with values in a single object. For example, the equation formula_22 (which defines a scheme) has no real solutions, but it has complex solutions, namely formula_23. It also has one solution modulo 2 and two modulo 5, 13, 29, etc. (all primes that are 1 modulo 4). Just taking the real solutions would give no information whatsoever.
Relation with set theory.
The situation is analogous to the case where C is the category Set, of sets of actual elements. In this case, we have the "one-pointed" set {1}, and the elements of any set "S" are the same as the {1}-valued points of "S". In addition, though, there are the {1,2}-valued points, which are pairs of elements of "S", or elements of "S"×"S". In the context of sets, these higher points are extraneous: "S" is determined completely by its {1}-points. However, as shown above, this is special (in this case, it is because all sets are iterated coproducts of {1}).


Karin Melnick

Karen Melnick is a mathematician and associate professor at University of Maryland, College Park. She specializes in differential geometry and was most recently awarded the 2020-2021 Joan and Joseph Birman Fellowship for Women Scholars by the American Mathematical Society.
Research.
Melnick's primary research area is in differential-geometric aspects of rigidity, where she focuses on global and local results relating the automorphisms of a differential-geometric structure with the geometric and topological properties of the space. In addition, she is a leader in research in Lorentzian geometry and has done substantial work on the Lorentzian Lichnerowicz conjecture.
Melnick also has research interests in conformal pseudo-Riemannian structures, parabolic Cartan geometries in general, and smooth dynamics. Her research has earned her recognition as a strong collaborator on groundbreaking work with her fellow mathematicians.
Melnick has authored or co-authored fourteen academic papers in her field and has been cited by over sixty authors. She has also presented her work in a variety of seminars and colloquia internationally.
Education.
Melnick received her Ph.D. in Mathematics from the University of Chicago in 2006, where she also earned her Master of Science in Mathematics in 2000, while working under the guidance of doctoral advisor Benson Farb. Her dissertation research focused on compact Lorentz manifolds with local symmetry. Prior to her graduate studies, Melnick received her Bachelor of Arts, also in Mathematics, from Reed College in 1999.
Career.
Melnick currently works as an associate professor in the Department of Mathematics at University of Maryland, College Park, where she worked also worked as an assistant professor beginning in 2009 prior to her promotion. Before working at UMD, Melnick held the position of Gibbs Assistant Professor at Yale University while on an NSF Postdoctoral Research Fellowship.
In her career, Melnick has contributed significant service to the mathematics profession. In spring 2020, she taught a graduate topics course in exterior differential systems. In addition, she is and has been involved in organizing multiple workshops and seminars, including a two-week workshop on Cartan Connections, Geometry of Homogeneous Spaces, and Dynamics at the Erwin Schrödinger Institute in 2011, the weekly Geometry-Topology Seminar at University of Maryland, and an upcoming conference on geometric structures at Goethe University Frankfurt in 2021. She is also an advocate for women in her field, speaking at events such as the 2014 Nebraska Conference for Undergraduate Women in Mathematics.
Honors.
Most recently, Melnick was awarded the 2020-2021 Joan and Joseph Birman Fellowship for Women Scholars by the American Mathematical Society for her research on differential-geometric aspects of rigidity. This prestigious fellowship, founded by mathematician Joan Birman and physicist Joseph L. Birman, is awarded to talented, mid-career women with a significant record of research in a core area of mathematics.
Melnick has previously been awarded an NSF CAREER Grant (2013), an AMS Centennial Fellowship (2012-2013), and a Junior Research Fellowship from the Erwin Schrödinger Institute (2009).


Kneser graph

In graph theory, the Kneser graph (alternatively ) is the graph whose vertices correspond to the -element subsets of a set of elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint. Kneser graphs are named after Martin Kneser, who first investigated them in 1955.
Examples.
The Kneser graph is the complete graph on vertices.
The Kneser graph is the complement of the line graph of the complete graph on vertices.
The Kneser graph is the odd graph ; in particular is the Petersen graph (see top right figure).
The Kneser graph , visualized on the right.
Properties.
Chromatic number.
As conjectured, the chromatic number of the Kneser graph for formula_4 is exactly ; for instance, the Petersen graph requires three colors in any proper coloring. This conjecture was proved in several ways.
When formula_5, the chromatic number of is 1.
Diameter.
Spectrum
Related graphs.
The Johnson graph is the graph whose vertices are the -element subsets of an -element set, two vertices being adjacent when they meet in a -element set. The Johnson graph is the complement of the Kneser graph . Johnson graphs are closely related to the Johnson scheme, both of which are named after Selmer M. Johnson.
The generalized Kneser graph has the same vertex set as the Kneser graph , but connects two vertices whenever they correspond to sets that intersect in or fewer items . Thus .
The bipartite Kneser graph has as vertices the sets of and items drawn from a collection of elements. Two vertices are connected by an edge whenever one set is a subset of the other. Like the Kneser graph it is vertex transitive with degree formula_18 The bipartite Kneser graph can be formed as a bipartite double cover of in which one makes two copies of each vertex and replaces each edge by a pair of edges connecting corresponding pairs of vertices . The bipartite Kneser graph is the Desargues graph and the bipartite Kneser graph is a crown graph.


Free functor


Max Noether

Max Noether (24 September 1844 – 13 December 1921) was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century". He was the father of Emmy Noether.
Biography.
Max Noether was born in Mannheim in 1844, to a Jewish family of wealthy wholesale hardware dealers. His grandfather, Elias Samuel, had started the business in Bruchsal in 1797. In 1809 the Grand Duchy of Baden established a "Tolerance Edict", which assigned a hereditary surname to the male head of every Jewish family which did not already possess one. Thus the Samuels became the Noether family, and as part of this Christianization of names, their son Hertz (Max's father) became Hermann. Max was the third of five children Hermann had with his wife Amalia Würzburger.
At 14, Max contracted polio and was afflicted by its effects for the rest of his life. Through self-study, he learned advanced mathematics and entered the University of Heidelberg in 1865. He served on the faculty there for several years, then moved to the University of Erlangen in 1888. While there, he helped to found the field of algebraic geometry.
In 1880 he married Ida Amalia Kaufmann, the daughter of another wealthy Jewish merchant family. Two years later they had their first child, named Amalia ("Emmy") after her mother. Emmy Noether went on to become a central figure in abstract algebra. In 1883 they had a son named Alfred, who later studied chemistry before dying in 1918. Their third child, Fritz, was born in 1884. Like Emmy, Fritz Noether also found prominence as a mathematician. Little is known about their fourth child, Gustav Robert, born in 1889. He suffered from continual illness and died in 1928.
Noether served as an "Ordinarius" (full professor) at Erlangen for many years, and died there on 13 December 1921.
Work on algebraic geometry.
Brill and Max Noether developed alternative proofs using algebraic methods for much of Riemann's work on Riemann surfaces. Brill–Noether theory went further by estimating the dimension of the space of maps of given degree "d" from an algebraic curve to projective space P"n". In birational geometry, Noether introduced the fundamental technique of blowing up in order to prove resolution of singularities for plane curves.
Noether made major contributions to the theory of algebraic surfaces. Noether's formula is the first case of the Riemann-Roch theorem for surfaces. The Noether inequality is one of the main restrictions on the possible discrete invariants of a surface. The Noether-Lefschetz theorem (proved by Lefschetz) says that the Picard group of a very general surface of degree at least 4 in P3 is generated by the restriction of the line bundle "O"(1).
Noether and Castelnuovo showed that the Cremona group of birational automorphisms of the complex projective plane is generated by the "quadratic transformation"
together with the group "PGL"(3,C) of automorphisms of P2. Even today, no explicit generators are known for the group of birational automorphisms of P3.


Steven Kleiman

Steven Lawrence Kleiman (born March 31, 1942) is an American mathematician.
Professional career.
Kleiman is a Professor of Mathematics at the Massachusetts Institute of Technology. Born in Boston, he did his undergraduate studies at MIT. He received his Ph.D. from Harvard University in 1965, after studying there with Oscar Zariski and David Mumford, and joined the MIT faculty in 1969. Kleiman held the prestigious NATO Postdoctoral Fellowship (1966-1967), Sloan Fellowship (1968), and Guggenheim Fellowship (1979).
Contributions.
Kleiman is known for his work in algebraic geometry and commutative algebra. He has made seminal contributions in motivic cohomology, moduli theory, intersection theory and enumerative geometry. A 2002 study of 891 academic collaborations in enumerative geometry and intersection theory covered by Mathematical Reviews found that he was not only the most prolific author in those areas, but also the one with the most collaborative ties, and the most central author of the field in terms of closeness centrality; the study's authors proposed to name the collaboration graph of the field in his honor.
Awards and honors.
In 1989 the University of Copenhagen awarded him an honorary doctorate and in May 2002 the Norwegian Academy of Science and Letters hosted a conference in honor of his 60th birthday and elected him as a foreign member. In 1992 Kleiman was elected foreign member of the Royal Danish Academy of Sciences and Letters.
In 2012 he became a fellow of the American Mathematical Society. He was an invited speaker at the International Congress of Mathematics at Nice in 1970.


Continuous q-Jacobi polynomials

In mathematics, the continuous "q"-Jacobi polynomials "P"("x"|"q"), introduced by , are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme. give a detailed list of their properties.
Definition.
The polynomials are given in terms of basic hypergeometric functions and the q-Pochhammer symbol by


Substitution matrix

In bioinformatics and evolutionary biology, a substitution matrix either describes the rate at which a character in a nucleotide sequence or a protein sequence changes to other character states over evolutionary time or it describes the log odds of finding two specific character states aligned. It is an application of a stochastic matrix. Substitution matrices are usually seen in the context of amino acid or DNA sequence alignments, where the similarity between sequences depends on their divergence time and the substitution rates as represented in the matrix.
Background.
In the process of evolution, from one generation to the next the amino acid sequences of an organism's proteins are gradually altered through the action of DNA mutations. For example, the sequence
 ALEIRYLRD
could mutate into the sequence
 ALEINYLRD
in one step, and possibly
 AQEINYQRD
over a longer period of evolutionary time. Each amino acid is more or less likely to mutate into various other amino acids. For instance, a hydrophilic residue such as arginine is more likely to be replaced by another hydrophilic residue such as glutamine, than it is to be mutated into a hydrophobic residue such as leucine. (Here, a residue refers to an amino acid stripped of a hydrogen and/or a hydroxyl group and inserted in the polymeric chain of a protein.) This is primarily due to redundancy in the genetic code, which translates similar codons into similar amino acids. Furthermore, mutating an amino acid to a residue with significantly different properties could affect the folding and/or activity of the protein. This type of disruptive substitution is likely to be removed from populations by the action of purifying selection because the substitution has a higher likelihood of rendering a protein nonfunctional.
If we have two amino acid sequences in front of us, we should be able to say something about how likely they are to be derived from a common ancestor, or homologous. If we can line up the two sequences using a sequence alignment algorithm such that the mutations required to transform a hypothetical ancestor sequence into both of the current sequences would be evolutionarily plausible, then we'd like to assign a high score to the comparison of the sequences.
To this end, we will construct a 20x20 matrix where the formula_1th entry is equal to the probability of the formula_2th amino acid being transformed into the formula_3th amino acid in a certain amount of evolutionary time. There are many different ways to construct such a matrix, called a substitution matrix. Here are the most commonly used ones:
Identity matrix.
The simplest possible substitution matrix would be one in which each amino acid is considered maximally similar to itself, but not able to transform into any other amino acid. This matrix would look like
This identity matrix will succeed in the alignment of very similar amino acid sequences but will be miserable at aligning two distantly related sequences. We need to figure out all the probabilities in a more rigorous fashion. It turns out that an empirical examination of previously aligned sequences works best.
Log-odds matrices.
We express the probabilities of transformation in what are called log-odds scores. The scores matrix S is defined as
where formula_6 is the probability that amino acid formula_2 transforms into amino acid formula_3, and formula_9, formula_10 are the frequencies of amino acids "i" and "j". The base of the logarithm is not important, and the same substitution matrix is often expressed in different bases.
PAM.
One of the first amino acid substitution matrices, the PAM "(Point Accepted Mutation)" matrix was developed by Margaret Dayhoff in the 1970s. This matrix is calculated by observing the differences in closely related proteins. Because the use of very closely related homologs, the observed mutations are not expected to significantly change the common functions of the proteins. Thus the observed substitutions (by point mutations) are considered to be accepted by natural selection.
One PAM unit is defined as 1% of the amino acid positions that have been changed. To create a PAM1 substitution matrix, a group of very closely related sequences with mutation frequencies corresponding to one PAM unit is chosen. Based on collected mutationaldata from this group of sequences, a substitution matrix can be derived. This PAM1 matrix estimates what rate of substitution would be expected if 1% of the amino acids had changed. The PAM1 matrix is used as the basis for calculating other matrices by assuming that repeated mutations would follow the same pattern as those in the PAM1 matrix, and multiple substitutions can occur at the same site. Using this logic, Dayhoff derived matrices as high as PAM250. Usually the PAM 30 and the PAM70 are used.
A matrix for more distantly related sequences can be calculated from a matrix for closely related sequences by taking the second matrix to a power. For instance, we can roughly approximate the WIKI2 matrix from the WIKI1 matrix by saying formula_11 where formula_12 is WIKI1 and formula_13 is WIKI2. This is how the PAM250 matrix is calculated.
BLOSUM.
Dayhoff's methodology of comparing closely related species turned out not to work very well for aligning evolutionarily divergent sequences. Sequence changes over long evolutionary time scales are not well approximated by compounding small changes that occur over short time scales. The BLOSUM "(BLOck SUbstitution Matrix)" series of matrices rectifies this problem. Henikoff & Henikoff constructed these matrices using multiple alignments of evolutionarily divergent proteins. The probabilities used in the matrix calculation are computed by looking at "blocks" of conserved sequences found in multiple protein alignments. These conserved sequences are assumed to be of functional importance within related proteins and will therefore have lower substitution rates than less conserved regions. To reduce bias from closely related sequences on substitution rates, segments in a block with a sequence identity above a certain threshold were clustered, reducing the weight of each such cluster (Henikoff and Henikoff). For the BLOSUM62 matrix, this threshold was set at 62%. Pairs frequencies were then counted between clusters, hence pairs were only counted between segments less than 62% identical. One would use a higher numbered BLOSUM matrix for aligning two closely related sequences and a lower number for more divergent sequences.
It turns out that the BLOSUM62 matrix does an excellent job detecting similarities in distant sequences, and this is the matrix used by default in most recent alignment applications such as BLAST.
Extensions and improvements.
Many specialized substitution matrices have been developed that describe the amino acid substitution rates in specific structural or sequence contexts, such as in transmembrane alpha helices, for combinations of secondary structure states and solvent accessibility states, or for local sequence-structure contexts. These context-specific substitution matrices lead to generally improved alignment quality at some cost of speed but are not yet widely used. Recently, sequence context-specific amino acid similarities have been derived that do not need substitution matrices but that rely on a library of sequence contexts instead. Using this idea, a context-specific extension of the popular BLAST program has been demonstrated to achieve a twofold sensitivity improvement for remotely related sequences over BLAST at similar speeds (CS-BLAST).
Terminology.
Although "transition matrix" is often used interchangeably with "substitution matrix" in fields other than bioinformatics, the former term is problematic in bioinformatics. With regards to nucleotide substitutions, "transition" is also used to indicate those substitutions that are between the two-ring purines (A → G and G → A) or are between the one-ring pyrimidines (C → T and T → C). Because these substitutions do not require a change in the number of rings, they occur more frequently than the other substitutions. "Transversion" is the term used to indicate the slower-rate substitutions that change a purine to a pyrimidine or vice versa (A ↔ C, A ↔ T, G ↔ C, and G ↔ T).


Reductive group

In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group "G" over a perfect field is reductive if it has a representation with finite kernel which is a direct sum of irreducible representations. Reductive groups include some of the most important groups in mathematics, such as the general linear group "GL"("n") of invertible matrices, the special orthogonal group "SO"("n"), and the symplectic group "Sp"(2"n"). Simple algebraic groups and (more generally) semisimple algebraic groups are reductive.
Claude Chevalley showed that the classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams, as in the theory of compact Lie groups or complex semisimple Lie algebras. Reductive groups over an arbitrary field are harder to classify, but for many fields such as the real numbers R or a number field, the classification is well understood. The classification of finite simple groups says that most finite simple groups arise as the group "G"("k") of "k"-rational points of a simple algebraic group "G" over a finite field "k", or as minor variants of that construction.
Reductive groups have a rich representation theory in various contexts. First, one can study the representations of a reductive group "G" over a field "k" as an algebraic group, which are actions of "G" on "k"-vector spaces. But also, one can study the complex representations of the group "G"("k") when "k" is a finite field, or the infinite-dimensional unitary representations of a real reductive group, or the automorphic representations of an adelic algebraic group. The structure theory of reductive groups is used in all these areas.
Definitions.
A linear algebraic group over a field "k" is defined as a smooth closed subgroup scheme of "GL"("n") over "k", for some positive integer "n". Equivalently, a linear algebraic group over "k" is a smooth affine group scheme over "k".
With the unipotent radical.
A connected linear algebraic group formula_1 over an algebraically closed field is called semisimple if every smooth connected solvable normal subgroup of formula_1 is trivial. More generally, a connected linear algebraic group formula_1 over an algebraically closed field is called reductive if the largest smooth connected unipotent normal subgroup of formula_1 is trivial. This normal subgroup is called the unipotent radical and is denoted formula_5. (Some authors do not require reductive groups to be connected.) A group formula_1 over an arbitrary field "k" is called semisimple or reductive if the base change formula_7 is semisimple or reductive, where formula_8 is an algebraic closure of "k". (This is equivalent to the definition of reductive groups in the introduction when "k" is perfect.) Any torus over "k", such as the multiplicative group "G""m", is reductive.
With representation theory.
Another equivalent definition of a reductive group is a connected group formula_1 admitting a faithful semisimple representation which remains semisimple over its algebraic closure formula_10 page 383.
Simple reductive groups.
A linear algebraic group "G" over a field "k" is called simple (or "k"-simple) if it is semisimple, nontrivial, and every smooth connected normal subgroup of "G" over "k" is trivial or equal to "G". (Some authors call this property "almost simple".) This differs slightly from the terminology for abstract groups, in that a simple algebraic group may have nontrivial center (although the center must be finite). For example, for any integer "n" at least 2 and any field "k", the group "SL"("n") over "k" is simple, and its center is the group scheme μ"n" of "n"th roots of unity.
A central isogeny of reductive groups is a surjective homomorphism with kernel a finite central subgroup scheme. Every reductive group over a field admits a central isogeny from the product of a torus and some simple groups. For example, over any field "k",
It is slightly awkward that the definition of a reductive group over a field involves passage to the algebraic closure. For a perfect field "k", that can be avoided: a linear algebraic group "G" over "k" is reductive if and only if every smooth connected unipotent normal "k"-subgroup of "G" is trivial. For an arbitrary field, the latter property defines a pseudo-reductive group, which is somewhat more general.
Split-reductive groups.
A reductive group "G" over a field "k" is called split if it contains a split maximal torus "T" over "k" (that is, a split torus in "G" whose base change to formula_8 is a maximal torus in formula_7). It is equivalent to say that "T" is a split torus in "G" that is maximal among all "k"-tori in "G". These kinds of groups are useful because their classification can be described through combinatorical data called root data.
Examples.
GLn and SLn.
A fundamental example of a reductive group is the general linear group formula_14 of invertible "n" × "n" matrices over a field "k", for a natural number "n". In particular, the multiplicative group "G""m" is the group "GL"(1), and so its group "G""m"("k") of "k"-rational points is the group "k"* of nonzero elements of "k" under multiplication. Another reductive group is the special linear group "SL"("n") over a field "k", the subgroup of matrices with determinant 1. In fact, "SL"("n") is a simple algebraic group for "n" at least 2.
O(n), SO(n), and Sp(n).
An important simple group is the symplectic group "Sp"(2"n") over a field "k", the subgroup of "GL"(2"n") that preserves a nondegenerate alternating bilinear form on the vector space "k"2"n". Likewise, the orthogonal group "O"("q") is the subgroup of the general linear group that preserves a nondegenerate quadratic form "q" on a vector space over a field "k". The algebraic group "O"("q") has two connected components, and its identity component "SO"("q") is reductive, in fact simple for "q" of dimension "n" at least 3. (For "k" of characteristic 2 and "n" odd, the group scheme "O"("q") is in fact connected but not smooth over "k". The simple group "SO"("q") can always be defined as the maximal smooth connected subgroup of "O"("q") over "k".) When "k" is algebraically closed, any two (nondegenerate) quadratic forms of the same dimension are isomorphic, and so it is reasonable to call this group "SO"("n"). For a general field "k", different quadratic forms of dimension "n" can yield non-isomorphic simple groups "SO"("q") over "k", although they all have the same base change to the algebraic closure formula_8.
Tori.
The group formula_16 and products of it are called the algebraic tori. They are examples of reductive groups since they embed in formula_14 through the diagonal, and from this representation, their unipotent radical is trivial. For example, formula_18 embeds in formula_19 from the mapformula_20
Non-examples.
Associated reductive group.
Note that the normality of the unipotent radical formula_5 implies that the quotient group formula_27 is reductive. For example,formula_28
Other characterizations of reductive groups.
Every compact connected Lie group has a complexification, which is a complex reductive algebraic group. In fact, this construction gives a one-to-one correspondence between compact connected Lie groups and complex reductive groups, up to isomorphism. For a compact Lie group "K" with complexification "G", the inclusion from "K" into the complex reductive group "G"(C) is a homotopy equivalence, with respect to the classical topology on "G"(C). For example, the inclusion from the unitary group "U"("n") to "GL"("n",C) is a homotopy equivalence.
For a reductive group "G" over a field of characteristic zero, all finite-dimensional representations of "G" (as an algebraic group) are completely reducible, that is, they are direct sums of irreducible representations. That is the source of the name "reductive". Note, however, that complete reducibility fails for reductive groups in positive characteristic (apart from tori). In more detail: an affine group scheme "G" of finite type over a field "k" is called linearly reductive if its finite-dimensional representations are completely reducible. For "k" of characteristic zero, "G" is linearly reductive if and only if the identity component "G"o of "G" is reductive. For "k" of characteristic "p">0, however, Masayoshi Nagata showed that "G" is linearly reductive if and only if "G"o is of multiplicative type and "G"/"G"o has order prime to "p".
Roots.
The classification of reductive algebraic groups is in terms of the associated root system, as in the theories of complex semisimple Lie algebras or compact Lie groups. Here is the way roots appear for reductive groups.
Let "G" be a split reductive group over a field "k", and let "T" be a split maximal torus in "G"; so "T" is isomorphic to ("G""m")"n" for some "n", with "n" called the rank of "G". Every representation of "T" (as an algebraic group) is a direct sum of 1-dimensional representations. A weight for "G" means an isomorphism class of 1-dimensional representations of "T", or equivalently a homomorphism "T" → "G""m". The weights form a group "X"("T") under tensor product of representations, with "X"("T") isomorphic to the product of "n" copies of the integers, Z"n".
The adjoint representation is the action of "G" by conjugation on its Lie algebra formula_29. A root of "G" means a nonzero weight that occurs in the action of "T" ⊂ "G" on formula_29. The subspace of formula_29 corresponding to each root is 1-dimensional, and the subspace of formula_29 fixed by "T" is exactly the Lie algebra formula_33 of "T". Therefore, the Lie algebra of "G" decomposes into formula_33 together with 1-dimensional subspaces indexed by the set Φ of roots:
For example, when "G" is the group "GL"("n"), its Lie algebra formula_36 is the vector space of all "n" × "n" matrices over "k". Let "T" be the subgroup of diagonal matrices in "G". Then the root-space decomposition expresses formula_36 as the direct sum of the diagonal matrices and the 1-dimensional subspaces indexed by the off-diagonal positions ("i", "j"). Writing "L"1...,"L""n" for the standard basis for the weight lattice "X"("T") ≅ Z"n", the roots are the elements "L""i" − "L""j" for all "i" ≠ "j" from 1 to "n".
The roots of a semisimple group form a root system; this is a combinatorial structure which can be completely classified. More generally, the roots of a reductive group form a root datum, a slight variation. The Weyl group of a reductive group "G" means the quotient group of the normalizer of a maximal torus by the torus, "W" = "N""G"("T")/"T". The Weyl group is in fact a finite group generated by reflections. For example, for the group "GL"("n") (or "SL"("n")), the Weyl group is the symmetric group "S""n".
There are finitely many Borel subgroups containing a given maximal torus, and they are permuted simply transitively by the Weyl group (acting by conjugation). A choice of Borel subgroup determines a set of positive roots Φ+ ⊂ Φ, with the property that Φ is the disjoint union of Φ+ and −Φ+. Explicitly, the Lie algebra of "B" is the direct sum of the Lie algebra of "T" and the positive root spaces:
For example, if "B" is the Borel subgroup of upper-triangular matrices in "GL"("n"), then this is the obvious decomposition of the subspace formula_39 of upper-triangular matrices in formula_36. The positive roots are "L""i" − "L""j" for 1 ≤ "i" < "j" ≤ "n".
A simple root means a positive root that is not a sum of two other positive roots. Write Δ for the set of simple roots. The number "r" of simple roots is equal to the rank of the commutator subgroup of "G", called the semisimple rank of "G" (which is simply the rank of "G" if "G" is semisimple). For example, the simple roots for "GL"("n") (or "SL"("n")) are "L""i" − "L""i"+1 for 1 ≤ "i" ≤ "n" − 1.
Root systems are classified by the corresponding Dynkin diagram, which is a finite graph (with some edges directed or multiple). The set of vertices of the Dynkin diagram is the set of simple roots. In short, the Dynkin diagram describes the angles between the simple roots and their relative lengths, with respect to a Weyl group-invariant inner product on the weight lattice. The connected Dynkin diagrams (corresponding to simple groups) are pictured below.
For a split reductive group "G" over a field "k", an important point is that a root α determines not just a 1-dimensional subspace of the Lie algebra of "G", but also a copy of the additive group "G"a in "G" with the given Lie algebra, called a root subgroup "U"α. The root subgroup is the unique copy of the additive group in "G" which is normalized by "T" and which has the given Lie algebra. The whole group "G" is generated (as an algebraic group) by "T" and the root subgroups, while the Borel subgroup "B" is generated by "T" and the positive root subgroups. In fact, a split semisimple group "G" is generated by the root subgroups alone.
Parabolic subgroups.
For a split reductive group "G" over a field "k", the smooth connected subgroups of "G" that contain a given Borel subgroup "B" of "G" are in one-to-one correspondence with the subsets of the set Δ of simple roots (or equivalently, the subsets of the set of vertices of the Dynkin diagram). Let "r" be the order of Δ, the semisimple rank of "G". Every parabolic subgroup of "G" is conjugate to a subgroup containing "B" by some element of "G"("k"). As a result, there are exactly 2"r" conjugacy classes of parabolic subgroups in "G" over "k". Explicitly, the parabolic subgroup corresponding to a given subset "S" of Δ is the group generated by "B" together with the root subgroups "U"−α for α in "S". For example, the parabolic subgroups of "GL"("n") that contain the Borel subgroup "B" above are the groups of invertible matrices with zero entries below a given set of squares along the diagonal, such as:
By definition, a parabolic subgroup "P" of a reductive group "G" over a field "k" is a smooth "k"-subgroup such that the quotient variety "G"/"P" is proper over "k", or equivalently projective over "k". Thus the classification of parabolic subgroups amounts to a classification of the projective homogeneous varieties for "G" (with smooth stabilizer group; that is no restriction for "k" of characteristic zero). For "GL"("n"), these are the flag varieties, parametrizing sequences of linear subspaces of given dimensions "a"1...,"a""i" contained in a fixed vector space "V" of dimension "n":
For the orthogonal group or the symplectic group, the projective homogeneous varieties have a similar description as varieties of isotropic flags with respect to a given quadratic form or symplectic form. For any reductive group "G" with a Borel subgroup "B", "G"/"B" is called the flag variety or flag manifold of "G".
Classification of split reductive groups.
Chevalley showed in 1958 that the reductive groups over any algebraically closed field are classified up to isomorphism by root data. In particular, the semisimple groups over an algebraically closed field are classified up to central isogenies by their Dynkin diagram, and the simple groups correspond to the connected diagrams. Thus there are simple groups of types A"n", B"n", C"n", D"n", E6, E7, E8, F4, G2. This result is essentially identical to the classifications of compact Lie groups or complex semisimple Lie algebras, by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. In particular, the dimensions, centers, and other properties of the simple algebraic groups can be read from the list of simple Lie groups. It is remarkable that the classification of reductive groups is independent of the characteristic. For comparison, there are many more simple Lie algebras in positive characteristic than in characteristic zero.
The exceptional groups "G" of type G2 and E6 had been constructed earlier, at least in the form of the abstract group "G"("k"), by L. E. Dickson. For example, the group "G"2 is the automorphism group of an octonion algebra over "k". By contrast, the Chevalley groups of type F4, E7, E8 over a field of positive characteristic were completely new.
More generally, the classification of "split" reductive groups is the same over any field. A semisimple group "G" over a field "k" is called simply connected if every central isogeny from a semisimple group to "G" is an isomorphism. (For "G" semisimple over the complex numbers, being simply connected in this sense is equivalent to "G"(C) being simply connected in the classical topology.) Chevalley's classification gives that, over any field "k", there is a unique simply connected split semisimple group "G" with a given Dynkin diagram, with simple groups corresponding to the connected diagrams. At the other extreme, a semisimple group is of adjoint type if its center is trivial. The split semisimple groups over "k" with given Dynkin diagram are exactly the groups "G"/"A", where "G" is the simply connected group and "A" is a "k"-subgroup scheme of the center of "G".
For example, the simply connected split simple groups over a field "k" corresponding to the "classical" Dynkin diagrams are as follows:
The outer automorphism group of a split reductive group "G" over a field "k" is isomorphic to the automorphism group of the root datum of "G". Moreover, the automorphism group of "G" splits as a semidirect product:
where "Z" is the center of "G". For a split semisimple simply connected group "G" over a field, the outer automorphism group of "G" has a simpler description: it is the automorphism group of the Dynkin diagram of "G".
Reductive group schemes.
A group scheme "G" over a scheme "S" is called reductive if the morphism "G" → "S" is smooth and affine, and every geometric fiber formula_7 is reductive. (For a point "p" in "S", the corresponding geometric fiber means the base change of "G" to an algebraic closure formula_8 of the residue field of "p".) Extending Chevalley's work, Michel Demazure and Grothendieck showed that split reductive group schemes over any nonempty scheme "S" are classified by root data. This statement includes the existence of Chevalley groups as group schemes over Z, and it says that every split reductive group over a scheme "S" is isomorphic to the base change of a Chevalley group from Z to "S".
Real reductive groups.
In the context of Lie groups rather than algebraic groups, a real reductive group is a Lie group "G" such that there is a linear algebraic group "L" over R whose identity component (in the Zariski topology) is reductive, and a homomorphism "G" → "L"(R) whose kernel is finite and whose image is open in "L"(R) (in the classical topology). It is also standard to assume that the image of the adjoint representation Ad("G") is contained in Int("g"C) = Ad("L"0(C)) (which is automatic for "G" connected).
In particular, every connected semisimple Lie group (meaning that its Lie algebra is semisimple) is reductive. Also, the Lie group R is reductive in this sense, since it can be viewed as the identity component of "GL"(1,R) ≅ R*. The problem of classifying the real reductive groups largely reduces to classifying the simple Lie groups. These are classified by their Satake diagram; or one can just refer to the list of simple Lie groups (up to finite coverings).
Useful theories of admissible representations and unitary representations have been developed for real reductive groups in this generality. The main differences between this definition and the definition of a reductive algebraic group have to do with the fact that an algebraic group "G" over R may be connected as an algebraic group while the Lie group "G"(R) is not connected, and likewise for simply connected groups.
For example, the projective linear group "PGL"(2) is connected as an algebraic group over any field, but its group of real points "PGL"(2,R) has two connected components. The identity component of "PGL"(2,R) (sometimes called "PSL"(2,R)) is a real reductive group that cannot be viewed as an algebraic group. Similarly, "SL"(2) is simply connected as an algebraic group over any field, but the Lie group "SL"(2,R) has fundamental group isomorphic to the integers Z, and so "SL"(2,R) has nontrivial covering spaces. By definition, all finite coverings of "SL"(2,R) (such as the metaplectic group) are real reductive groups. On the other hand, the universal cover of "SL"(2,R) is not a real reductive group, even though its Lie algebra is reductive, that is, the product of a semisimple Lie algebra and an abelian Lie algebra.
For a connected real reductive group "G", the quotient manifold "G"/"K" of "G" by a maximal compact subgroup "K" is a symmetric space of non-compact type. In fact, every symmetric space of non-compact type arises this way. These are central examples in Riemannian geometry of manifolds with nonpositive sectional curvature. For example, "SL"(2,R)/"SO"(2) is the hyperbolic plane, and "SL"(2,C)/"SU"(2) is hyperbolic 3-space.
For a reductive group "G" over a field "k" that is complete with respect to a discrete valuation (such as the p-adic numbers Q"p"), the affine building "X" of "G" plays the role of the symmetric space. Namely, "X" is a simplicial complex with an action of "G"("k"), and "G"("k") preserves a CAT(0) metric on "X", the analog of a metric with nonpositive curvature. The dimension of the affine building is the "k"-rank of "G". For example, the building of "SL"(2,Q"p") is a tree.
Representations of reductive groups.
For a split reductive group "G" over a field "k", the irreducible representations of "G" (as an algebraic group) are parametrized by the dominant weights, which are defined as the intersection of the weight lattice "X"("T") ≅ Z"n" with a convex cone (a Weyl chamber) in R"n". In particular, this parametrization is independent of the characteristic of "k". In more detail, fix a split maximal torus and a Borel subgroup, "T" ⊂ "B" ⊂ "G". Then "B" is the semidirect product of "T" with a smooth connected unipotent subgroup "U". Define a highest weight vector in a representation "V" of "G" over "k" to be a nonzero vector "v" such that "B" maps the line spanned by "v" into itself. Then "B" acts on that line through its quotient group "T", by some element λ of the weight lattice "X"("T"). Chevalley showed that every irreducible representation of "G" has a unique highest weight vector up to scalars; the corresponding "highest weight" λ is dominant; and every dominant weight λ is the highest weight of a unique irreducible representation "L"(λ) of "G", up to isomorphism.
There remains the problem of describing the irreducible representation with given highest weight. For "k" of characteristic zero, there are essentially complete answers. For a dominant weight λ, define the Schur module ∇(λ) as the "k"-vector space of sections of the "G"-equivariant line bundle on the flag manifold "G"/"B" associated to λ; this is a representation of "G". For "k" of characteristic zero, the Borel–Weil theorem says that the irreducible representation "L"(λ) is isomorphic to the Schur module ∇(λ). Furthermore, the Weyl character formula gives the character (and in particular the dimension) of this representation.
For a split reductive group "G" over a field "k" of positive characteristic, the situation is far more subtle, because representations of "G" are typically not direct sums of irreducibles. For a dominant weight λ, the irreducible representation "L"(λ) is the unique simple submodule (the socle) of the Schur module ∇(λ), but it need not be equal to the Schur module. The dimension and character of the Schur module are given by the Weyl character formula (as in characteristic zero), by George Kempf. The dimensions and characters of the irreducible representations "L"(λ) are in general unknown, although a large body of theory has been developed to analyze these representations. One important result is that the dimension and character of "L"(λ) are known when the characteristic "p" of "k" is much bigger than the Coxeter number of "G", by Henning Andersen, Jens Jantzen, and Wolfgang Soergel (proving Lusztig's conjecture in that case). Their character formula for "p" large is based on the Kazhdan–Lusztig polynomials, which are combinatorially complex. For any prime "p", Simon Riche and Geordie Williamson conjectured the irreducible characters of a reductive group in terms of the "p"-Kazhdan-Lusztig polynomials, which are even more complex, but at least are computable.
Non-split reductive groups.
As discussed above, the classification of split reductive groups is the same over any field. By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field. Some examples among the classical groups are:
As a result, the problem of classifying reductive groups over "k" essentially includes the problem of classifying all quadratic forms over "k" or all central simple algebras over "k". These problems are easy for "k" algebraically closed, and they are understood for some other fields such as number fields, but for arbitrary fields there are many open questions.
A reductive group over a field "k" is called isotropic if it has "k"-rank greater than 0 (that is, if it contains a nontrivial split torus), and otherwise anisotropic. For a semisimple group "G" over a field "k", the following conditions are equivalent:
For "k" perfect, it is also equivalent to say that "G"("k") contains a unipotent element other than 1.
For a connected linear algebraic group "G" over a local field "k" of characteristic zero (such as the real numbers), the group "G"("k") is compact in the classical topology (based on the topology of "k") if and only if "G" is reductive and anisotropic. Example: the orthogonal group "SO"("p","q") over R has real rank min("p","q"), and so it is anisotropic if and only if "p" or "q" is zero.
A reductive group "G" over a field "k" is called quasi-split if it contains a Borel subgroup over "k". A split reductive group is quasi-split. If "G" is quasi-split over "k", then any two Borel subgroups of "G" are conjugate by some element of "G"("k"). Example: the orthogonal group "SO"("p","q") over R is split if and only if |"p"−"q"| ≤ 1, and it is quasi-split if and only if |"p"−"q"| ≤ 2.
Structure of semisimple groups as abstract groups.
For a simply connected split semisimple group "G" over a field "k", Robert Steinberg gave an explicit presentation of the abstract group "G"("k"). It is generated by copies of the additive group of "k" indexed by the roots of "G" (the root subgroups), with relations determined by the Dynkin diagram of "G".
For a simply connected split semisimple group "G" over a perfect field "k", Steinberg also determined the automorphism group of the abstract group "G"("k"). Every automorphism is the product of an inner automorphism, a diagonal automorphism (meaning conjugation by a suitable formula_8-point of a maximal torus), a graph automorphism (corresponding to an automorphism of the Dynkin diagram), and a field automorphism (coming from an automorphism of the field "k").
For a "k"-simple algebraic group "G", Tits's simplicity theorem says that the abstract group "G"("k") is close to being simple, under mild assumptions. Namely, suppose that "G" is isotropic over "k", and suppose that the field "k" has at least 4 elements. Let "G"("k")+ be the subgroup of the abstract group "G"("k") generated by "k"-points of copies of the additive group "G""a" over "k" contained in "G". (By the assumption that "G" is isotropic over "k", the group "G"("k")+ is nontrivial, and even Zariski dense in "G" if "k" is infinite.) Then the quotient group of "G"("k")+ by its center is simple (as an abstract group). The proof uses Jacques Tits's machinery of BN-pairs.
The exceptions for fields of order 2 or 3 are well understood. For "k" = F2, Tits's simplicity theorem remains valid except when "G" is split of type "A"1, "B"2, or "G"2, or non-split (that is, unitary) of type "A"2. For "k" = F3, the theorem holds except for "G" of type "A"1.
For a "k"-simple group "G", in order to understand the whole group "G"("k"), one can consider the Whitehead group "W"("k","G")="G"("k")/"G"("k")+. For "G" simply connected and quasi-split, the Whitehead group is trivial, and so the whole group "G"("k") is simple modulo its center. More generally, the Kneser–Tits problem asks for which isotropic "k"-simple groups the Whitehead group is trivial. In all known examples, "W"("k","G") is abelian.
For an anisotropic "k"-simple group "G", the abstract group "G"("k") can be far from simple. For example, let "D" be a division algebra with center a "p"-adic field "k". Suppose that the dimension of "D" over "k" is finite and greater than 1. Then "G" = "SL"(1,"D") is an anisotropic "k"-simple group. As mentioned above, "G"("k") is compact in the classical topology. Since it is also totally disconnected, "G"("k") is a profinite group (but not finite). As a result, "G"("k") contains infinitely many normal subgroups of finite index.
Lattices and arithmetic groups.
Let "G" be a linear algebraic group over the rational numbers Q. Then "G" can be extended to an affine group scheme "G" over Z, and this determines an abstract group "G"(Z). An arithmetic group means any subgroup of "G"(Q) that is commensurable with "G"(Z). (Arithmeticity of a subgroup of "G"(Q) is independent of the choice of Z-structure.) For example, "SL"("n",Z) is an arithmetic subgroup of "SL"("n",Q).
For a Lie group "G", a lattice in "G" means a discrete subgroup Γ of "G" such that the manifold "G"/Γ has finite volume (with respect to a "G"-invariant measure). For example, a discrete subgroup Γ is a lattice if "G"/Γ is compact. The Margulis arithmeticity theorem says, in particular: for a simple Lie group "G" of real rank at least 2, every lattice in "G" is an arithmetic group.
The Galois action on the Dynkin diagram.
In seeking to classify reductive groups which need not be split, one step is the Tits index, which reduces the problem to the case of anisotropic groups. This reduction generalizes several fundamental theorems in algebra. For example, Witt's decomposition theorem says that a nondegenerate quadratic form over a field is determined up to isomorphism by its Witt index together with its anisotropic kernel. Likewise, the Artin–Wedderburn theorem reduces the classification of central simple algebras over a field to the case of division algebras. Generalizing these results, Tits showed that a reductive group over a field "k" is determined up to isomorphism by its Tits index together with its anisotropic kernel, an associated anisotropic semisimple "k"-group.
For a reductive group "G" over a field "k", the absolute Galois group Gal("k""s"/"k") acts (continuously) on the "absolute" Dynkin diagram of "G", that is, the Dynkin diagram of "G" over a separable closure "k"s (which is also the Dynkin diagram of "G" over an algebraic closure formula_54). The Tits index of "G" consists of the root datum of "G""k""s", the Galois action on its Dynkin diagram, and a Galois-invariant subset of the vertices of the Dynkin diagram. Traditionally, the Tits index is drawn by circling the Galois orbits in the given subset.
There is a full classification of quasi-split groups in these terms. Namely, for each action of the absolute Galois group of a field "k" on a Dynkin diagram, there is a unique simply connected semisimple quasi-split group "H" over "k" with the given action. (For a quasi-split group, every Galois orbit in the Dynkin diagram is circled.) Moreover, any other simply connected semisimple group "G" over "k" with the given action is an inner form of the quasi-split group "H", meaning that "G" is the group associated to an element of the Galois cohomology set "H"1("k","H"/"Z"), where "Z" is the center of "H". In other words, "G" is the twist of "H" associated to some "H"/"Z"-torsor over "k", as discussed in the next section.
Example: Let "q" be a nondegenerate quadratic form of even dimension 2"n" over a field "k" of characteristic not 2, with "n" ≥ 5. (These restrictions can be avoided.) Let "G" be the simple group "SO"("q") over "k". The absolute Dynkin diagram of "G" is of type D"n", and so its automorphism group is of order 2, switching the two "legs" of the D"n" diagram. The action of the absolute Galois group of "k" on the Dynkin diagram is trivial if and only if the signed discriminant "d" of "q" in "k"*/("k"*)2 is trivial. If "d" is nontrivial, then it is encoded in the Galois action on the Dynkin diagram: the index-2 subgroup of the Galois group that acts as the identity is formula_55. The group "G" is split if and only if "q" has Witt index "n", the maximum possible, and "G" is quasi-split if and only if "q" has Witt index at least "n" − 1.
Torsors and the Hasse principle.
A torsor for an affine group scheme "G" over a field "k" means an affine scheme "X" over "k" with an action of "G" such that formula_56 is isomorphic to formula_7 with the action of formula_7 on itself by left translation. A torsor can also be viewed as a principal G-bundle over "k" with respect to the fppf topology on "k", or the étale topology if "G" is smooth over "k". The pointed set of isomorphism classes of "G"-torsors over "k" is called "H"1("k","G"), in the language of Galois cohomology.
Torsors arise whenever one seeks to classify forms of a given algebraic object "Y" over a field "k", meaning objects "X" over "k" which become isomorphic to "Y" over the algebraic closure of "k". Namely, such forms (up to isomorphism) are in one-to-one correspondence with the set "H"1("k",Aut("Y")). For example, (nondegenerate) quadratic forms of dimension "n" over "k" are classified by "H"1("k","O"("n")), and central simple algebras of degree "n" over "k" are classified by "H"1("k","PGL"("n")). Also, "k"-forms of a given algebraic group "G" (sometimes called "twists" of "G") are classified by "H"1("k",Aut("G")). These problems motivate the systematic study of "G"-torsors, especially for reductive groups "G".
When possible, one hopes to classify "G"-torsors using cohomological invariants, which are invariants taking values in Galois cohomology with "abelian" coefficient groups "M", "H""a"("k","M"). In this direction, Steinberg proved Serre's "Conjecture I": for a connected linear algebraic group "G" over a perfect field of cohomological dimension at most 1, "H"1("k","G") = 1. (The case of a finite field was known earlier, as Lang's theorem.) It follows, for example, that every reductive group over a finite field is quasi-split.
Serre's Conjecture II predicts that for a simply connected semisimple group "G" over a field of cohomological dimension at most 2, "H"1("k","G") = 1. The conjecture is known for a totally imaginary number field (which has cohomological dimension 2). More generally, for any number field "k", Martin Kneser, Günter Harder and Vladimir Chernousov (1989) proved the Hasse principle: for a simply connected semisimple group "G" over "k", the map
is bijective. Here "v" runs over all places of "k", and "k""v" is the corresponding local field (possibly R or C). Moreover, the pointed set "H"1("k""v","G") is trivial for every nonarchimidean local field "k""v", and so only the real places of "k" matter. The analogous result for a global field "k" of positive characteristic was proved earlier by Harder (1975): for every simply connected semisimple group "G" over "k", "H"1("k","G") is trivial (since "k" has no real places).
In the slightly different case of an adjoint group "G" over a number field "k", the Hasse principle holds in a weaker form: the natural map
is injective. For "G" = "PGL"("n"), this amounts to the Albert–Brauer–Hasse–Noether theorem, saying that a central simple algebra over a number field is determined by its local invariants.
Building on the Hasse principle, the classification of semisimple groups over number fields is well understood. For example, there are exactly three Q-forms of the exceptional group E8, corresponding to the three real forms of E8.


Coxeter matroid

In mathematics, Coxeter matroids are generalization of matroids depending on a choice of a Coxeter group "W" and a parabolic subgroup "P". Ordinary matroids correspond to the case when "P" is a maximal parabolic subgroup of a symmetric group "W". They were introduced by , who named them after H. S. M. Coxeter.
 give a detailed account of Coxeter matroids.
Definition.
Suppose that "W" is a Coxeter group, generated by a set "S" of involutions, and "P" is a parabolic subgroup (the subgroup generated by some subset of "S"). A Coxeter matroid is a subset "M" of "W"/"P" that for every "w" in "W", "M" contains a unique minimal element with respect to the "w"-Bruhat order.
Relation to matroids.
Suppose that the Coxeter group "W" is the symmetric group "S""n" and "P" is the parabolic subgroup "S""k"×"S""n"–"k". Then "W"/"P" can be identified with the "k"-element subsets of the "n"-element set {1,2...,"n"} and the elements "w" of "W" correspond to the linear orderings of this set. A Coxeter matroid consists of "k" elements sets such that for each "w" there is a unique minimal element in the corresponding Bruhat ordering of "k"-element subsets. This is exactly the definition of a matroid of rank "k" on an "n"-element set in terms of bases: a matroid can be defined as some "k"-element subsets called bases of an "n"-element set such that for each linear ordering of the set there is a unique minimal base in the Gale ordering of "k"-element subsets.


David A. Cox

David Archibald Cox (born September 23, 1948 in Washington, D.C.) is a retired American mathematician, working in algebraic geometry.
Cox graduated from Rice University with a bachelor's degree in 1970 and his Ph.D. in 1975 at Princeton University, under the supervision of Eric Friedlander ("Tubular Neighborhoods in the Etale Topology"). From 1974 to 1975, he was assistant professor at Haverford College and at Rutgers University from 1975 to 1979. In 1979, he became assistant professor and in 1988 professor at Amherst College.
He studies, among other things, étale homotopy theory, elliptic surfaces, computer-based algebraic geometry (such as Gröbner basis), Torelli sets and toric varieties, and history of mathematics. He is also known for several textbooks. He is a fellow of the American Mathematical Society.
From 1987 to 1988 he was a guest professor at Oklahoma State University. In 2012, he received the Lester Randolph Ford Award for "Why Eisenstein Proved the Eisenstein Criterion and Why Schönemann Discovered It First".


Category of preordered sets

In mathematics, the category Ord has preordered sets as objects and order-preserving functions as morphisms. This is a category because the composition of two order-preserving functions is order preserving and the identity map is order preserving.
The monomorphisms in Ord are the injective order-preserving functions.
The empty set (considered as a preordered set) is the initial object of Ord, and the terminal objects are precisely the singleton preordered sets. There are thus no zero objects in Ord.
The categorical product in Ord is given by the product order on the cartesian product.
We have a forgetful functor Ord → Set that assigns to each preordered set the underlying set, and to each order-preserving function the underlying function. This functor is faithful, and therefore Ord is a concrete category. This functor has a left adjoint (sending every set to that set equipped with the equality relation) and a right adjoint (sending every set to that set equipped with the total relation).
2-category structure.
The set of morphisms (order-preserving functions) between two preorders actually has more structure than that of a set. It can be made into a preordered set itself by the pointwise relation:
This preordered set can in turn be considered as a category, which makes Ord a 2-category (the additional axioms of a 2-category trivially hold because any equation of parallel morphisms is true in a posetal category).
With this 2-category structure, a pseudofunctor F from a category "C" to Ord is given by the same data as a 2-functor, but has the relaxed properties:
where "x" ≃ "y" means "x" ≤ "y" and "y" ≤ "x".


Friedrich Bachmann

Friedrich Bachmann (born 11 February 1909 in Wernigerode, died 1 October 1982 in Kiel.) was a German mathematician who specialised in geometry and group theory.
Life.
Bachmann was the son of a Lutheran minister Hans Bachmann. Bachmann came from an intellectual family, his paternal grandfather was the number theorist Paul Gustav Heinrich Bachmann. Bachmann took his early education at the Gymnasium in Münster. After attending the Gymnasium, he attended the University of Münster and the Humboldt University of Berlin and graduated in 1927. While there he was a member of the Münster Wingolfs.
In 1933, Bachman was promoted to D.Phil with a thesis titled, "Studies on the foundation of arithmetic with special reference to Dedekind, Frege, and Russell" (). His advisor was Heinrich Scholz.
His doctoral students include Andreas Dress and Rolf Lingenberg.
Bachmann was married to a great-granddaughter of Otto von Bismarck, Alexandra von Bredow. They had a son together, Sebastian
Career.
In 1935, Bachmann moved to the University of Marburg where assisted Kurt Reidemeister. Bachman habilitated at Marburg before becoming a Privatdozent in 1939.
From 1941 Bachmann became a private lecturer at University of Königsberg and from 1943 at the Humboldt University of Berlin.
On 1 March 1949, he was promoted to full professor (Ordentlicher Professor) at the University of Kiel. He became known for his work on geometry, especially for his axiomatic justification of elementary geometry with mirroring operations, previously begun by Johannes Hjelmslev and others.
After the war, Bachmann worked with the mathematician Karl-Heinrich Weise to rebuild the mathematics department at the University of Kiel.
From 1960 Bachman was editor of the series titled, "Grundzüge der Mathematik für Lehrer an Gymnasien sowie für Mathematiker in Industrie und Wirtschaft" (Basic features of mathematics for teachers at grammar schools as well as for mathematicians in industry and business).
In 1962–1963, Bachmann was elected Dean of the mathematics department of the Faculty of Philosophy of Kiel University.
Bachmann retired on 31 March 1977.
Bibliography.
The following are articles, written or co-authored by Bachmann.
The following is books or monographs written by Bachmann


Monoidal adjunction

Suppose that formula_1 and formula_2 are two monoidal categories. A monoidal adjunction between two lax monoidal functors
is an adjunction formula_5 between the underlying functors, such that the natural transformations
are monoidal natural transformations.
Lifting adjunctions to monoidal adjunctions.
Suppose that
is a lax monoidal functor such that the underlying functor formula_9 has a right adjoint formula_10. This adjunction lifts to a monoidal adjunction formula_11⊣formula_12 if and only if the lax monoidal functor formula_11 is strong.


Alternativity

In abstract algebra, alternativity is a property of a binary operation. A magma "G" is said to be left alternative if ("xx")"y" = "x"("xy") for all "x" and "y" in "G" and right alternative if "y"("xx") = ("yx")"x" for all "x" and "y" in "G". A magma that is both left and right alternative is said to be alternative.
Any associative magma (i.e., a semigroup) is alternative. More generally, a magma in which every pair of elements generates an associative submagma must be alternative. The converse, however, is not true, in contrast to the situation in alternative algebras. In fact, an alternative magma need not even be power-associative.


SystemDrive (environment variable)


Integrally closed domain

In commutative algebra, an integrally closed domain "A" is an integral domain whose integral closure in its field of fractions is "A" itself. Spelled out, this means that if "x" is an element of the field of fractions of "A" which is a root of a monic polynomial with coefficients in "A," then "x" is itself an element of "A." Many well-studied domains are integrally closed: fields, the ring of integers Z, unique factorization domains and regular local rings are all integrally closed.
Note that integrally closed domains appear in the following chain of class inclusions:
Basic properties.
Let "A" be an integrally closed domain with field of fractions "K" and let "L" be a field extension of "K". Then "x"∈"L" is integral over "A" if and only if it is algebraic over "K" and its minimal polynomial over "K" has coefficients in "A". In particular, this means that any element of "L" integral over "A" is root of a monic polynomial in "A"["X"] that is irreducible in "K"["X"].
If "A" is a domain contained in a field "K," we can consider the integral closure of "A" in "K" (i.e. the set of all elements of "K" that are integral over "A"). This integral closure is an integrally closed domain.
Integrally closed domains also play a role in the hypothesis of the Going-down theorem. The theorem states that if "A"⊆"B" is an integral extension of domains and "A" is an integrally closed domain, then the going-down property holds for the extension "A"⊆"B".
Examples.
The following are integrally closed domains.
To give a non-example, let "k" be a field and formula_8 ("A" is the subalgebra generated by "t"2 and "t"3.) "A" is not integrally closed: it has the field of fractions formula_9, and the monic polynomial formula_10 in the variable "X" has root "t" which is in the field of fractions but not in "A." This is related to the fact that the plane curve formula_11 has a singularity at the origin.
Another domain which is not integrally closed is formula_12; it does not contain the element formula_13 of its field of fractions, which satisfies the monic polynomial formula_14.
Noetherian integrally closed domain.
For a noetherian local domain "A" of dimension one, the following are equivalent.
Let "A" be a noetherian integral domain. Then "A" is integrally closed if and only if (i) "A" is the intersection of all localizations formula_15 over prime ideals formula_16 of height 1 and (ii) the localization formula_15 at a prime ideal formula_16 of height 1 is a discrete valuation ring.
A noetherian ring is a Krull domain if and only if it is an integrally closed domain.
In the non-noetherian setting, one has the following: an integral domain is integrally closed if and only if it is the intersection of all valuation rings containing it.
Normal rings.
Authors including Serre, Grothendieck, and Matsumura define a normal ring to be a ring whose localizations at prime ideals are integrally closed domains. Such a ring is necessarily a reduced ring, and this is sometimes included in the definition. In general, if "A" is a Noetherian ring whose localizations at maximal ideals are all domains, then "A" is a finite product of domains. In particular if "A" is a Noetherian, normal ring, then the domains in the product are integrally closed domains. Conversely, any finite product of integrally closed domains is normal. In particular, if formula_19 is noetherian, normal and connected, then "A" is an integrally closed domain. (cf. smooth variety)
Let "A" be a noetherian ring. Then (Serre's criterion) "A" is normal if and only if it satisfies the following: for any prime ideal formula_16,
Item (i) is often phrased as "regular in codimension 1". Note (i) implies that the set of associated primes formula_29 has no embedded primes, and, when (i) is the case, (ii) means that formula_30 has no embedded prime for any non-zerodivisor "f". In particular, a Cohen-Macaulay ring satisfies (ii). Geometrically, we have the following: if "X" is a local complete intersection in a nonsingular variety; e.g., "X" itself is nonsingular, then "X" is Cohen-Macaulay; i.e., the stalks formula_31 of the structure sheaf are Cohen-Macaulay for all prime ideals p. Then we can say: "X" is normal (i.e., the stalks of its structure sheaf are all normal) if and only if it is regular in codimension "1".
Completely integrally closed domains.
Let "A" be a domain and "K" its field of fractions. An element "x" in "K" is said to be almost integral over "A" if the subring "A"["x"] of "K" generated by "A" and "x" is a fractional ideal of "A"; that is, if there is a formula_32 such that formula_33 for all formula_34. Then "A" is said to be completely integrally closed if every almost integral element of "K" is contained in "A". A completely integrally closed domain is integrally closed. Conversely, a noetherian integrally closed domain is completely integrally closed.
Assume "A" is completely integrally closed. Then the formal power series ring formula_35 is completely integrally closed. This is significant since the analog is false for an integrally closed domain: let "R" be a valuation domain of height at least 2 (which is integrally closed.) Then formula_36 is not integrally closed. Let "L" be a field extension of "K". Then the integral closure of "A" in "L" is completely integrally closed.
An integral domain is completely integrally closed if and only if the monoid of divisors of "A" is a group.
See also: Krull domain.
"Integrally closed" under constructions.
The following conditions are equivalent for an integral domain "A":
1 → 2 results immediately from the preservation of integral closure under localization; 2 → 3 is trivial; 3 → 1 results from the preservation of integral closure under localization, the exactness of localization, and the property that an "A"-module "M" is zero if and only if its localization with respect to every maximal ideal is zero.
In contrast, the "integrally closed" does not pass over quotient, for Z[t]/(t2+4) is not integrally closed.
The localization of a completely integrally closed need not be completely integrally closed.
A direct limit of integrally closed domains is an integrally closed domain.
Modules over an integrally closed domain.
Let "A" be a Noetherian integrally closed domain.
An ideal "I" of "A" is divisorial if and only if every associated prime of "A"/"I" has height one.
Let "P" denote the set of all prime ideals in "A" of height one. If "T" is a finitely generated torsion module, one puts:
which makes sense as a formal sum; i.e., a divisor. We write formula_38 for the divisor class of "d". If formula_39 are maximal submodules of "M", then formula_40 and formula_41 is denoted (in Bourbaki) by formula_42.


Wei-Liang Chow

Chow Wei-Liang (; October 1, 1911, Shanghai – August 10, 1995, Baltimore) was a Chinese mathematician and stamp collector born in Shanghai, known for his work in algebraic geometry.
Biography.
Chow was a student in the USA, graduating from the University of Chicago in 1931. In 1932 he attended the University of Göttingen, then transferred to the Leipzig University where he worked with van der Waerden. They produced a series of joint papers on intersection theory, introducing in particular the use of what are now generally called Chow coordinates (which were in some form familiar to Arthur Cayley).
He married Margot Victor in 1936, and took a position at the National Central University in Nanjing. His mathematical work was seriously affected by the wartime situation in China. He taught at the National Tung-Chi University in Shanghai in the academic year 1946–47, and then went to the Institute for Advanced Study in Princeton, where he returned to his research. From 1948 to 1977 he was a professor at Johns Hopkins University.
He was also a stamp collector, known for his book "Shanghai Large Dragons, The First Issue of The Shanghai Local Post", published in 1996.
Research.
According to Shiing-Shen Chern,
"Wei-Liang was an original and versatile mathematician, although his major field was algebraic geometry. He made several fundamental contributions to mathematics:


Irena Peeva

Irena Vassileva Peeva is a professor of mathematics at Cornell University, specializing in commutative algebra. She disproved the Eisenbud–Goto regularity conjecture jointly with Jason McCullough.
Education and career.
Peeva did her graduate studies at Brandeis University, earning a Ph.D. in 1995 under the supervision of David Eisenbud with a thesis entitled "Free Resolutions". She was a postdoctoral researcher at the University of California, Berkeley and a C. L. E. Moore instructor at the Massachusetts Institute of Technology
before joining the Cornell Department of Mathematics faculty in 1998.
Peeva is an editor of the Transactions of AMS.
Books.
Peeva is the author of:
Recognition.
In 2014 Peeva was elected as a fellow of the American Mathematical Society "for contributions to commutative algebra and its applications."
In 2019/2020 and in 2012/2013 Peeva was a Simons Foundation Fellow. During 1999-2001 she was a Sloan Foundation Fellow and was a Sloan Doctoral Dissertation Fellow in 1994/1995.


Exact completion

In category theory, a branch of mathematics, the exact completion constructs a Barr-exact category from any finitely complete category. It is used to form the effective topos and other realizability toposes.
Construction.
Let "C" be a category with finite limits. Then the "exact completion" of "C" (denoted "C""ex") has for its objects pseudo-equivalence relations in "C". A pseudo-equivalence relation is like an equivalence relation except that it need not be jointly monic. An object in "C""ex" thus consists of two objects "X"0 and "X"1 and two parallel morphisms "x"0 and "x"1 from "X"1 to "X"0 such that there exist a reflexivity morphism "r" from "X"0 to "X"1 such that "x"0"r" = "x"1"r" = 1"X"0; a symmetry morphism "s" from "X"1 to itself such that "x"0"s" = "x"1 and "x"1"s" = "x"0; and a transitivity morphism "t" from "X"1 × "x"1, "X"0, "x"0 "X"1 to "X"1 such that "x"0"t" = "x"0"p" and "x"1"t" = "x"1"q", where "p" and "q" are the two projections of the aforementioned pullback. A morphism from ("X"0, "X"1, "x"0, "x"1) to ("Y"0, "Y"1, "y"0, "y"1) in "C""ex" is given by an equivalence class of morphisms "f"0 from "X"0 to "Y"0 such that there exists a morphism "f"1 from "X"1 to "Y"1 such that "y"0"f"1 = "f"0"x"0 and "y"1"f"1 = "f"0"x"1, with two such morphisms "f"0 and "g"0 being equivalent if there exists a morphism "e" from "X"0 to "Y"1 such that "y"0"e" = "f"0 and "y"1"e" = "g"0.


Vladimir Abramovich Rokhlin

Vladimir Abramovich Rokhlin ( Russian: Влади́мир Абра́мович Ро́хлин) (23 August 1919 – 3 December 1984) was a Soviet mathematician, who made numerous contributions in algebraic topology, geometry, measure theory, probability theory, ergodic theory and entropy theory.
Life.
Vladimir Abramovich Rokhlin was born in Baku, Azerbaijan, to a wealthy Jewish family. His mother, Henrietta Emmanuilovna Levenson, had studied medicine in France (she died in Baku in 1923, believed to have been killed during civil unrest provoked by an epidemic). His maternal grandmother, Clara Levenson, had been one of the first female doctors in Russia. His maternal grandfather Emmanuil Levenson was a wealthy businessman (he was also the illegitimate father of Korney Chukovsky, who was thus Henrietta's half-brother). Vladimir Rokhlin's father Abram Veniaminovich Rokhlin was a well-known social democrat (he was imprisoned during Stalin's Great Purge, and executed in 1941).
Vladimir Rokhlin entered Moscow State University in 1935. His advisor was Abraham Plessner. He volunteered for the army in 1941, leading to four years as a prisoner of a German war camp. During this time he was able to hide his Jewish origins from the Nazis. Rokhlin was liberated by the Soviet military in January 1945. He then served as a German language translator for the 5th Army of the Belorussian front. In May 1945 he was sent to a Soviet 'verification camp' for former prisoners of war. In January 1946 he was transferred to another camp to determine if he was an "enemy of the Soviet." Rokhlin was cleared in June 1946 but was forced to remain in the camp as a guard. Due to intercession by mathematicians Andrey Kolmogorov and Lev Pontryagin, he was released in December 1946 and allowed to return to Moscow, after which he returned to mathematics.
In 1959 Rokhlin joined Leningrad State University as a faculty member. He died in 1984 in Leningrad. His students include Viatcheslav Kharlamov, Yakov Eliashberg, Mikhail Gromov, Nikolai V. Ivanov, Anatoly Vershik and Oleg Viro.
Work.
Rokhlin's contributions to topology include Rokhlin's theorem, a result of 1952 on the signature of 4-manifolds. He also worked in the theory of characteristic classes, homotopy theory, cobordism theory, and in the topology of real algebraic varieties.
In measure theory, Rokhlin introduced what are now called Rokhlin partitions. He introduced the notion of standard probability space, and characterised such spaces up to isomorphism mod 0. He also proved the famous Rokhlin lemma.
Family.
His son Vladimir Rokhlin, Jr. is a well-known mathematician and computer scientist at Yale University.
Rokhlin's uncle was Korney Chukovsky, a well-known Russian poet, most famous for his popular children's books.


Free strict monoidal category


Regular Hadamard matrix

In mathematics a regular Hadamard matrix is a Hadamard matrix whose row and column sums are all equal. While the order of a Hadamard matrix must be 1, 2, or a multiple of 4, regular Hadamard matrices carry the further restriction that the order be a square number. The excess, denoted "E"("H"), of a Hadamard matrix "H" of order "n" is defined to be the sum of the entries of "H". The excess satisfies the bound
Parameters.
If "n" = 4"u"2 is the order of a regular Hadamard matrix, then the excess is ±8"u"3 and the row and column sums all equal ±2"u". It follows that each row has 2"u"2 ± "u" positive entries and 2"u"2 ∓ "u" negative entries. The orthogonality of rows implies that any two distinct rows have exactly "u"2 ± "u" positive entries in common. If "H" is interpreted as the incidence matrix of a block design, with 1 representing incidence and −1 representing non-incidence, then "H" corresponds to a symmetric 2-("v","k","λ") design with parameters (4"u"2, 2"u"2 ± "u", "u"2 ± "u"). A design with these parameters is called a Menon design.
Construction.
A number of methods for constructing regular Hadamard matrices are known, and some exhaustive computer searches have been done for regular Hadamard matrices with specified symmetry groups, but it is not known whether every even perfect square is the order of a regular Hadamard matrix. Bush-type Hadamard matrices are regular Hadamard matrices of a special form, and are connected with finite projective planes.
History and naming.
Like Hadamard matrices more generally, regular Hadamard matrices are named after Jacques Hadamard. Menon designs are named after P Kesava Menon, and Bush-type Hadamard matrices are named after Kenneth A. Bush.


Vector triple product


Tensor-hom adjunction

In mathematics, the tensor-hom adjunction is that the tensor product formula_1 and hom-functor formula_2 form an adjoint pair:
This is made more precise below. The order of terms in the phrase "tensor-hom adjunction" reflects their relationship: tensor is the left adjoint, while hom is the right adjoint.
General statement.
Say "R" and "S" are (possibly noncommutative) rings, and consider the right module categories (an analogous statement holds for left modules):
Fix an ("R","S")-bimodule "X" and define functors "F": "D" → "C" and "G": "C" → "D" as follows:
Then "F" is left adjoint to "G". This means there is a natural isomorphism
This is actually an isomorphism of abelian groups. More precisely, if "Y" is an ("A", "R") bimodule and "Z" is a ("B", "S") bimodule, then this is an isomorphism of ("B", "A") bimodules. This is one of the motivating examples of the structure in a closed bicategory.
Counit and unit.
Like all adjunctions, the tensor-hom adjunction can be described by its counit and unit natural transformations. Using the notation from the previous section, the counit
has components
given by evaluation: For
The components of the unit
are defined as follows: For "y" in "Y",
is a right "S"-module homomorphism given by
The counit and unit equations can now be explicitly verified. For
"Y" in "C",
is given on simple tensors of "Y"⊗"X" by
Likewise,
For φ in "Hom"S("X", "Z"),
is a right "S"-module homomorphism defined by
and therefore
The Ext and Tor functors.
The Hom functor formula_22 commutes with arbitrary limits, while the tensor product formula_23 functor commutes with arbitrary colimits that exist in their domain category. However, in general, formula_22 fails to commute with colimits, and formula_23 fails to commute with limits; this failure occurs even among finite limits or colimits. This failure to preserve short exact sequences motivates the definition of the Ext functor and the Tor functor.


Morphism of algebraic stacks

In algebraic geometry, given algebraic stacks formula_1 over a base category "C", a morphism formula_2 of algebraic stacks is a functor such that formula_3.
More generally, one can also consider a morphism between prestacks; for this, see prestack#Morphisms (a stackification would be an example.)
Types.
One particular important example is a presentation of a stack, which is widely used in the study of stacks.
An algebraic stack "X" is said to be smooth of dimension "n" - "j" if there is a smooth presentation formula_4 of relative dimension "j" for some smooth scheme "U" of dimension "n". For example, if formula_5 denotes the moduli stack of rank-"n" vector bundles, then there is a presentation formula_6 given by the trivial bundle formula_7 over formula_8.
A quasi-affine morphism between algebraic stacks is a morphism that factorizes as a quasi-compact open immersion followed by an affine morphism.


Main diagonal

In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, or major diagonal) of a matrix formula_1 is the collection of entries formula_2 where formula_3. All off-diagonal elements are zero in a diagonal matrix. The following three matrices have their main diagonals indicated by red 1's:
Antidiagonal.
The antidiagonal (sometimes Harrison diagonal, secondary diagonal, trailing diagonal, minor diagonal, or bad diagonal) of a dimension formula_5 square matrix, formula_6, is the collection of entries formula_7 such that formula_8 for all formula_9. That is, it runs from the top right corner to the bottom left corner.


McKay conjecture

In mathematics, specifically in the field of group theory, the McKay Conjecture is a conjecture of equality between the number of irreducible complex characters of degree not divisible by a prime number formula_1 to that of the normalizer of a Sylow formula_1-subgroup. It is named after Canadian mathematician John McKay.
Statement.
Suppose formula_1 is a prime number, formula_4 is a finite group, and formula_5 is a Sylow formula_1-subgroup. Define
where formula_8 denotes the set of complex irreducible characters of the group formula_4. The McKay conjecture claims the equality
where formula_11 is the normalizer of formula_12 in formula_4.


Hans Schneider (mathematician)

Hans Schneider (24 January 1927 in Vienna, Austria - 28 October 2014) was a British-American mathematician, and James Joseph Sylvester Emeritus Professor at the University of Wisconsin–Madison. He was the first president of the International Matrix Group (1987-1990) and its successor, the International Linear Algebra Society (1990 – 1996), which established the triennial Hans Schneider Prize in 1993. Schneider was a founding editor (1968-1972) and then editor-in-chief of "Linear Algebra and Its Applications (1972 - 2012)" and an Advisory Editor of the "Electronic Journal of Linear Algebra".
He received his Ph.D. from the University of Edinburgh in 1952; his advisor was Alexander Craig Aitken. Following his doctorate, he taught at the Queen's University of Belfast until 1959, when he moved to the University of Wisconsin. He retired in 1993. He was the author of over 160 research papers. His research covered many topics in linear algebra, such as Perron Frobenius theory and related topics, inertia theory. and lately max algebra. For his experiences in the years 1938 - 1940 see
the Kindertransport site.
Schneider died of cancer at the age of 87.
He was the grandfather of YouTube musician / producer Kurt Hugo Schneider.


Faithfully flat descent

Faithfully flat descent is a technique from algebraic geometry, allowing one to draw conclusions about objects on the target of a faithfully flat morphism. Such morphisms, that are flat and surjective, are common, one example coming from an open cover.
In practice, from an affine point of view, this technique allows one to prove some statement about a ring or scheme after faithfully flat base change.
"Vanilla" faithfully flat descent is generally false; instead, faithfully flat descent is valid under some finiteness conditions (e.g., quasi-compact or locally of finite presentation).
A faithfully flat descent is a special case of Beck's monadicity theorem.
Basic form.
Let formula_1 be a faithfully flat ring homomorphism. Given an formula_2-module formula_3, we get the formula_4-module formula_5 and because formula_1 is faithfully flat, we have the inclusion formula_7. Moreover, we have the isomorphism formula_8 of formula_9-modules that is induced by the isomorphism formula_10 and that satisfies the cocycle condition:
where formula_12 are given as:
with formula_16. Note the isomorphisms formula_12 are determined only by formula_18 and do not involve formula_19
Now, the most basic form of faithfully flat descent says that the above construction can be reversed; i.e., given a formula_4-module formula_21 and a formula_9-module isomorphism formula_8 such that formula_11, an invariant submodule:
is such that formula_26.
Zariski descent.
The Zariski descent refers simply to the fact that a quasi-coherent sheaf can be obtained by gluing those on a (Zariski-)open cover. It is a special case of a faithfully flat descent but is frequently used to reduce the descent problem to the affine case.
In details, let formula_27 denote the category of quasi-coherent sheaves on a scheme "X". Then Zariski descent states that, given quasi-coherent sheaves formula_28 on open subsets formula_29 with formula_30 and isomorphisms formula_31 such that (1) formula_32 and (2) formula_33 on formula_34, then exists a unique quasi-coherent sheaf formula_35 on "X" such that formula_36 in a compatible way (i.e., formula_37 restricts to formula_38).
In a fancy language, the Zariski descent states that, with respect to the Zariski topology, formula_39 is a stack; i.e., a category formula_40 equipped with the functor formula_41 the category of (relative) schemes that has an effective descent theory. Here, let formula_39 denote the category consisting of pairs formula_43 consisting of a (Zariski)-open subset "U" and a quasi-coherent sheaf on it and formula_44 the forgetful functor formula_45.
Descent for quasi-coherent sheaves.
There is a succinct statement for the major result in this area: (the prestack of quasi-coherent sheaves over a scheme "S" means that, for any "S"-scheme "X", each "X"-point of the prestack is a quasi-coherent sheaf on "X".)
The proof uses Zariski descent and the faithfully flat descent in the affine case.
Here "quasi-compact" cannot be eliminated; see https://mathoverflow.net/questions/127362/counter-example-to-faithfully-flat-descent/


CommonProgramFiles (environment variable)


Quadratic function

In algebra, a quadratic function, a quadratic polynomial, a polynomial of degree 2, or simply a quadratic, is a polynomial function with one or more variables in which the highest-degree term is of the second degree.
For example, a "univariate" (single-variable) quadratic function has the form
in the single variable "x". The graph of a univariate quadratic function is a parabola whose axis of symmetry is parallel to the -axis, as shown at right.
If the quadratic function is set equal to zero, then the result is a quadratic equation. The solutions to the univariate equation are called the roots of the univariate function.
The bivariate case in terms of variables "x" and "y" has the form
with at least one of "a, b, c" not equal to zero, and an equation setting this function equal to zero gives rise to a conic section (a circle or other ellipse, a parabola, or a hyperbola).
A quadratic function in three variables "x", "y," and "z" contains exclusively terms "x"2, "y"2, "z"2, "xy", "xz", "yz", "x", "y", "z", and a constant:
with at least one of the coefficients "a, b, c, d, e," or "f" of the second-degree terms being non-zero.
In general there can be an arbitrarily large number of variables, in which case the resulting surface of setting a quadratic function to zero is called a quadric, but the highest degree term must be of degree 2, such as "x"2, "xy", "yz", etc.
Etymology.
The adjective "quadratic" comes from the Latin word "quadrātum" ("square"). A term like is called a square in algebra because it is the area of a "square" with side .
Terminology.
Coefficients.
The coefficients of a polynomial are often taken to be real or complex numbers, but in fact, a polynomial may be defined over any ring.
Degree.
When using the term "quadratic polynomial", authors sometimes mean "having degree exactly 2", and sometimes "having degree at most 2". If the degree is less than 2, this may be called a "degenerate case". Usually the context will establish which of the two is meant.
Sometimes the word "order" is used with the meaning of "degree", e.g. a second-order polynomial.
Variables.
A quadratic polynomial may involve a single variable "x" (the univariate case), or multiple variables such as "x", "y", and "z" (the multivariate case).
The one-variable case.
Any single-variable quadratic polynomial may be written as
where "x" is the variable, and "a", "b", and "c" represent the coefficients. In elementary algebra, such polynomials often arise in the form of a quadratic equation formula_5. The solutions to this equation are called the roots of the quadratic polynomial, and may be found through factorization, completing the square, graphing, Newton's method, or through the use of the quadratic formula. Each quadratic polynomial has an associated quadratic function, whose graph is a parabola.
Bivariate case.
Any quadratic polynomial with two variables may be written as
where "x" and "y" are the variables and "a", "b", "c", "d", "e", and "f" are the coefficients. Such polynomials are fundamental to the study of conic sections, which are characterized by equating the expression for "f" ("x", "y") to zero.
Similarly, quadratic polynomials with three or more variables correspond to quadric surfaces and hypersurfaces. In linear algebra, quadratic polynomials can be generalized to the notion of a quadratic form on a vector space.
Forms of a univariate quadratic function.
A univariate quadratic function can be expressed in three formats:
The coefficient is the same value in all three forms. To convert the standard form to factored form, one needs only the quadratic formula to determine the two roots and . To convert the standard form to vertex form, one needs a process called completing the square. To convert the factored form (or vertex form) to standard form, one needs to multiply, expand and/or distribute the factors.
Graph of the univariate function.
Regardless of the format, the graph of a univariate quadratic function formula_10 is a parabola (as shown at the right). Equivalently, this is the graph of the bivariate quadratic equation formula_11.
The coefficient controls the degree of curvature of the graph; a larger magnitude of gives the graph a more closed (sharply curved) appearance.
The coefficients and together control the location of the axis of symmetry of the parabola (also the -coordinate of the vertex and the "h" parameter in the vertex form) which is at
The coefficient controls the height of the parabola; more specifically, it is the height of the parabola where it intercepts the -axis.
Vertex.
The vertex of a parabola is the place where it turns; hence, it is also called the turning point. If the quadratic function is in vertex form, the vertex is . Using the method of completing the square, one can turn the standard form
into
so the vertex, , of the parabola in standard form is
If the quadratic function is in factored form
the average of the two roots, i.e.,
is the -coordinate of the vertex, and hence the vertex is
The vertex is also the maximum point if , or the minimum point if .
The vertical line
that passes through the vertex is also the axis of symmetry of the parabola.
Maximum and minimum points.
Using calculus, the vertex point, being a maximum or minimum of the function, can be obtained by finding the roots of the derivative:
 is a root of if
resulting in
with the corresponding function value
so again the vertex point coordinates, , can be expressed as
Roots of the univariate function.
Exact roots.
The roots (or "zeros"), and , of the univariate quadratic function
are the values of for which .
When the coefficients , , and , are real or complex, the roots are
Upper bound on the magnitude of the roots.
The modulus of the roots of a quadratic formula_27 can be no greater than formula_28 where formula_29 is the golden ratio formula_30
The square root of a univariate quadratic function.
The square root of a univariate quadratic function gives rise to one of the four conic sections, almost always either to an ellipse or to a hyperbola.
If formula_31 then the equation formula_32 describes a hyperbola, as can be seen by squaring both sides. The directions of the axes of the hyperbola are determined by the ordinate of the minimum point of the corresponding parabola formula_33. If the ordinate is negative, then the hyperbola's major axis (through its vertices) is horizontal, while if the ordinate is positive then the hyperbola's major axis is vertical.
If formula_34 then the equation formula_32 describes either a circle or other ellipse or nothing at all. If the ordinate of the maximum point of the corresponding parabola
formula_33 is positive, then its square root describes an ellipse, but if the ordinate is negative then it describes an empty locus of points.
Iteration.
To iterate a function formula_37, one applies the function repeatedly, using the output from one iteration as the input to the next.
One cannot always deduce the analytic form of formula_38, which means the "n"th iteration of formula_39. (The superscript can be extended to negative numbers, referring to the iteration of the inverse of formula_39 if the inverse exists.) But there are some analytically tractable cases.
For example, for the iterative equation
one has
where
So by induction,
can be obtained, where formula_46 can be easily computed as
Finally, we have
as the solution.
See Topological conjugacy for more detail about the relationship between "f" and "g". And see Complex quadratic polynomial for the chaotic behavior in the general iteration.
The logistic map
with parameter 2<"r"<4 can be solved in certain cases, one of which is chaotic and one of which is not. In the chaotic case "r"=4 the solution is
where the initial condition parameter formula_51 is given by formula_52. For rational formula_51, after a finite number of iterations formula_54 maps into a periodic sequence. But almost all formula_51 are irrational, and, for irrational formula_51, formula_54 never repeats itself – it is non-periodic and exhibits sensitive dependence on initial conditions, so it is said to be chaotic.
The solution of the logistic map when "r"=2 is
formula_58
for formula_59. Since formula_60 for any value of formula_61 other than the unstable fixed point 0, the term formula_62 goes to 0 as "n" goes to infinity, so formula_54 goes to the stable fixed point formula_64
Bivariate (two variable) quadratic function.
A bivariate quadratic function is a second-degree polynomial of the form
where "A, B, C, D", and "E" are fixed coefficients and "F" is the constant term.
Such a function describes a quadratic surface. Setting formula_66 equal to zero describes the intersection of the surface with the plane formula_67, which is a locus of points equivalent to a conic section.
Minimum/maximum.
If formula_68 the function has no maximum or minimum; its graph forms a hyperbolic paraboloid.
If formula_69 the function has a minimum if "A">0, and a maximum if "A"<0; its graph forms an elliptic paraboloid. In this case the minimum or maximum occurs at formula_70 where:
If formula_73 and formula_74 the function has no maximum or minimum; its graph forms a parabolic cylinder.
If formula_73 and formula_76 the function achieves the maximum/minimum at a line—a minimum if "A">0 and a maximum if "A"<0; its graph forms a parabolic cylinder.