$p$-bases and differential operators on varieties defined over a non-perfect field
Carlos Abad
math.AC 13N35, 13N05, 14B05




Let k be a possibly non-perfect field of characteristic p > 0. In this work we prove the local existence of absolute p-bases for regular algebras of finite type over k. Namely, consider a regular variety Z over k. Kimura and Niitsuma proved that, for every ξ∈ Z, the local ring _Z,ξ has a p-basis over _Z,ξ^p. Here we show that, for every ξ∈ Z, there exists an open affine neighborhood of ξ, say ξ∈(A) ⊂ Z, so that A admits a p-basis over A^p.



This passage from the local ring to an affine neighborhood of ξ has geometrical consequences, some of which will be discussed in the second part of the article. As we will see, given a p-basis  of the algebra A over A^p, there is a family of differential operators on A naturally associated to . These differential operators will enable us to give a Jacobian criterion for regularity for varieties defined over k, as well as a method to compute the order of an ideal I ⊂ A.







Undulator-Based Positron Source at 250 GeV CM Energy with Different 
	Optical Matching Devices: Pulsed Flux Concentrator and Quarter Wave
	TransformerTalk presented at the International Workshop on Future Linear
	Colliders (LCWS2017), Strasbourg, France, 23-27 October 2017. C17-10-23.2.
    Andriy Ushakov^1 andriy.ushakov@uni-hamburg.de,
	Gudrid Moortgat-Pick^1,2 and Sabine Riemann^3

 
 
 ^1University of Hamburg, Luruper Chaussee 149, D-22761 Hamburg, Germany 

 ^2Deutsches Elektronen-Synchrotron (DESY), Notkestrasse 85, D-22607 Hamburg, Germany

 ^3Deutsches Elektronen-Synchrotron (DESY), Platanenallee 6, D-15738 Zeuthen, Germany
    
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§ INTRODUCTION



Differential methods have been widely used in the study of singularities of algebraic varieties.




While some of these methods are well suited for varieties defined over a perfect field k, they tend to fail when k is non-perfect. The aim of this article is to provide some tools and methods based on the use of differential operators which enable us to study the singularities of an embedded variety defined over a non-perfect field.

The difficulties of the non-perfect case already arise when considering a hypersurface embedded in the affine space. Let f ∈ k[x_1, …, x_n] be a polynomial defined over a field k, and let

    H = (k[x_1, …, x_n] / ⟨ f ⟩)
    	⊂𝔸^n_k

be the hypersurface defined by f in 𝔸^n_k. When the field k is perfect, the singular locus of H is given by

    (H)
    	= ( ⟨ f,
    		∂ f/∂ x_1,
    		…, ∂ f/∂ x_n⟩)
    	⊂𝔸^n_k .

However, this characterization of the singular locus of H fails in general when k is non-perfect.
For instance, assume that k = 𝔽_p(v), where 𝔽_p represents the prime field of characteristic p > 0 and v is a transcendental element over 𝔽_p, and consider the polynomial f = x^p + v y^p ∈ k[x,y]. Let H = (k[x,y] / ⟨ f ⟩) be the hypersurface defined by f in 𝔸^2_k. It can be proved that H is regular everywhere except at the point of the origin. However, ∂ f/∂ x = ∂ f/∂ y = 0, and thus the partial derivatives of f do not help to compute the singular locus of H.
As we will see in section <ref>, one way to overcome this difficulty is to consider the partial derivative of f with respect to the transcendental element v, say ∂ f/∂ v = y^p (see Example <ref>).

In order to extend some of the existing differential methods to the case of varieties defined over a non-perfect field, we will work with what we call absolute p-bases. Let A be a ring of characteristic p > 0. An absolute p-basis of A will be a p-basis of A over the subring A^p ⊂ A. That is, an absolute p-basis of A will be a subset ⊂ A which has the following properties:


  * The elements of  are p-independent, i.e., the monomials of the form b_1^α_1·…· b_r^α_r with b_1, …, b_r ∈ and 0 ≤α_i < p are linearly independent over A^p;

  * The ring A, regarded as an A^p-algebra, is generated by . That is, it is required that A = A^p[].




It is known that every field of characteristic p > 0 admits an absolute p-basis. However, in general, given a ring C of characteristic p > 0, there is no an absolute p-basis of C. Kimura and Niitsuma prove in <cit.> that, for a regular variety Z defined over a field k of characteristic p > 0, the local ring _Z,ξ admits an absolute p-basis for every ξ∈ Z. In this work we prove the local existence of p-basis for finitely generated algebras over a non-necessarily perfect field k. Namely, given regular variety Z over k, we prove that, for every ξ∈ Z, there exists an open affine neighborhood of Z at ξ, say (A) ⊂ Z, so that A admits an absolute p-basis (see Theorem <ref>).



As we will see in section <ref>, the notion of p-basis is closely related to that of differential basis. Given an algebra A, we will say that a set ⊂ A is a differential basis of A if the module of Kähler differentials of A, say _A, is a free A-module with the set {db ∈_A | b ∈} as a basis. It is not hard to see that an absolute p-basis of A is a differential basis of A (the converse fails in general). Thus, given an absolute p-basis A, say , one can define a family of derivatives on A which act as partial derivatives with respect to the elements of . Namely, for each b ∈, one can define a derivative ∂/∂ b : A → A so that ∂/∂ b(b') = δ_bb' (Kronecker delta) for every b' ∈.

Similarly, in section <ref> we will show that, when A is a regular domain, there is a family of differential operators of higher order naturally associated to the p-basis  which behave as the differential operators in a polynomial ring (see Proposition <ref> and Corollary <ref>). These derivatives and differential operators will be used along sections <ref> and <ref> to give two applications of p-bases to the study of singularities on varieties defined over a non-perfect field. This applications are explained in the following lines.






 §.§ Jacobian criterion for regularity


Let k be a field of characteristic p > 0, and consider the polynomial ring A = k[x_1, …, x_n]. Let J = ⟨ f_1, …, f_s ⟩⊂ A be an ideal, and set C = A / J. Fix a prime ⊂ C, and let P denote the preimage of  in A. Assume that (JA_P) = r, in such a way that (C_) = n-r. The usual formulation of the Jacobian criterion says that C_ is smooth over k if and only if the Jacobian matrix

    [ ∂ f_1/∂ x_1             ∂ f_1/∂ x_n;           ⋮           ⋱           ⋮; ∂ f_s/∂ x_1             ∂ f_s/∂ x_n ]

has rank r modulo P. When k is a perfect field, this is equivalent to say that C_ is regular. However, the notions of regularity and smoothness differ when k is non-perfect. Namely, when k is non-perfect, it may happen that C_ is simultaneously smooth over k and non-regular (see <cit.>). 
Note that, as opposed to smoothness, which is a relative notion with respect to the field k, regularity is an intrinsic property of the local ring C_.


In section <ref> we formulate a Jacobian criterion for regularity which works over non-perfect fields. Let k be a non-perfect field of characteristic p > 0, and let A = k[x_1, …, x_n], J = ⟨ f_1, …, f_s ⟩, and C = A / J be as in the previous paragraph. Fix an absolute p-basis of k, say _0. Then it is not hard to check that = _0 ∪{x_1, …, x_n} is an absolute p-basis of A. Thus, for each b ∈_0, one can define a partial derivative on A associated to b which we denote by ∂/∂ b : A → A. As  can also be regarded as a differential basis of A, one has that ∂ f_1/∂ b = … = ∂ f_s/∂ b = 0 for almost every element b ∈_0, except for a finite number of them, say b_1, …, b_m ∈_0. Fix a prime ⊂ C, let P denote its preimage in A, and set r = (JA_P). Under these hypotheses, Zariski's Jacobian criterion <cit.> says that C_ is regular if and only if the extended Jacobian matrix

    [ ∂ f_1/∂ b_1             ∂ f_1/∂ b_m ∂ f_1/∂ x_1             ∂ f_1/∂ x_n;           ⋮           ⋱           ⋮           ⋮           ⋱           ⋮; ∂ f_s/∂ b_1             ∂ f_s/∂ b_m ∂ f_s/∂ x_1             ∂ f_s/∂ x_n ]

has rank r modulo P.




In Proposition <ref> we will give a more general formulation of Zariski's criterion in which A is not assumed to be a polynomial ring, but simply a regular ring that admits an absolute p-basis.






































 §.§ Differential operators and order of ideals


Consider a regular variety Z over a field k. Given a coherent ideal ⊂_Z, the order of  at a point ξ∈ Z is defined by

    _ξ()
    	= max{ n ∈ℕ|_ξ⊂_ξ^n },

where _ξ denotes the maximal ideal of _Z,ξ. For instance, when Z is affine, say Z = (A), and  is a principal ideal generated by an element f ∈ A, then _ξ() coincides with the maximum integer n such that f ∈_ξ^n. Fix an integer N ≥ 1. From the point of view of resolution of singularities, it is interesting to study the set of points of Z where a given ideal ⊂_Z has at least order N, as this set coincides with the singular locus of the idealistic exponent (,N) as defined by Hironaka <cit.>. The purpose of section <ref> is to find a description of this set of points or, more precisely, to find coherent ideal 𝒥_N ⊂_Z such that

    {ξ∈ Z |_ξ() ≥ N }
    	=  (𝒥_N)
    	⊂ Z .

When Z is a regular variety defined over a perfect field k, such ideal can be constructed by applying differential operators of order at most N-1 over k to the elements of  (see <cit.> and <cit.>). However, this approach fails in general when k is non-perfect for the following reasons. On the one hand, it is not enough to consider k-linear differential operators on Z. Instead, we shall work with absolute differential operators. That is, with differential operators on Z which are relative to the prime field of k, say 𝔽_p, rather than those which are relative to k. On the other hand, the sheaf of absolute differential operators of order at most N-1 may not be finitely generated or quasi-coherent over Z. Still, given a regular variety Z over a non-perfect field k, a coherent ideal ⊂_Z, and an integer N ≥ 1, we will show that it is possible to construct a coherent ideal 𝒥_N ⊂_Z as that in (<ref>) by applying absolute differential operators to the elements of  (see Proposition <ref> and Theorem <ref>).












































§ DEFINITION AND FIRST PROPERTIES OF P-BASES




In this section we review the notion of p-basis, as well as some examples and some of the main properties of p-bases.



Let k be a ring containing 𝔽_p, let A be an arbitrary k-algebra, and consider a (possibly infinite) subset ⊂ A. A p-monomial in  is a monomial of the form b_1^α_1·…· b_r^α_r, with b_1, …, b_r ∈, r ∈ℕ, and 0 ≤α_i < p. The subset ⊂ A is said to be p-independent over k if the set of p-monomials in , say

    { b_1^α_1·…· b_r^α_r| b_i ∈,   0 ≤α_i < p },

is linearly independent over the subring A^p[k]. The subset  is said to be a p-basis of A over k if it is p-independent over k and

    A = A^p[k][].

A p-basis of A over the prime field 𝔽_p (or equivalently over A^p) is also called an absolute p-basis of A.




Observe that, if  is a p-basis of A over k, then A can be regarded as a free A^p[k]-module with the set of p-monomials in  as a basis, and vice-versa. Note also that, given a set of variables, say { t_b}_b ∈, there is a natural morphism of A^p[k]-algebras from A^p[k][t_b | b ∈] to A mapping each variable t_b to the corresponding element b. Thus one can characterize a p-basis of A over k as follows.

  
A subset ⊂ A is a p-basis of A over k if and only if

    A ≃ A^p[k][t_b | b ∈]
    		/ ⟨ t_b^p - b^p | b ∈⟩ ,

where { t_b}_b ∈ represents a set of variables.



Given an arbitrary set ⊂ A, the ideal ⟨ t_b^p - b^p | b ∈⟩ is clearly contained in kernel of the natural map from A^p[k][t_b | b ∈] to A which maps t_b to b. Since

    A^p[k][t_b | b ∈] / ⟨ t_b^p - b^p | b ∈⟩

is a free A^p[k]-module with the set of p-monomials in { t_b}_b ∈, say

    {t_b_1^α_1·…· t_b_r^α_r| b_i ∈,   0 ≤α_i < p },

as a basis, one readily checks that  is a p-basis of A over k if and only if the natural map

    A^p[k][t_b | b ∈] / ⟨ t_b^p - b^p | b ∈⟩[r]    
    	A

is an isomorphism.




 §.§ On the existence of p-bases

In general, given a ring k and an arbitrary k-algebra R, the algebra R does not admit a p-basis over k. For instance, if k is a field and [k : k^p] = ∞, then the power series ring k[[x]] does not admit a p-basis over k or over 𝔽_p (see <cit.>). However, there are some cases in which it is known that a ring R does admit a p-basis over k.

Consider a field extension K / k. In this case, one can check that a maximal p-independent set over k, say ⊂ K, is a p-basis of K over k, and conversely (see <cit.>). 
Thus, by Zorn's lemma, K admits a p-basis over k. Unfortunately, as the following example shows, this argument does not work in a wider setting.


Consider the polynomial ring 𝔽_2[x,y], where x, y represent variables.  It can be checked that { x, xy } is a maximal p-independent set over 𝔽_2, but it is not a p-basis of 𝔽_2[x,y] over 𝔽_2.


For a polynomial ring k[x_1, …, x_n], the set of variables {x_1, …, x_n} is always a p-basis of k[x_1, …, x_n] over k. This property can be generalized as follows.

  
Let k be a ring over 𝔽_p, and let A_0 be an arbitrary k-algebra which admits a p-basis over k, say _0. Consider the polynomial ring A = A_0[x_1, …, x_n]. Then the set = _0 ∪{x_1, …, x_n} is a p-basis of A over k.



Observe that A^p[k] = A_0^p[k][x_1^p, …, x_n^p]. Since A_0 is a free A_0^p[k]-module with the the set of monomials

    {b_1^α_1·…· b_r^α_r| b_i ∈_0,   0 ≤α_i < p }

as a basis, it follows that A is a free A^p[k]-module with the set of monomials

    {b_1^α_1·…· b_r^α_r
    		x_1^β_1·…· x_n^β_n| b_i ∈,   0 ≤α_i < p,   0 ≤β_j < p }

as a basis. Hence  is a p-basis of A over k.



The reader should be aware that, in general, under the hypotheses of the lemma, the set _0 ∪{x_1, …, x_n} is not a p-basis of A_0 [[ x_1, …, x_n ]] over k (see <cit.>).



  
Let k be a ring over 𝔽_p and let A be an arbitrary k-algebra which has a p-basis over k, say . Consider a multiplicative subset S ⊂ A which does not contain nilpotent elements. Then the image of  in S^-1A is a p-basis of S^-1A over k.



Set S^p = { s^p | s ∈ S }, which is a multiplicative subset of A^p[k]. Observe that

    S^-1A
    	= (S^p)^-1 A
    	= A ⊗_A^p[k] (S^p)^-1 (A^p[k]).

Since A is a free A^p[k]-module with the set of monomials of the form b_1^α_1·…· b_r^α_r with b_i ∈ and 0 ≤α_i < p as basis, the image of these monomials is also a basis of S^-1A over (S^p)^-1 (A^p[k]). Hence the image of  is a p-basis of S^-1A over k.




Let  be a p-basis of an algebra A over a ring k. Suppose that a p-monomial in , say b_1^α_1·…· b_r^α_r, has a zero-divisor in A, say a · b_1^α_1·…· b_r^α_r = 0. In this case,

    a^p · b_1^α_1·…· b_r^α_r = 0.

Then, by the property of p-independence of , we have that a^p = 0. Thus we see that all the zero-divisors of the p-monomials in  are nilpotent.

















§ DIFFERENTIAL BASES, P-BASES, AND DERIVATIVES




Fix an arbitrary ring k containing 𝔽_p, and let A be an arbitrary k-algebra. The module of Kähler differentials of A over k is defined by _A/k = I_A/k / I_A/k^2, where I_A/k represents the kernel of the diagonal morphism δ : A ⊗_k A → A given by δ(a_1 ⊗ a_2) = a_1 a_2. We shall denote the universal derivation of A over k by d_A/k : A →_A/k. Recall that, regarded as an A-module, _A/k is generated by the elements of the form d_A/k(a) with a ∈ A.





  
We will say that a subset ⊂ A is a differential basis of A over k if _A/k is a free A-module with the set d_A/k() = {d_A/k(b) | b ∈} as a basis. A differential basis of A over the prime field 𝔽_p will also be called an absolute differential basis of A.


As it will be shown in the following lines, if  is a p-basis of A over k, then  is a differential basis of A over k (see Proposition <ref>). Conversely, if  is a differential basis of A over k, then  is p-independent over k (see Lemma <ref>). However, in general, a differential basis of A over k might not be a p-basis of A, as A^p[k][] could be a proper subset of A.



 
Let A be an algebra over a ring k containing 𝔽_p. If  is a p-basis of A over k, then _A/k is a free A-module with the set d_A/k() as basis i.e.,  is a differential basis of A over k.




Set C_0 = A^p[k]. Consider the polynomial ring C = C_0[t_b | b ∈]  and the ideal J = ⟨ t_b^p - b^p | b ∈⟩⊂ C. Recall that, by Lemma <ref>, A = C / J. Note also that _A/k = _A/C_0. Then, by

<cit.>,
there is an exact sequence of A-modules

    J / J^2  [rr]^-d_C / C_0      _C / C_0⊗_C A  [r]    _A / k[r]    
    	0.

Since t_b^p ∈ C^p and b^p ∈ C_0, one readily checks that J / J^2 is mapped to zero via d_C / C_0. Thus we get a natural isomorphism, say

    _C / C_0⊗_C A  [r]^-∼   _A / k,

which maps the class of d_C / C_0(t_b) to d_A/k(b) for each b ∈. Hence _A/k is a free A-module with the set d_A/k() = { d_A/k(b) | b ∈} as a basis.






  
Let A be an algebra over a ring k containing 𝔽_p. Assume that A admits a p-basis over k, say . Then, for each b ∈, there exists a unique k-linear derivative from A to A, which we shall denote by ∂/∂ b : A → A, such that ∂/∂ b (c) = δ_bc Kronecker delta for all c ∈.



_k(A) ≃_A(_A/k, A) (cf. <cit.>).





Let  be a p-basis of A over k as in the previous corollary. Then, for each f ∈ A, there is a finite number of elements in , say b_1, …, b_m ∈, such that ∂/∂ b_i(f) ≠ 0.





Let  be a p-basis of A over k as in Proposition <ref>, and consider a k-linear derivative δ∈_k(A). Since _A/k is a free A-module with the set { d_A/k(b) | b ∈} as a basis, one readily checks that

    δ
    	= ∑_b ∈δ(b) ·∂/∂ b.

Note that, in principle, this might be an infinite sum. However, given a fixed element f ∈ A, the previous corollary says that there is a finite number of elements b_1, …, b_m ∈ such that ∂/∂ b_i(f) ≠ 0. Thus δ(f) = ∑_i = 1^m δ(b_i) ·∂/∂ b_i(f),




which is a finite sum.



  
Let A be an algebra over a ring k containing 𝔽_p. Assume that a subset  is a differential basis of A over k. Then  is p-independent over k.



We will proceed by contradiction. Assume that there exists a non-trivial relation of p-dependence in , say F(b_1, …, b_r) = 0, where b_1, …, b_r ∈, and

    F(b_1, …, b_r)
    	= ∑_0 ≤α_1, …, α_r < p
    		a_α_1, …, α_r
    		b_1^α_1·…· b_r^α_r∈ A^p[k][b_1, …, b_r].

Then choose α_1, …, α_r so that α_1 + … + α_r is maximum with a_α_1, …, α_r≠ 0. Since  is a differential basis of A over k, one can define derivatives ∂/∂ b_1, …, ∂/∂ b_r from A to A such that ∂/∂ b_i(b_j) = δ_ij (Kronecker delta) for all i,j. Note that all these derivatives are A^p[k]-linear. Thus one readily checks that

    1/α_1 ! …α_r !·∂^α_1 + … + α_r/∂ b_1^α_1…∂ b_r^α_r
    	F(b_1, …, b_r)
    	= a_α_1, …, α_r≠ 0.

Since F(b_1, …, b_r) = 0, this leads to a contradiction.

















§ ABSOLUTE P-BASES ON REGULAR VARIETIES






Consider regular variety Z defined over a possibly non-perfect field k of characteristic p > 0. In this section we will prove that, for every ξ∈ Z, there exists an affine neighborhood of Z at ξ, say ξ∈(A) ⊂ Z, so that A admits an absolute p-basis (see Theorem <ref>).



Let R be regular local ring of characteristic p > 0. Assume that R has an absolute differential basis, say  that is,  is a differential basis of R over 𝔽_p. Then  is an absolute p-basis of R.


As we show below, this result can be easily extended to the case of regular domains which are not necessarily local.

 
Let A be a regular domain of characteristic p > 0. Suppose that A has an absolute differential basis, say . Then  is an absolute p-basis of A.



The set  is p-independent by Lemma <ref>. Thus we just need to check that A^p[] = A. To this end, let us regard the algebras A and A^p[] as modules over A^p. Then, in order to check that A^p[] = A, it suffices to see that

    A^p[] ⊗_A^p (A^p)_
    	= A ⊗_A^p (A^p)_

for every prime ideal ⊂ A^p (cf. <cit.>).

Fix a prime ideal ⊂ A^p. Note that the primes of A are in natural correspondence with those of A^p. Hence  must be of the form = Q ∩ A^p for some prime ideal Q ⊂ A. In order to prove (<ref>), observe that (A^p ∖) = (A ∖ Q)^p. Thus one readily checks that A ⊗_A^p (A^p)_ = A_Q and (A^p)_ = (A_Q)^p. On the other hand, note that  is also a differential basis of A_Q, since _A_Q = _A ⊗_A A_Q (see <cit.>). 
Hence (A_Q)^p[] = A_Q by Theorem <ref>. In this way,

    A^p[] ⊗_A^p (A^p)_
    	= (A^p)_[]
    	= (A_Q)^p[]
    	= A_Q
    	= A ⊗_A^p (A^p)_ ,

which proves (<ref>). 

Since the previous argument works for any ⊂ A_p, we conclude that A^p[] = A, and hence  is an absolute p-basis of A.

























Let (R,,K) be a noetherian local ring containing a field. Then there is a natural short exact sequence

    0 [r]     / ^2 [r]^-d_R   _R ⊗ K [r]    _K [r]    
    	0.




Since K is separable over its prime field, the result follows from <cit.>.





  
Let (R,,K) be a regular local ring that has an absolute p-basis, say ^*. Then, for any regular system of parameters of R, say z_1, …, z_d, there exists a subset _0 ⊂^* so that _0 ∪{z_1, …, z_d} is an absolute p-basis of R.



Note that, given a regular system of parameters of R, say z_1, …, z_d, the set {z_1, …, z_d} is a basis of / ^2. In addition, by Proposition <ref>, we know that _R is a free R-module with {d_R(b) | b ∈^*} as a basis, and

    d_R (z_i)
    	= ∑_b ∈^*∂ z_i/∂ b· d_R(b)

for i = 1, …, d. Since the induced map d_R :  / ^2 →_R ⊗ K is injective (Lemma <ref>), there should be elements b_1, …, b_d ∈^* such that the Jacobian matrix

    J = [ ∂ z_1/∂ b_1             ∂ z_1/∂ b_d;           ⋮           ⋱           ⋮; ∂ z_d/∂ b_1             ∂ z_d/∂ b_d ]∈_d × d(R)

has rank d modulo  (i.e., as a matrix over K = R/). Set _0 = ^* ∖{b_1, …, b_d}. Then we claim that the set = _0 ∪{z_1, …, z_d} is an absolute p-basis of R.

In order to prove the claim, we shall show that {d_R(b) | b ∈} is a basis of _R (recall that, as ^* is differential basis of R, we already know that _R is a free R-module). By Theorem <ref>, this proves that  is an absolute p-basis of R.

Note that the matrix J has non-zero determinant modulo , as it has rank d modulo . Hence the determinant of J is a unit in R, and therefore J is invertible over R. Since

    {d_R(b) | b ∈^* }
    	= {d_R(b) | b ∈_0}∪{d_R(b_1), …, d_R(b_d)}

is a basis of _R, and J is invertible, one readily checks that

    {d_R(b) | b ∈}
    	= {d_R(b) | b ∈_0}∪{d_R(z_1), …, d_R(z_d)}

is also a basis of _R. Hence = _0 ∪{z_1, …, z_d} is an absolute p-basis of R.





Fix a regular local ring (R,,K), a regular system of parameters z_1, …, z_d, and an absolute p-basis of R of the form _0 ∪{z_1, …, z_d} as in Proposition <ref>. Note that, attending to the short exact sequence of Lemma <ref>, {d_K (b) | b ∈_0 } should be basis of _K, and hence the image of _0 in K is an absolute p-basis of K. 
By <cit.>, this implies that the image of _0 in K is algebraically independent over the prime field 𝔽_p. As a consequence, 𝔽_p[_0] can be regarded as polynomial ring contained in R and, since 𝔽_p[_0] ∩ = { 0}, we may write 𝔽_p(_0) ⊂ R. Note also that, since the image of _0 in K is an absolute p-basis of K, the field K is formally étale over 𝔽_p(_0) (cf. <cit.>). 
Hence 𝔽_p(_0) can be extended to a unique coefficient field of R (the completion of R with respect to ). This proves that, under the hypotheses of Proposition <ref>, the subfield 𝔽_p(_0) ⊂ R is a quasi-coefficient field of R.


























Let _0 ∪{z_1, …, z_d} be an absolute p-basis of a regular local ring (R,,K) as in Proposition <ref>. In the previous remark we showed that 𝔽_p(_0) ⊂ R is a quasi-coefficient field of R. Therefore 𝔽_p(_0) can be extended to a unique coefficient field of R, say K_0 ⊂R, for which _0 is an absolute p-basis. In addition,
Cohen's structure theorem 
says that R≃ K_0 [[z_1, …, z_d]. Nevertheless, it should be noted that, in general, _0 ∪{z_1, …, z_d} is not a p-basis of R. Indeed, it can be proved that _0 ∪{z_1, …, z_d} is an absolute p-basis of R if and only if _0 is finite (cf. <cit.>).











 
Let A be a regular ring of characteristic p > 0 that has an absolute p-basis, say . Fix a prime ideal P ⊂ A. Then, for any regular system of parameters z_1, …, z_d of A_P, there exists a subset _0 ⊂ and an element f ∈ A ∖ P so that z_1, …, z_d ∈ A_f, and _0 ∪{z_1, …, z_d} is an absolute p-basis of A_f.



Arguing as in the proof of Proposition <ref>, one can find a finite collection of elements b_1, …, b_d ∈ A_P such that the Jacobian matrix

    J = [ ∂ z_1/∂ b_1             ∂ z_1/∂ b_d;           ⋮           ⋱           ⋮; ∂ z_d/∂ b_1             ∂ z_d/∂ b_d ]∈_d × d(A_P)

is invertible over A_P. That is, so that (J) is a unit in A_P. Then choose f so that z_1, …, z_d ∈ A_f and (J) is invertible in A_f, and set _0 = ∖{b_1, …, b_d}. We claim that _0 ∪{z_1, …, z_d} is an absolute p-basis of A_f.

In virtue of Corollary <ref>, in order to prove the claim we just need to show that _0 ∪{z_1, …, z_d} is an absolute differential basis of A_f. That is, that _A_f = _A ⊗ A_f is a free module with {d_A(b) | b ∈_0 }∪{d_A(z_1), … d_A(z_d) } as a basis.

Since _A_f = _A ⊗ A_f, one has that _A_f is free. On the other hand, observe that J can also be regarded as a matrix over A_f. As (J) is invertible in A_f, it follows that J is also invertible as a matrix over A_f. Thus one can check that {d_A(b) | b ∈_0 }∪{d_A(z_1), … d_A(z_d) } is a basis of _A_f.





  
Let A be a regular ring containing 𝔽_p that admits an absolute p-basis, and let C = A / J be a regular quotient of A. Then, for each prime ideal ⊂ C, there exists an element g ∈ C ∖ so that C_g admits an absolute p-basis.




Let P denote the preimage of  in A. Since C is regular, one can find a regular system of parameters of A_P, say z_1, …, z_d, so that JA_P = ⟨ z_1, …, z_r ⟩, with r = (JA_P). In this setting, according to Lemma <ref>, it is possible to choose an element f ∈ A ∖ P in such a way that z_1, …, z_d ∈ A_f, JA_f = ⟨ z_1, …, z_r ⟩, and A_f has an absolute p-basis of the form _0 ∪{z_1, …, z_d}. Choose g as the image of f in C. Then we claim that the image of _0 ∪{z_r+1, …, z_d} is an absolute p-basis of C_g = A _f / J A_f.

In order to prove the claim, we may assume without loss of generality that A = A_f, J = J A_f, and C = C_g. Recall that, by Proposition <ref>, _A is a free A-module with the set d_A(_0) ∪{d_A(z_1), …, d_A(z_d)} as a basis. In virtue of

<cit.>,
there is a natural exact sequence

    J / J^2 [r]^-d_A   _A⊗ C [r]^-υ   _C[r]    
    	0.

Since J is generated by z_1, …, z_r, the kernel of  υ is generated by the images of d_A(z_1), …, d_A(z_r) in _A ⊗ C. Thus we see that _C is a free C-module with the image of the set d_A(_0) ∪{ d_A(z_r+1), …, d_A(z_d) } as a basis. Hence the image of _0 ∪{z_r+1, …, z_d} in C_g is an absolute p-basis of C_g by Corollary <ref>































































































As a consequence of Lemma <ref> one has that, if R is a regular local ring of characteristic p > 0 that admits an absolute p-basis and R / J is a regular quotient of R, then R/J also admits an absolute p-basis.






































































































































 
Let Z_0 be a regular scheme of characteristic p > 0 that can be covered by open affine charts of the form (A) ⊂ Z_0 so that A admits an absolute p-basis. Then any regular scheme Z of finite type over Z_0 can be covered by open affine charts of the form (C) ⊂ Z so that C admits an absolute p-basis.



Observe that Z can be covered by affine charts of the form

    ( A[x_1, …, x_n] / J ),

where (A) is an affine chart of Z_0, and J is an ideal of the polynomial ring A[x_1, …, x_n]. Then the result follows from Lemma <ref> and Lemma <ref>.





 
Let Z be a regular variety over a possibly non-perfect field k of characteristic p > 0. For each ξ∈ Z, there exists an open affine neighborhood of Z at ξ, say (C) ⊂ Z, so that C admits an absolute p-basis.





By assumption Z is endowed with a finite type morphism Z →(k). Since every field admits an absolute p-basis, the claim follows from the previous proposition.





It is an open problem to determine the class of schemes that admit p-bases or, more precisely, that can be covered by open affine charts of the form (A), so that A admits an absolute p-basis.

Note that a noetherian domain A that has an absolute p-basis can be regarded as a free A^p-module (see Lemma <ref>), and hence it is regular by a theorem of Kunz <cit.>.

In addition, a noetherian domain that has an absolute p-basis is excellent by <cit.>. Thus the class of noetherian domains that admit an absolute p-basis is contained in the category of regular excellent domains.

On the other hand, there are examples of regular excellent domains of characteristic p > 0 which do not admit an absolute p-basis. Indeed, <cit.> exhibits a complete regular local ring of dimension one which does not admit an absolute p-basis.


























§ JACOBIAN CRITERION FOR REGULARITY




Let A be a regular algebra of characteristic p > 0 that admits an absolute p-basis, and let C = A / J be a quotient of A. In this section we give a Jacobian criterion for the regularity of C at a prime ∈(C) (Proposition <ref>). This can be regarded as a generalization of Zariski's Jacobian criterion <cit.>, which characterizes the regular points of an algebra that is given as the quotient of a polynomial ring (see Corollary <ref>).



  
Let A be a regular ring containing 𝔽_p that admits an absolute p-basis, say . Let J = ⟨ f_1, …, f_s ⟩ be an ideal in A, and consider a prime ideal P ⊂ A such that J ⊂ P. Set C = A / J and = P / J ⊂ C. Then:



  * There is a finite subset {b_1, …, b_m}⊂ so that ∂ f_1/∂ b' = … = ∂ f_s/∂ b' = 0[The notation ∂/∂ b' was introduced in Corollary <ref>.] for every b' ∈∖{b_1, …, b_m};


  * Assume that (JA_P) = r. Then C_ is regular if and only if the Jacobian matrix


    (f; ∂∂ b )
    	:= [ ∂ f_1/∂ b_1             ∂ f_1/∂ b_m;           ⋮           ⋱           ⋮; ∂ f_s/∂ b_1             ∂ f_s/∂ b_m ]


has rank r modulo P that is, if and only if it contains an r × r minor whose determinant is non-zero modulo P.





Since p-bases are compatible with localization (Lemma <ref>), we may assume without loss of generality that A = A_P and P = P A_P. Note that, in this setting, C = C_ and =  C_.

Let K denote the residue field of (A,P) and (C,), and set d = (A) = _K(P / P^2). As A is catenary (see <cit.>), one has that

    (C) = (A) - (J) = d-r .

Thus we see that C_ is regular if and only if _K( / ^2) = d-r.


In virtue of

<cit.>
and Lemma <ref>, one can construct a natural commutative diagram as follows,

    J / J^2 ⊗ K  @=[r] [d]^-λ   
    		J / J^2 ⊗ K  [d]^-δ   
    		0  [d]    
    
    	0 [r]    
    		P / P^2  [r]^-d_A[d]^-μ   _A⊗ K  [r] [d]    _K  [r] [d]    
    		0 
    
    	0 [r]     / ^2  [r]^-d_C[d]    _C⊗ K  [r] [d]    _K  [r] [d]    
    		0, 
       
    		0    
    		0    
    		0

where all the rows and columns are exact. According to this diagram, δ = d_A∘λ, and hence

    _K ( / ^2)
    	
    	= d - _K (μ)
    	= d - _K δ(J / J^2 ⊗ K)
    	
    	.

Note that the linear map δ is given by the matrix ( f; ∂∂ b) modulo P. Then the rank of δ(J / J^2 ⊗ K) coincides with that of the matrix ( f; ∂∂ b) modulo P. Thus we see that C_ is regular if and only if ( f; ∂∂ b) has rank r modulo P.





Let A be a regular ring of characteristic p > 0 which has an absolute p-basis, and let C be an algebra of finite type over A. Then ((C)) is a possibly empty open subset of (C).



Assume that C = A[x_1, …, x_n] / J. By Lemma <ref>, if  is an absolute p-basis of A, then ∪{x_1 …, x_n} is an absolute p-basis of A[x_1, …, x_n]. Thus the result follows easily from the proposition.
































Consider a polynomial ring k[x_1, …, x_n] over a field k of characteristic p > 0, and let _0 be an absolute p-basis of k. In virtue of Corollary <ref>, there is family of derivatives in k canonically associated to _0, say {∂/∂ b| b ∈_0 }⊂(k). Each of these derivatives can be extended (non-canonically) to a derivative on k[x_1, …, x_n] by setting ∂ x_1/∂ b = … = ∂ x_n/∂ b = 0. For the sake of readability, in the following proposition we shall abuse our notation and use the same symbol to refer to this extension of ∂/∂ b to k[x_1, …, x_n].




Let A = k[x_1, …, x_n] be a polynomial ring over a possibly non-perfect field k of characteristic p > 0. Fix an absolute p-basis of k, say _0. Let J = ⟨ f_1, …, f_s ⟩ be an ideal in A, and consider a prime ideal P ⊂ A such that J ⊂ P. Set C = A / J and = P / J ⊂ C. Then:





  * The set _0 ∪{x_1, …, x_n} is an absolute p-basis of A, and there is a finite subset {b_1, …, b_m}⊂_0 so that ∂ f_1/∂ b' = … = ∂ f_s/∂ b' = 0 for every b' ∈∖{b_1, …, b_m};


  * Assume that (JA_P) = r. Then C_ is regular if and only if the Jacobian matrix

    ( f;
    		∂∂ b,
    		∂∂ x )
    	:= [ ∂ f_1/∂ b_1             ∂ f_1/∂ b_m ∂ f_1/∂ x_1             ∂ f_1/∂ x_n;           ⋮           ⋱           ⋮           ⋮           ⋱           ⋮; ∂ f_s/∂ b_1             ∂ f_s/∂ b_m ∂ f_s/∂ x_1             ∂ f_s/∂ x_n ]

has rank r modulo P that is, if and only if it contains an r × r minor whose determinant is non-zero modulo P.




By Lemma <ref>, the set = _0 ∪{x_1, …, x_n} is an absolute p-basis of A. Thus the result follows immediately from Proposition <ref>.



 
Consider the field 𝔽_p(v), where v represents a transcendental element, the polynomial f = x^p + v y^p ∈ k[x,y], and the variety

    H = ( k[x,y] / ⟨ f ⟩ )
    	⊂𝔸^2_k .

Observe that H has codimension 1 in 𝔸^2_k, and that a p-basis of k[x,y] is given by { v, x, y } (see Lemma <ref>). Thus, according to Corollary <ref>, in order to compute the singular locus of H, we shall calculate the Jacobian matrix

    ( f;
    		∂∂ v,
    		∂∂ x,
    		∂∂ y )
    	= [ ∂ f/∂ v ∂ f/∂ x ∂ f/∂ y ]
    	= [ y^p   0   0 ] .

Clearly, this matrix has rank 1 everywhere in H except at the point of the origin. Hence the singular locus of H consists only of the point at the origin.
















§ ABSOLUTE P-BASES AND DIFFERENTIAL OPERATORS OF HIGHER ORDER




Let A be a regular ring of characteristic p > 0. In this section we show that, given an absolute p-basis of A, say , there is a natural family of differential operators on A associated to  which behaves mostly like the family of differential operators with respect to the variables on a polynomial ring. The following lines are intended to fix the setting and notation for Proposition <ref>.



Consider a ring k, a k-algebra A, and an A-module M.
A map Δ : A → M is said to be a differential operator of order zero from A to M over k if it is A-linear. For n > 0, a map Δ : A → M is said to be a differential operator of order n over k if it is k-linear and, for every a ∈ A,

    [a,Δ] : A    ⟶ M 
    
    	f    ⟼ a Δ(f) - Δ(af)

is a differential operator of order n-1 over k.





Let k be a ring of characteristic p > 0, let A be an arbitrary k-algebra, and let M be an A-module. If Δ : A → M is a differential operator of order n over k, then Δ is k[A^p^e]-linear for every p^e > n.





 §.§ Multi-index notation
 Let Λ be a possibly infinite set of indices. We shall denote by ℕ^⊕Λ the set of Λ-tuples with almost all entries equal to zero (i.e., except for a finite number of them). Given a tuple β = (β_λ) ∈ℕ^⊕Λ, we define the order of β as |β| := ∑_λ∈Λβ_λ. Given two tuples α = (α_λ) and β = (β_λ), we will say that α≤β if α_λ≤β_λ for all λ∈Λ. Similarly, we will say that α < β if α≤β and α≠β. For two tuples α, β∈ℕ^⊕Λ, we shall define the binomial coefficient

    αβ
    	:= ∏_λ∈Λα_λβ_λ∈ℕ,

taking α_λβ_λ = 0 whenever α_λ < β_λ. In addition, given a ring R and a Λ-indexed subset of R, say T = {t_λ|λ∈Λ}⊂ R, for each β∈ℕ^⊕Λ we shall define the monomial

    T^β
    	:= ∏_λ∈Λ t_λ^β_λ∈ A.





 §.§ Differential operators on a polynomial ring
 Given a ring A and possibly infinite set of variables X = {x_λ|λ∈Λ}, consider the polynomial ring A[X]. Let T = {t_λ|λ∈Λ} be a different set of variables, and set X+T := { x_λ + t_λ|λ∈Λ}. For a polynomial F(X) ∈ A[X], let us consider the following expansion of F(X+T) ∈ A[X,T]:

    F(X+T)
    	= ∑_α∈ℕ^⊕Λ
    		F_α(X) · T^α,
    	     F_α(X) ∈ A[X].

Then, for each β∈ℕ^⊕Λ, we shall define the Taylor operator ^β : A[X] → A[X] by ^β(F(X)) = F_β(X). Note that ^β is A-linear for every β. In fact, it can be proved that ^β is a differential operator of order |β| over A for every β∈ℕ^⊕ (see <cit.>). In addition, one can check that, for α∈ℕ^⊕Λ,

    ^β(X^α)
    	= αβ X^α - β.



































  
Let A be a reduced ring containing 𝔽_p that has an absolute p-basis, say . Then, for each β∈ℕ^⊕, there exists a differential operator of order β over 𝔽_p, say D^[;β]_A : A → A, such that

    D^[;β]_A (^α) = αβ^α - β

for all α∈ℕ^⊕.








Fix β∈ℕ^⊕, choose e > 0 so that p^e > β, and let T = {t_b | b ∈} be a set of variables. By definition of p-basis, we have that

    A ≃ A^p [T]
    		/ ⟨ t_b^p - b^p | b ∈⟩.

Since A is reduced, A ≃ A^p via the Frobenius isomorphism. Thus, by induction on e, we get

    A ≃ A^p^e [T]
    		/ ⟨ t_b^p^e - b^p^e| b ∈⟩.

Set J = ⟨ t_b^p^e - b^p^e| b ∈⟩⊂, in such a way that A ≃ / J. Regarding  as a polynomial ring over A^p^e, and according to the discussion preceding this proposition, there is an A^p^e-linear differential operator ^β : → such that

    ^β(T^α) = αβ T^α - β

for α∈ℕ^⊕. We will show that ^β induces a natural differential operator D^[;β]_A : A → A of order β over 𝔽_p via the quotient map → A ≃ / J.

By composing ^β with the map → A, we obtain a differential operator ^β : → A. Note that ^β(T^α) = αβ^α - β for α∈ℕ^⊕. In order to see that ^β induces a differential operator D^[;β]_A : A → A, we shall verify that it annihilates the ideal J ⊂.

Recall that ^β : → is a differential operator of order β < p^e over A^p^e. Therefore, by Lemma <ref>, ^β is linear over the subring

    A^p^e[ t_b^p^e| b ∈]
    	⊂
    	.

Thus we see that, for each element b ∈ and each polynomial f ∈,

    ^β ( ( t_b^p^e - b^p^e) · f  )
    	= ( t_b^p^e - b^p^e) ·^β(f)
    	∈ J.

This implies that ^β annihilates J, and hence it induces a differential operator on A, say D^[;β]_A : A → A, as required.








At first glance, it may seem that the definition D^[;β]_A depends on the choice of the integer e ≫ 0. However, as follows from Corollary <ref> below, there is a unique differential operator on A of order β over 𝔽_p satisfying condition (<ref>) of Proposition <ref>. Thus D^[;β]_A is well-defined, in the sense that it does not depend on the choice of the integer e.






  
Let A be a reduced ring containing 𝔽_p that admits an absolute p-basis, say . Then A is differentially smooth over 𝔽_p in the sense of Grothendieck <cit.>. Moreover, the family of differential operators D^[;β] as defined in Proposition <ref> has the following properties:


  * For all β, β' ∈ℕ^⊕,

    D_A^[;β]∘ D_A^[;β']
    	= D_A^[;β']∘ D_A^[;β]
    	= (β + β')!/β! β'!· D_A^[;β + β'] .


  * For any element f ∈ A, there exists a finite number of indices β∈ℕ^⊕ so that D_A^[;β] (f) ≠ 0.

  * Every differential operator from A to an A-module M of order n over 𝔽_p, say Δ : A → M, has a decomposition of the form

    Δ
    	= ∑_β≤ n c_β· D_A^[;β] ,

with

    c_β
    	= ∑_γ≤ββγ
    		(-1)^γ·^γ·Δ(^β - γ)
    	∈ M

for β≤ n note that, although β runs over a possibly infinite set, Δ(f) is a finite sum for each f ∈ A by the previous condition.





By Proposition <ref>, _A is a free A-module with {d_A(b) | b ∈} as basis. Thus the differential smoothness and properties i), ii), and iii) follow from <cit.>.













































§ DIFFERENTIAL SATURATION AND ORDER OF IDEALS





In this section we intend to characterize the set of points of a variety where a given ideal has order greater than or equal to certain integer N. To this end, we shall make use of the differential operators introduced in the previous section.



Let A be a regular ring and fix a prime ideal ⊂ A. The order of an element f ∈ A at  is defined by

    _(f)
    	= max{ n ∈ℕ| f ∈^n A_}.

Similarly, the order of an ideal I ⊂ A at  is defined by

    _(I)
    	= max{ n ∈ℕ| I A_⊂^n A_} .

Given a regular variety Z and a coherent ideal sheaf ⊂_Z, we define the order of  at a point ξ∈ Z as

    _ξ()
    	= max{ n ∈ℕ|_ξ⊂_ξ^n }.






Let A be a regular ring and, for an integer s ≥ 0, let ^s(A) denote the module of absolute differential operators of order at most s from A to itself. Fix an element f ∈ A and a prime ideal ⊂ A. It is known that, if _(f) ≥ N, then Δ(f) ∈ for every Δ∈^N-1(A) (cf. <cit.>). Moreover, in the case that A is a polynomial ring over field k, say A = k[x_1, …, x_r], the previous implication turns out to be an equivalence (see <cit.>). That is, when A is a polynomial ring, _(f) ≥ N if and only if Δ(f) ∈ for every Δ∈^N-1(A). This result has a natural generalization to the case in which A admits an absolute p-basis, as shown in the following proposition.

  
Let A be a regular ring over a field of characteristic p > 0 which has an absolute p-basis. Then, for any element f ∈ A and any prime ideal ⊂ A, the following conditions are equivalent:


  * _(f) ≥ N.

  * Δ(f) ∈ for every Δ∈^N-1(A).






i) ⇒ ii) follows from <cit.>. For the converse we proceed by contradiction. Namely, assume that ii) holds and that _(f) = n with n < N.

Fix a regular system of parameters of A_, say z_1, …, z_d, and let A_ denote the completion of A_ with respect to its maximal ideal. Recall that, by Cohen's structure theorem,

A_≃ k()[[z_1, …, z_d]], where k() represents the residue field of A_. Since A_ is faithfully flat over A_, the order of f in A_ coincides with its order in A_. Hence f has order n in A_, and therefore there should be a differential operator in A_ of order at most n over k(), say Δ^* : A_→A_, so that Δ^*(f) is a unit in A_. That is, so that Δ^*(f) ∉A_.



Next, fix an absolute p-basis of A, say . Recall that, in virtue of Proposition <ref>, there is family of differential operators on A naturally associated to , say { D_A^[;β]|β∈ℕ^⊕( }, where D_A^[;β]∈^β(A) for each β∈ℕ^⊕. Let ι : A →A_ denote the natural homomorphism from A to A_. By composition, Δ^* ∘ι is an absolute differential operator of order n from A to A_. Thus, by Corollary <ref> ii) and iii), there exists a finite collection of indices, say β_1, …, β_r ∈ℕ^⊕, with β_i≤ n, such that

    (Δ^* ∘ι)(f)
    	= ∑_i=1^r Δ(^β_i) D_A^[;β_i](f).

Since Δ^*(f) ∉A_, it follows that D_A^[;β_i](f) ∉ for some i ∈{1, …, r}, which contradicts ii).






 §.§ Differential saturation of ideals


Consider a regular ring A and an ideal I ⊂ A, and fix an integer n ≥ 0. We shall define the saturation of I by differential operators of order at most n by

    ^n (A)(I)
    	= ⟨Δ(f)
    		| f ∈ I,  Δ∈^n(A) ⟩.

Assume that the ring A has an absolute p-basis. In this case, attending to Proposition <ref>, it is easy to see that I has order at least N at a prime ideal ⊂ A if and only if ^N-1(I) ⊂. Thus, when A has an absolute p-basis,

    {∈(A) |_(I) ≥ N }
    	= ( ^N-1(A)(I) ).

This gives a characterization of the set of points of (A) where the ideal I has order at least N. The purpose of the rest of this section is to find a similar characterization for the set of points of a given variety Z where a coherent ideal ⊂_Z has certain order. To this end, we shall first check that the ideal ^n (A)(I) introduced above is compatible with localization.



  
Let A be regular ring of characteristic p > 0 which has an absolute p-basis, say , and let I ⊂ A be an ideal. Fix an integer n ≥ 0. Then, for any multiplicative subset S ⊂ A,

    ^n (S^-1A) (S^-1I)
    	= ^n (A)(I) ⊗_A S^-1A .









Fix a collection of generators of I, say I = ⟨ f_1, …, f_r ⟩. For each f_i, consider the ideal

    ^n(A)(f_i)
    	= ⟨Δ(f_i)
    		|Δ∈^n(A) ⟩.

Note that, with this notation,

    ^n (A)(I)
    	= ^n (A)(f_1) + … + ^n (A)(f_r).

Moreover, by Corollary <ref> iii), one has that

    ^n (A)(f_i)
    	= ⟨ D^[;β]_A (f_i)
    		| β≤ n ⟩ ,

and hence

    ^n (A)(I)
    	= ⟨ D^[;β]_A (f_i)
    		| i ∈{1, …, r};  β≤ n ⟩.

Paralleling the previous argument, and taking into account that  can also be regarded as p-basis of S^-1A (Lemma <ref>), one can see that

    ^n (S^-1A) (S^-1I)
    	= ⟨ D^[;β]_S^-1A (f_i)
    		| i ∈{1, …, r};  β≤ n ⟩.

In this way, as the image of D^[;β]_A (f_i) in S^-1A coincides with D^[;β]_S^-1A (f_i), one readily checks that (<ref>) holds.






Next, given a regular variety Z defined over a field positive characteristic k and a coherent ideal ⊂_Z, we would like to define some sort of saturation of  by differential operators on _Z. Note that, in general, the sheaf of absolute differential operators of order at most n on Z, say ^n(_Z), is not finitely generated or quasi-coherent (namely, this occurs whenever [k:k^p] = ∞). Thus, it is not clear a priori whether one can construct a coherent differential saturation of .



  
Let Z be a regular variety over a possibly non-perfect field k of characteristic p > 0, and let  be a coherent sheaf of ideals over Z. Then, for each integer n ≥ 0, there exists a unique coherent sheaf of ideals over Z, which we shall denote by ^n(_Z)(), with the following properties:


  * For every open subset W ⊂ Z, every element f ∈Γ(,W), and every differential operator Δ∈^n ( Γ(_Z,W) ),

    Δ(f)
    	∈Γ( ^n(_Z)(), W );


  * For every ξ∈ Z,

    ( ^n(_Z)() )_ξ
    	= ^n (_Z,ξ)(_ξ).






We shall start by constructing the sheaf ^n(_Z)(). After that we will check that it satisfies conditions i), ii), and iii).

Let 𝔄 denote the family of open affine subschemes of Z of the form U = (A) such that A admits an absolute p-basis. Recall that, by Theorem <ref>, Z is covered by these charts. For each chart U ∈𝔄, say U = (A), consider the sheaf

    _U := [ ^n(A) (Γ(,U)) ]^∼

defined over U. By Lemma <ref>,

    (_U)_ξ
    	= ^n (_Z,ξ)(_ξ)

for every ξ∈ U. Thus we see that the collection of sheafs {_U}_U ∈𝔄 can be patched in order to obtain a coherent sheaf of ideals over Z. This will be the sheaf ^n(_Z)().

To check property i), fix an open subset W ⊂ Z, an element f ∈Γ(,W), and a differential operator Δ∈^n ( Γ(_Z,W) ). Choose a finite collection of charts, say U_1, …, U_m ∈𝔄, with U_i = (A_i), so that W = ⋃_i=1^m U_i. Note that, for each i, the element f maps to a section f_i ∈ A_i, and Δ maps to a differential operator Δ_i ∈^n(A_i), in such a way that Δ(f) maps to the section Δ_i(f_i) ∈ A_i. By construction,

    Δ_i(f_i)
    	∈Γ( ^n(_Z)(), U_i )
    	= ^n(A_i) (Γ(,U_i)),

and hence the section Δ(f) belongs to Γ( ^n(_Z)(), W ). This proves i). Property ii) follows from the construction and condition (<ref>). 




  
Let Z be a regular variety over a possibly non-perfect field k of characteristic p > 0, and let  be a coherent sheaf of ideals over Z. Then, for every integer N ≥ 1,

    {ξ∈ Z |_ξ() ≥ N }
    	= ( ^N-1(_Z)() )
    	⊂ Z,

where ^N-1(_Z)() ⊂_Z denotes the coherent ideal sheaf constructed in Proposition <ref>.



This is a consequence of Proposition <ref> ii) and Proposition <ref>.















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