as a special case), as well as the snake lemma (and the nine lemma as a special case). Semi-simple Abelian categories An abelian category is called semi-simple if there is a collection of objects called simple objects (meaning the only sub-objects of any are the zero object and itself) such that an object can be decomposed as a direct sum (denoting the coproduct of the abelian category)This technical condition is rather strong and excludes many natural examples of abelian categories found in nature. For example, most module categories over a ring are not semi-simple; in fact, this is the case if and only if is a semisimple ring. Examples Some Abelian categories found in nature are semi-simple, such as Category of finite-dimensional vector spaces over a fixed field By Maschke's theorem the category of representations of a finite group over a field whose characteristic does not divide is a semi-simple abelian category. The category of coherent sheaves on a Noetherian scheme is semi-simple if and only if is a finite disjoint union of irreducible points. This is equivalent to a finite coproduct of categories of vector spaces over different fields. Showing this is true in the forward direction is equivalent to showing all groups vanish, meaning the cohomological dimension is 0. This only happens when the skyscraper sheaves at a point have Zariski tangent space equal to zero, which is isomorphic to using local algebra for such a scheme. Non-examples There do exist some natural counter-examples of abelian categories which are not semi-simple, such as certain categories of representations. For example, the category of representations of the Lie group has the representationwhich only has one subrepresentation of dimension . In fact, this is true for any unipotent grouppg 112. Subcategories of abelian categories There are numerous types of (full, additive) subcategories of abelian categories that occur in nature, as well as some conflicting terminology. Let A be an abelian category, C a full, additive subcategory, and I the inclusion functor. C is an exact subcategory if it is itself an exact category and the inclusion I is an exact functor. This occurs if and only if C is closed under pullbacks of epimorphisms and pushouts of monomorphisms. The exact sequences in C are thus the exact sequences in A for which all objects lie in C. C is an abelian subcategory if it is itself an abelian category and the