7. CONSULTATION INTERVIEWS WITH KEY STAKEHOLDERS
7.4 Activity within work
Example 7.9. Letkbe a commutative ring,Aak-algebra and ΣAthe one-object k-Mod-category defined by A. Then, [(ΣA)op, k-Mod] is the category of left A-modules. When A is finitely generated as a k-module, ΦΣA is the category of finitely generatedA-modulesA-Modf.
Example7.10. SupposeV =k-Mod, the category of modules over a commutative ringk. IfAandB arek-algebras of finite rank,A-Modf⊠B-Modf can be taken to be A⊗B-Modf. To see this, define aV-functor right exact in each variable A-Modf⊗B-Modf →A⊗B-Modf sending (M, N) to theA⊗B-moduleM⊗N. IfC is aV-category with finite colimits, a functor F :A-Modf ⊗B-Modf →C right exact in each variable is the same, up to an equivalence, as an objectC of C with an action of A and an action ofB, each one commuting with the other;
in other words, it is the same as aV-functor Σ(A⊗B)→C, or a right exactV -functorA⊗B-Modf →C. Therefore there is an equivalence and an isomorphism as depicted in the following diagram.
A-Modf ⊗B-Modf //
⊗TTTTTTTTT**
TT TT TT
T A-Modf ⊠B-Modf
≃
∼=
A⊗B-Modf
cocompleteA ⊠B defined by Proposition 6.31, at leasta priori.
Although in [19] the tensor product is defined for arbitrary abelian catego-ries, it is only shown to exist for certain special abelian categocatego-ries, namely those satisfying the following conditions.
Condition 1. The ground commutative ringkis a field, all the objects have finite length and the homs are finite-dimensional.
We shall show that for this special type of abelian categories,A ⊠Bhas the defining property of A •B. In other words, for the kind of abelian categories that A •B is shown to exist in [19], A ⊠B and A •B coincide. This gives evidence that the right product to consider would be⊠.
Recall that asub-quotient of an object in an abelian category is a quotient of a subobject. An abelian subcategory is closed under sub-quotients exactly when it is closed under subobjects, or dually, when it is closed under quotients.
Observation 7.11. IfD is an abelian category satisfying Condition 1 above, the inclusion of a full abelian subcategory closed under sub-quotientsi:C →D has a left adjointiℓ; the left adjoint sends an objectXofD to the greatest quotient of X lying inC. WhenD has a projective generatorP,D is canonically equivalent toD(P, P)-Modf via X7→D(P, X). Moreover,iℓ(P) is a projective generator in C, so that C ≃C(iℓ(P), iℓ(P))-Modf. There is an isomorphism
C
≃ //
∼=
C(iℓP, iℓP)-Modf
D ≃ //D(P, P)-Modf
with componentsC(iℓP, X)∼=D(P, iX); here the functor on the right hand side is the one induced by the morphism of algebrasiℓP,P :D(P, P)→C(iℓP, iℓP).
We consider the base category U = k-Mod equipped with chosen finite co-products and coequalizers given by the usual constructions. This gives chosen finite colimits by the usual construction of colimits out of coproducts and co-equalizers. For any k-algebra A, the category A-Modf of finitely presentable A-modules inherits a choice of finite colimits.
Observation 7.12. Given algebrak-morphismsf :A→A′ and g:B →B′ call f∗andg∗ the functors given by restriction of scalars. If the four algebras involved are finitely presentable as k-modules and Noetherian as algebras, we can prove thatf∗⊠g∗ is an exact functor.
Consider the diagram
A′-Modf⊗B′-Modf //
f∗⊗g∗
∼=
A′-Modf ⊠B′-Modf ≃ //
f∗⊠g∗
A′⊗B′-Modf (f⊗g)∗
A-Modf⊗B-Modf //A-Modf ⊠B-Modf ≃ //A⊗B-Modf
where the equivalences are the ones of Example 7.10. If A is a Noetherian al-gebra, the category of finitely presentedA-modules A-Modf is not only finitely cocomplete but it is closed under kernels in A-Mod. Hence it makes sense to say that f∗ ⊠g∗ is exact (as a tensor product of Noetherian algebras is again Noetherian). Since the outside rectangle commutes up to an isomorphism, we deduce that there exist an isomorphism filling in the square on the right hand side. Therefore, the exactness off∗⊠g∗ follows from the exactness of (f⊗g)∗.
Now suppose thatA is an abelian category satisfying Condition 1 above. Us-ing [19, 5.12]A can be shown to be a filtered colimit of full abelian subcategories Ai closed under sub-quotients, such that each Ai is equivalent to category of modules of finite rank over ak-algebra of finite rank (depending oni). Following the notation of [19], denote byhXi the full subcategory closed non-empty finite direct sums and under subquotients of A generated by the object X. Define a filtered categoryJ with obJ = obA and an arrow X → Y if an only if X is a direct summand of Y inA, and a functor J → U-Cat by sending X → Y to the inclusionhXi֒→ hYi. Clearly, A is a (filtered) colimit of this functor. By [19, 2.14, 2.17], each categoryhXi has a projective generatorPX and there is an equivalencehXi ≃A(PX, PX)-Modf (see Observation 7.11).
Let R be the 2-monad on U-Cat whose algebras are theU-categories with chosen finite colimits. Suppose that the abelian category A in the paragraph above is equipped with chosen finite colimits. Then each subcategory Ai is a subobject ofA in the category (R-Algs)0 of categories with chosen finite colimits and functors strictly preserving them. Since R is finitary, R-Algs → U-Cat creates filtered colimits, and A is a filtered colimit of the subcategories Ai in R-Algs.
Theorem 7.13. Suppose A andB are abelian categories with chosen finite col-imits and satisfying Condition 1. ThenA ⊠Bnot only has chosen finite colimits but is also abelian. Therefore, the monoidal structure⊠coincides on such abelian categories with the tensor product defined in [19].
Proof. Suppose A,B are U-categories satisfying Condition 1 and with chosen finite colimits. We only need to prove thatA ⊠B is an abelian category.
As observed above, A and B are filtered colimit of filtered diagrams of sub-categoriesAi andBj inR-Algs, respectively. By Proposition 6.14,A′ is filtered colimit of the diagram Ai′. The 2-functor ⊘ : R-Algs ×R-Algs → R-Algs of Corollary 6.30 preserves filtered colimits in each variable (since R does so) and henceA ⊠B=A′⊘B will be the filtered colimit of the Ai⊠B =Ai′⊘Bj in R-Algs, and in U-Cat.
We have seen in Observation 7.11 that each inclusion Ai֒→Aj is, after com-posing with certain equivalences and up to isomorphism, a restriction of scalars functor between categories of finite-dimensional modules, and likewise for the Bj’s. Hence, each functor Ai⊠Bu →Aj ⊠Bv is exact by Observation 7.12. It follows that A ⊠B is abelian, since the colimit in U-Cat of a filtered diagram of abelian categories and exact functors is an abelian category.