Let F be a field, and consider an F-linear abelian categoryA.
Recall that we bypass set-theoretical difficulties by assuming the logical axiom of existence of universes, which is independent of (ZFC).
Definition 23.1. The category indA of ind-objects of A is the following. An object of indA is a filtered direct system (Xi)i∈I of objects ofA. Given two such objects (Xi)i∈I and (Yj)j∈J, we set
We have a natural functor A → indA, mapping an object X of A to the object (Xi)i∈I∅ given by I∅ := {∅}and X∅ := X.
Recall thatA is called cocomplete if it contains all colimits, which is equiva-lent to requiringA to contain all direct sums.
Lemma 23.2. (a) indA is a cocomplete F-linear abelian category.
(b) The functorA →indA is F-linear, exact, and fully faithful.
(c) IfA is Noetherian, thenA is closed under subquotients in indA, and we may describe every object of indA as a union of objects in the essential image ofA →indA.
Proof. [Del87,§4.1 and Lemme 4.2.1]. ∴
Let F0/F be a field extension.
Definition 23.3. An F0-module inA is a pair X = (X, φ) consisting of an object X of A, and an F-linear ring homomorphism φ : F0 → EndA(X). Given two F0-modules X and Y in A, we let HomAF0(X,Y) be the subset of HomA(X,Y) consisting of those homomorphisms that commute with the respective actions of F0. In this way, we obtain the F0-linear abelian categoryAF0of F0-modules inA. Note that AF0 may consist only of trivial F0-modules, for instance if A is F-finite and [F0: F] is infinite.
Definition 23.4. Consider an element X ∈ indA, and let E ⊂ EndindA(X) be a subring. For a free right E-module M, the external tensor product M ⊗E X is defined (abusing language slightly) to be “the” object representing the functor Y 7→ HomE(M,HomindA(X,Y)) on indA, i.e., equipped with a natural isomor-phism
HomE(M,HomindA(X,Y))−−−→ HomindA(M⊗E X,Y).
It may be identified with a direct sum of rkE(M) copies of X.
The external tensor product is an exact F-linear functor in its first variable if we fix X and E, and in its second variable if we let E = F and fix M.
Remark 23.5. In the situation of Definition 23.4, if M is a free right E-module of finite rank, then M⊗E X has a second universal property, namely it represents the functor Z 7→V ⊗E HomindA(X,Z) on indA, so one has a natural isomorphism
M⊗E HomindA(Z,X)−−−→ HomindA(Z,M⊗E X).
For every object X ∈ indA, we may consider F0 ⊗F X as an F0-module in indA by using the natural action of F0 on itself by multiplicationµ. In this way, we obtain an exact F-linear functor
t : indA −→ (indA)F0, X7→ (F0⊗F X, µ⊗id). (23.6) Lemma 23.7. For every X in indA and Y = (Y, ψ) in (indA)F0, the following natural homomorphism is an isomorphism:
HomindA(X,Y)−→Hom(indA)F0(F0⊗F X,Y).
In other words, the functor t of (23.6) is left adjoint to the forgetful functor from F0-modules in indA to indA.
Proof. We start by making explicit the natural homomorphism in the statement of this lemma. An element h∈ Hom(X,Y) is mapped to the unique homomorphism e(h)∈Hom(indAF0(F0⊗F X,Y) which corresponds via the injection
Hom(indA)F0(F0⊗F X,Y)⊂HomindA(F0⊗F X,Y)HomF F0,HomindA(X,Y) to the homomorphism mapping f0 ∈F0to the homomorphism
X −−−h→Y ψ( f
0)
−−−−−→ Y.
By construction, e(h) is a homomorphism of F0-modules.
The inverse to e is given by mapping an element of Hom(indA)F0(F0⊗FX,Y) to its restriction to X via the injection X F⊗F X ⊂ F0⊗F X. ∴ Lemma 23.8. IfA is finite, then t : A →(indA)F0 is F0/F-fully faithful.
Proof. We must show that forA finite and X,Y ∈A the natural homomorphism F0⊗F HomindA(X,Y) −→Hom(indA)F0(F0⊗F X,F0⊗F Y)
is an isomorphism. By Lemma 23.7, the target of this isomorphism coincides with HomindA(X,F0⊗FY), so we must show that the natural homomorphism
F0⊗F HomindA(X,Y)−→HomindA(X,F0⊗F Y) is an isomorphism.
Injectivity: Given an element eh ∈ F0 ⊗F Hom(X,Y), there exists a finite F-dimensional subspace V ⊂ F0 such that eh arises from an element of V ⊗F
Hom(X,Y). By Remark 23.5, we have a natural isomorphism V ⊗F Hom(X,Y) Hom(X,V ⊗F Y). Now the commutative diagram
V⊗F Hom(X,Y) //
Surjectivity: Consider an element h of Hom(X,F0 ⊗F Y). Since X is finite, the image im(h) of h is finite. The object F0 ⊗F Y is the union over all finite F-dimensional subspaces V ⊂ F0 of its subobjects V ⊗F Y. It follows that im(h) ⊂ V⊗F Y for some finite F-dimensional V ⊂F0.
Therefore, h lies in Hom(X,V ⊗F Y). By Remark 23.5, we have a natural isomorphism V ⊗F Hom(X,Y)Hom(X,V⊗F Y), so h arises from an element of V ⊗F Hom(X,Y) ⊂ F0 ⊗F Hom(X,Y) as desired, since again the diagram (23.9)
commutes. ∴
Remark 23.10. IfA is not finite, then t need not be F0/F-fully faithful. Here is a counter-example: Let F be a field, andA the category of all F-vector spaces.
Consider X := L
j∈NF and Y :=F. Choose a field extension F0 ⊃F such that F0 is isomorphic, as F-vector space, toL
i∈NF. We claim that the homomorphism F0⊗F Hom(X,Y) →Hom(X,F0⊗FY)
i∈NF. The latter strictly contains the former.
Definition 23.11. (a) An object X0 ∈A generates an F0-module X in indA if there exists an epimorphism F0⊗F X0 → X of F0-modules.
(b) If A is F-finite, the scalar extension of A from F to F0 is the full sub-categoryA ⊗F F0of (indA)F0 consisting of those F0-modules X in indA generated by objects ofA.
It is clear thatA ⊗F F0 is an F0-linear additive category, and that the functor A → (indA)F0 restricts to an exact F-linear functor
t : A →A ⊗F F0, X 7→F0⊗F X
which is F0/F-fully faithful by Lemma 23.8. But, whereas inA ⊗F F0 all coker-nels exist by definition, the same is not true for kercoker-nels. Therefore, in general it is not clear whetherA ⊗F F0 is abelian.
Lemma 23.12. LetA be F-finite.
(a) If [F0 : F] is finite, thenA ⊗F F0 =AF0 is an abelian category.
(b) Every object of (indA)F0 is the union of subobjects lying inA ⊗F F0. Proof. (a): This is clear from the definitions. We state it for clarification.
(b): We consider an object X = (X, φ) of (indA)F0. By Lemma 23.2(c), we may write X = S
i∈IXi for objects Xi ∈A. To prove our claim, it suffices to find objects Yi ofA ⊗F F0 such that X = S
Yi. We can achieve this as follows: We put
Yi := X
f0∈F0
φ( f0) Xi),
this is an object of indA. By definition of Yi, the actionφof X maps Yiinto itself, so we have found objects Yi :=(Yi, φ|Yi) of (indA)F0 such that X= S
Yi.
It remains to show that each Yiis an object ofA ⊗FF0. However, the inclusion Xi ⊂ Yiinduces an epimorphism F0⊗F Xi →Yiby the very definition of Yi, which
shows that Xi generates Yi. We are done. ∴
Before we can study the question of whether or not A ⊗F F0 is abelian, we intersperse a discussion of the semisimplicity ofA → (indA)F0.
Definition 23.13. Given X∈indA, an object Y ∈indA is called X-isotypic if Y is isomorphic to a direct sum of copies of X.
Lemma 23.14. For X ∈ indA and E := EndindA(X), the functor − ⊗E X gives rise to an equivalence of categories between the category of free right E-modules and category of X-isotypic objects of indA.
Proof. For any index set I let (−)(I) denote the direct sum of I copies of −. We first show that− ⊗EX is well-defined. Since M is a free right E-module, M E(I) for some index set I. Then
M⊗E X E(I)⊗E X (E⊗E X)(I) X(I),
so M⊗E X is X-isotypic. We claim that HomindA(X,−) is a quasi-inverse functor, and start by showing that this functor is well-defined: If Y X(I) is X-isotypic, then
HomindA(X,Y) HomindA(X,X(I))HomindA(X,X)(I) E(I) is a free right E-module.
Similar calculations show that HomindA(X,M⊗E X) M if M is a free right E-module, and if Y is X-isotypic then HomindA(X,Y)⊗E X Y.
Clearly both − ⊗E X and HomindA(X,−) are additive functors. It remains to show that they are fully faithful. However, if M E(I)and N E(J) are two free right E-modules, then commutativity of the following natural diagram (which is easily checked) shows that− ⊗E X is fully faithful, and a similar argument shows that HomindA(X,−) is fully faithful:
HomE(M,N)
//HomindA(M⊗EV,N⊗E V)
MatJ×I(E) id //MatJ×I(E).
∴ Proposition 23.15. Let X be a simple object ofA, and set E := EndA(X). Then the functor− ⊗E X gives rise to an inclusion preserving bijection between the set of right ideals of F0⊗F E and the set of subobjects of F0⊗F X in (indA)F0. Proof. We set E0 := F0⊗F E and X0 := F0⊗F X. Since X is simple, E is a skew field over F. Note that we may regard X0 as an X-isotypic element of indA, and that E0is a free right E0-module.
Consider the following diagram of lattices:
right E-submodules of E0 oo //
( X-isotypic subobjects of X0
)
( F0-stable right E-submodules of E0
)
oo //(
F0-stable X-isotypic subobjects of X0
)
right ideals of E0 oo //
subobjects of X0
The upper row is a bijection by Lemma 23.14 and it preserves inclusions by con-struction. The second row corresponds to the F0-stable objects in the upper row, using the operations of F0 on E0 and V0, respectively. Since the bijection in the first row is functorial, it induces a bijection of the second row. Finally, we may clearly identify the objects of the second row with the objects of the third row. ∴ Definition 23.16. A semisimple algebra E is separable if for every simple F-algebra direct summand E0 ⊂ E the center of E0 is a separable field extension of F (cf. Definition 16.1 for the general notion of separable field extensions).
Remark 23.17. This definition of separability for algebras is equivalent to various others, cf. [Bou81, VIII.§7.5, D´efinition 1 and Proposition 6, Corollaire].
Proposition 23.18. Let X ∈ A be a semisimple object of finite length such that dimFEndA(X)<∞.
(a) F0⊗F X has finite length in (indA)F0.
(b) If F0/F is a separable field extension, or EndA(X) is a separable F-algebra, then F0⊗F X is semisimple.
Proof. We may assume that X is simple by applying the following proofs to each direct summand of X separately.
(a): Set E := EndA(X). Since E is finite F-dimensional, F0 ⊗F E is finite F0-dimensional, and the lattice of right ideals of F0 ⊗F E has finite length. By Proposition 23.15, this implies that the lattice of subobjects of F0⊗F X has finite length, so F0⊗F X has finite length.
(b): Since X is semisimple of finite length, E is a finite-dimensional semisim-ple F-algebra. Now [Bou81, §7, no. 6, Corollaire 3] proves that this, together with either the separability of F0/F or E/F, implies that F0⊗F E is a semisimple algebra. This implies that the radical of the lattice of right ideals of E0, i.e., the intersection of its maximal subobjects, is zero. Therefore, again by Proposition 23.15, the radical of F0⊗F X is zero. Since F0⊗F X has finite length by (a), this
shows that F0⊗F X is semisimple. ∴
Theorem 23.19. Assume thatA is F-finite.
(a) The objects ofA⊗FF0are precisely the F0-modules in indA of finite length.
(b) A ⊗F F0is a finite abelian category.
(c) If F0/F is separable, thenA is F0-finite andA → A ⊗F F0is semisimple.
Remark 23.20. If A is a finite F-linear abelian category, but not F-finite, then A ⊗F F0 may contain objects of infinite length. For instance, if F0/F is an in-finite field extension, consider the category VecF0 of finite-dimensional F0-vector spaces, with F0-linear homomorphisms. It is obviously F0-finite abelian, so it is a finite F-linear abelian category. The object F0⊗F F0 is an object of (VecF0)⊗F F0 of infinite length, as may be verified using Proposition 23.15.
Remark 23.21. Following discussions with Richard Pink, I am convinced that with a little more effort, dealing with inseparability, one should be able to show thatA ⊗F F0 is F0-finite for every F-finite abelian categoryA.
Proof. (a,b): We first show that all objects ofA ⊗F F0 have finite length. It is sufficient to show this for objects of the form F0 ⊗F X with X ∈ A, since every object ofA ⊗F F0 is a quotient of such an object. We may also assume that X is
simple, since X has finite length. Then Proposition 23.18(a) shows that X⊗F F0 has finite length.
Conversely (and here we paraphrase parts of [Del87, Lemme 4.5]), let X be a finite-length object of (indA)F0. By Lemma 23.12(b), X is a union of subobjects Y lying inA ⊗F F0. Since X has finite length, it equals one of these subobjects, so we have X ∈A ⊗F F0.
Clearly, the full subcategory of (the abelian category) (indA)F0 consisting of those objects having finite length is abelian, soA⊗FF0is a finite abelian category.
(c): The idea of the proof of F0-finiteness is the following: Given X,Y ∈ A ⊗F F0, choose X0 ∈A and an epimorpismπ : F0⊗F X0 −−→→ X. If we can find an object Y0 ∈A and a monomorphismι : Y ,→ F0⊗F Y00, then the assignment
f 7→ι◦ f ◦πgives rise to an F0-linear monomorphism
HomA⊗FF0(X,Y),→ HomA⊗FF0(F0⊗F X0,F0⊗FY0)= F0⊗F HomA(X0,Y0), which is a finite-dimensional F0-vectorspace sinceA is F-finite.
Assume that F0/F is separable. It is sufficient to show that EndA⊗FF0(X) is finite F0-dimensional for every X ∈ A ⊗F F0. By Proposition 22.1, we may assume that X is semisimple. Let an object X0 ∈ A and an epimorphism π : F0 ⊗F X0 −−→→ X be chosen. Since X is semisimple, rad(F0 ⊗F X0) ⊂ ker(π), soπ induces an epimorphism
$: F0⊗F X0/rad(F0⊗F X00)−−→→ X.
Since F0/F is separable, by Proposition 23.18(b) the functor A → A ⊗F F0 is semisimple, so by Theorem 3.4(c) we have rad(F0⊗FX0)= F0⊗Frad(X0). So X is a quotient of F0⊗F(X0/rad X0), a semisimple object ofA ⊗FF0since X0/rad(X0) is semisimple andA → A ⊗F F is semisimple. Hence $splits, we can choose an embeddingι : X ,→ (X0/rad X0), and may follow the method of proof given
above. ∴
Example 23.22. (a) If F0/F is any field extension, and VecF is the category of finite-dimensional F-vector spaces, then VecF⊗FF0is the category VecF0 of finite-dimensional F0-vector spaces.
(b) If G is an affine group scheme over F, and RepFG is the category of finite-dimensional representations of G over F, then RepFG⊗F F0 is the cate-gory RepF0(GF0) of finite-dimensional representations of GF0 over F0. This follows, for example, from [Wat79, Theorem 3.5].