Perfect category-graded algebras

In [19] the second author explored homological properties of algebras graded over a small category. Our interest in these algebras arose from our research on the homological properties of Schur algebras, but we believe that they play an important organizational role in representation theory in general. Recall that an abelian category C is called perfect if every object of C has a projective cover (see Section 2). The existence of projective covers for every object guarantees the existence of minimal projective resolutions for every object in the category. The category C is called semi-perfect if every finitely generated object has a projective cover. We say that a category-graded algebra A is (semi)-perfect if the category of A-modules is (semi)-perfect. In [19] it was given a criterion for category-graded algebras to be semi-perfect. This criterion is sufficient to ensure that all category-graded algebras which appear in [17] are semi-perfect. But this is not enough to prove the existence of a minimal projective resolution for some of them, as the kernel of a projective cover may not be finitely generated. In this article we fill this gap by giving a criterion for a category-graded algebra to be perfect and extend the results of [19] to algebras over an arbitrary commutative ring with identity. Next we introduce the notions related with category-graded algebras that will be needed and explain the main result in more detail. Let R be a commutative ring with identity. We will write ⊗ for the tensor product of two R-modules over R. Given a small category C, a C-graded R-algebra (see [19]) is a collection of R-modules A α parametrised by the arrows α of C, with preferred elements e s ∈ A 1s for every object s of C and a collection of R-module homomorphisms µ α,β : A α ⊗ A β → A αβ for every composable pair of morphisms α, β of C. For a ∈ A α and b ∈ A β we shall write ab for µ α,β (a ⊗ b). For every composable triple α, β, and γ of arrows in C and a ∈ A α , b ∈ A β , and c ∈ A γ we require associativity a(bc) = (ab)c. Suppose also that α : s → t. Then we require e t a = a = ae s , for any α ∈ A α . A C-graded module M over a C-graded R-algebra A is a collection of Rmodules M γ parametrised by the arrows γ of C with R-module homomorphisms r α,β : A α ⊗ M β → M αβ for every composable pair of morphisms in C. We shall write am instead of r α,β (a ⊗ m) for a ∈ A α and m ∈ M β . As always we will assume the usual module axioms: where α, β, and γ are composable; and e t m = m, where γ : s → t and m ∈ M γ . An A-homomorphism between two C-graded A-modules M and N is a collection of R-module homomorphisms f γ : M γ → N γ such that for every composable pair of morphisms α, β ∈ C We denote the category of all C-graded A-modules by A-mod. For morphisms β : s → u and γ : s → t in C define Note that A (γ : γ) is a ring with unit e t and the multiplication induced by the maps µ α,β . The main result of this paper is The main idea of the proof of Theorem 1.1 is to apply the general criterion of perfectness obtained in [8]. Therefore we start in Section 2 with a result on the radical of an abelian category and a recollection of notions used in that work. Section 3 is devoted to Harada's criterion and the study of perfectness of a class of abelian categories, which will be useful in the sequel. In Section 4 we prove the main result and in Section 5 we give examples and indicate connections with previously known results. Throughout this article R denotes a commutative ring with identity. For undefined notation the reader is referred to [19]. The notion of radical for general additive categories was introduced in [15]. Let C be an additive category. An ideal ). A radical of an additive category C is an ideal I of C such that for every object A of C we have I(A, A) = J(C(A, A)), where J denotes the Jacobson radical of the ring. Let C be an abelian category. Then C has a unique radical. This fact was used without explicit proof in [8]. For completeness we provide a proof in the appendix. We will also write J for the radical of C. Given two objects A, B of C we will denote by π A : A⊕B → A, π B : A⊕B → B, i A : A → A⊕ B, and i B : B → A⊕ B the canonical projections and inclusions associated with the definition of the direct sum A⊕B. We will need the following technical property of the radical of C. and the desired equality follows. Next we introduce some standard notation which will be used in the following sections. We say that X ⊂ Y is a small subobject of Y if for any S ⊂ Y such that X + S = Y we have S = Y . An epimorphism π : P ։ Y , where P is projective, is called a projective cover of Y whenever Ker π is a small subobject of P . Note that in a perfect abelian category every object has a (unique up to isomorphism) minimal projective resolution. By definition a minimal projective resolution of an object X is an exact complex (P • , d • ) with a map ε : P 0 ։ X, such that the maps d k : P k+1 → Ker(d k-1 ) and ε are projective covers. The existence of minimal projective resolutions in a perfect category can be shown by induction. In this section we give a sufficient and a necessary condition for a Grothendieck category C to be perfect. These are based on Harada's criterion of perfectness, Corollary 1 p.338 of [8]. The crucial ingredient of this criterion is the notion of T -nilpotent system. . . , and every small subobject X of M i1 there is a natural number m such that f m f m-1 . . . f 1 (X) = 0. Definition 3.2. Let C be an abelian category. We say that an object Note that our definition of semi-perfect object is different from the definition given in [8] on p. 330, but this does not interfere with the work. Let { P α | α ∈ I} be a generating set of semi-perfect objects of an abelian category C. Then each ring C(P α , P α ) is semi-perfect. By Theorem 27.6 of [1], for each α the ring C(P α , P α ) has a complete set of orthogonal idempotents e α,1 , e α,2 , . . . , e α,nα and for every α ∈ I and every 1 ≤ j ≤ n α the ring e α,j C(P α , P α )e α,j is local. We denote by P α,j the direct summand of P α that corresponds to e α,j . We also write π α,j for the canonical projection of P α on P α,j and i α,j for the canonical embedding of P α,j in P α . Proposition 3.1. The objects P α,j are completely indecomposable. Proof. Since e α,1 , . . . , e α,nα is a complete orthogonal set of idempotents the ring C(P α,j , P α,j ) ∼ = e α,j C(P α , P α )e α,j is local. Thus P α,j is completely indecomposable. Proof. In this proof we are going to apply Corollary 1 on p.338 of [8]. This claims that if C has a generating set of finitely generated objects and { Q β | β ∈ K} is a T -nilpotent generating set of completely indecomposable projective objects, then C is perfect. Thus we have to construct a T -nilpotent generating set of completely indecomposable projective objects. If we apply the construction described above to { P α | α ∈ I}, we get a generating set G = { P α,j | α ∈ I, j = 1, . . . , n α }. Every object P α,j is a direct summand of P α and so P α,j is projective. The object P α,j is also completely indecomposable by Proposition 3.1. Now we will show that G is T -nilpotent. Let P α1,j1 , P α2,j2 , . . . be a sequence of objects in G and f k ∈ J(P α k ,j k , P α k+1 ,j k+1 ). From Proposition 2.1 it follows that Let X be a small subobject of P α1,j1 . Then i α1,j1 (X) is a small subobject of To prove the next proposition we shall use the following consequence of Axiom of Choice (see example 1 to Theorem III.7.4.1 of [4]). Proof. By Theorem 7.2 of [19] every object P α is semi-perfect. Let {e α,1 , . . . , e α,nα } be a complete set of orthogonal idempotents for C (P α , P α ), α ∈ I. Denote by P α,j the direct summand that corresponds to e α,i . Then Then there is a sequence f k ∈ J P α k , P α k+1 and a small subobject X of P α1 such that for every m ∈ N f m . . . f 1 (X) = 0 which is the same as f m e αm,jm . . . e α2,j2 f 1 e α1,j1 (X) = 0. Denote by S m+1 the subset of {1, . . . , n α1 } × • • • × 1, . . . , n αm+1 of elements (j 1 , . . . , j m+1 ) such that e αm+1,jm+1 f m e αm,jm . . . e α2,j2 f 1 e α1,j1 (X) = 0. Then S m are finite non-empty sets for every m ∈ N, and we have maps From Lemma 3.3 it follows that there is a sequence (l k ) k∈N such that (l 1 , . . . , l m ) ∈ S m for every m. Define h m = π αm+1,lm+1 f m i αm,lm . Then h m ∈ J P αm,lm , P αm+1,lm+1 . Moreover, for every m ∈ N i αm+1,lm+1 h m . . . h 1 (π α1,l1 X) = e αm+1,lm+1 f m e αm,lm . . . e α2,l2 f 1 e α1,l1 (X) = 0. Let C be a small category. We define a C-graded R-module V to be a collection of R-modules V γ parametrized by the arrows γ ∈ C. A morphism from a Cgraded R-module V to a C-graded R-module W is a collection of R-module homomorphisms f γ : V γ → W γ . We will write V C for the category of C-graded R-modules and V for the category of R-modules. Next we indicate how the results of [19] can be extended from the case when R is a field to the case of a general commutative ring. Let A be a C-graded R-algebra with multiplication map µ and V a C-graded R-module. Consider the functor given on objects by the formula Repeating the proof of [19, Proposition 2.1] we get that the functor F A is a left adjoint to the forgetful functor U : A-mod → V C . The counit ε of this adjunction is given by the structure maps of A-modules. Namely, if M is an Amodule with structure maps r α,β then the (α, β) component of ε γ : From the existence of local units it follows that the maps ε γ are surjective for all γ ∈ C. The proofs of Propositions 3.1, 4.2, and 4.3 of [19] can be extended without any changes to the case of general R. As a consequence we get Proposition 4.1. Let A be a C-graded R-algebra. Then the category A-mod is Grothendieck. In particular, A-mod is a complete and cocomplete abelian category. We say that an object It is clear that every free C-graded R-module is projective, as the lifting condition must be verified componentwise. Given a C-graded R-algebra A, we say that an A-module M is free if there is a free C-graded R-module V such that F A (V ) ∼ = M in A-mod. Now we have an analog of [19, Proposition 5.1]. Proposition 4.2. Let A be a C-graded R-algebra and M a free A-module. Then M is projective. Proof. From Proposition 4.2 we know that the objects A [γ], γ ∈ C, are projective. By the reasoning on p.105 of [10] a projective object is finitely generated if and only if it is small. To check that A [γ] is small we have to show that for any family of A-modules { M i | i ∈ I} and every map of A-modules f : A [γ] → i∈I M i there is a finite subset I ′ of I such that f factorizes via i∈I ′ M i . From the adjunction described above we have for any subset whose horizontal arrows are isomorphisms and vertical arrows are induced by the natural inclusion of i∈I ′ M i into i∈I M i . Let f ′ : R → i∈I (M i ) γ be the map that corresponds to f . Then f ′ (1) ∈ i∈I ′ (M i ) γ for a finite subset It is left to show that X := { A [γ] | γ ∈ C} is a generating set for A-mod. Let M be an A-module. For every γ ∈ C there is a free R-module V γ and a surjective homomorphism of R-modules ψ γ : V γ → M γ . Then V = (V γ ) γ∈C is a free C-graded R-module and ψ = (ψ γ ) γ∈C is a surjection of C-graded Rmodules. Now F A (V ) is a direct sum of objects from X, since F A commutes with direct sums. Moreover, the composition is a surjecive homomorphism of A-modules. Therefore, M is a quotient of a direct sum of objects from X, which shows that X generates A-mod. The proof of the criterion of semi-perfectness that extends [19, Theorme 8.1] to the case of C-graded algebras over an arbitrary commutative ring is similar to one given in [19] and we skip it: Theorem 4.4. Let C be a small category and A a C-graded R-algebra. The category A-mod is semi-perfect if and only if for every arrow γ ∈ C the algebra A (γ : γ) is semi-perfect. We are now ready to prove the main theorem of the paper. Suppose first that the rings A (γ : γ) are left perfect for all arrows γ ∈ C. Just like in the proof of [19,Theorem 8.1] there is an isomorphism of rings A-mod(A [γ] , A[γ]) ∼ = A(γ : γ) op . Note that every left or right perfect ring is semi-perfect. Thus, A-mod(A[γ], A[γ]) is a semi-perfect ring. Hence A[γ] is a semi-perfect object. To prove that A-mod is perfect, by Proposition 3.2 it is enough to check that From the adjunction between F A and the forgetful functor we have an isomorphism of C-graded R-modules There are two possibilities to consider: 1) There are k and l such that A (β k : β k+l ) ∼ = 0. Then f k+l-1 . . . f k = 0 and 2) We have A (β n : β n+m ) ∼ = 0 for all n, m ∈ N. Then there is an arrow β ∈ C such that β = β n for infinitely many n ∈ N. Let n (k), k ∈ N, be an increasing sequence of natural numbers such that β n(k) = β for all k. Define Suppose now that A-mod is a perfect category. By Theorem 4.4 the rings A (β : β) are semi-perfect. By definition of semi-perfect ring the quotient ring A (β : β) J (A (β : β)) is semi-simple. Thus by Theorem 28.4(b) of [1] it is enough to show that the ideals J (A (β : β)) are left T -nilpotent for every map β ∈ C. We will show in fact that the ideals J (A (β : In this section we apply the main theorem to some classes of interesting rings. Let Γ be a monoid with unit e. We denote by ( * , Γ) the category with one object * and the set of morphisms given by Γ. Recall that a Γ-graded R-algebra is an R-algebra A with a fixed direct sum decomposition into R-submodules A γ , γ ∈ Γ such that e A ∈ A e and A α A β ⊂ A αβ . Analogously, a Γ-graded module M over a Γ-graded R-algebra A is defined as an A-module with a direct sum decomposition M = γ∈Γ M γ of R-submodules such that A α M β ⊂ M αβ . Homomorphisms of Γ-graded R-algebras (A-modules) are homomorphisms of R-algebras (A-modules) that preserve the components of the direct sum decomposition. It immediately follows that purely syntactical replacement of the sign γ∈Γ by the brackets ( ) γ∈Γ gives an equivalence between the category of Γ-graded algebras and the category ( * , Γ)-graded algebras. By the same argument, if A = γ∈Γ A γ is an Γ-graded R-algebra then the category A-gr of Γ-graded A-modules is equivalent to the category of A ′ -modules, where A ′ is the ( * , Γ)graded algebra that corresponds to A. Now we assume that Γ is a group and A is a Γ-graded algebra. Note that this is the most widely studied case of graded algebras (the standard reference book on the subject is [12]). We denote by Supp (A) the support of A, that is the set of arrows γ ∈ Γ such that A γ ∼ = 0. Proposition 5.1. Let Γ be a group and A a Γ-graded R-algebra with finite support. Then A-gr is perfect if and only if A e is a left perfect ring. Proof. Denote by A ′ the ( * , Γ)-graded algebra that corresponds to A. Then A ′ and ( * , Γ) satisfy the conditions of Theorem 1.1. In fact, let (β k ) k∈N be a sequence of elements in Γ such that A ′ (β k : β k+l ) ∼ = 0 for all k, l ≥ 1. We have for every n ≥ 2 Thus β 1 β -1 n lies in Supp(A). Since Supp (A) is finite at least one element repeats infinitely many times in the sequence This result was previously obtained in [11,Theorem 6(1,2)] by a different technique. Note also that in [3] it is proved that a Γ-graded ring A with finite support is left perfect as a usual ring if and only if A e is left perfect. In fact, if Γ is finite, it was also proved in [9] that a Γ-graded ring A is left perfect as usual ring both if and only if A e is left perfect, and if and only if the category A-mod is perfect. Chronologically the first results of this type are due to Renault [14] and Woods [18] who gave a criterion for perfectness of group algebras over a finite group. Their results were extended by Park in [13] to the case of skew group rings. Let A be a Γ-graded ring. The reader can find in [2] a characterization of perfectness for the categories of modules graded by Γ-sets. These categories do not fit in the general framework of the present paper. Recall that a poset (S, ≤) is called artinian if every descending sequence s 1 ≥ s 2 ≥ . . . of elements in S stabilizes. Proposition 5.2. Let Γ be an artinian ordered monoid such that e is the least element. Then the category of left Γ-graded A-modules is perfect if and only if the ring A e is left perfect. Proof. Let A ′ be the ( * , Γ)-graded R-algebra that corresponds to A under the equivalence described above. Let γ ∈ Γ. Then { α | αγ = γ} = {e}. In fact, suppose αγ = γ and α = e. Since e is the least element of Γ we have α > e, and, as Γ is an ordered monoid it follows that αγ > eγ = γ, a contradiction. Therefore for all γ ∈ Γ (A ′ )(γ : γ) = A ′ e = A e . It is left to check that ( * , Γ) satisfies the condition of Theorem 1.2. Suppose γ 1 , γ 2 , . . . is a sequence of elements in Γ such that γ k+1 is a right divisor of γ k . Since e is the least element of Γ we get that γ k > γ k+1 . Therefore γ 1 , γ 2 , . . . is a descending sequence and must stabilize as Γ is artinian. An example of a graded algebra in the conditions just described is the Kostant form of the universal enveloping algebra of the complex Lie algebra of strictly upper triangular matrices. In our work on Schur algebras [17], we were led to the construction of a minimal projective resolution of the trivial module of this Kostant form. Although this module is obviously finitely generated it can not to be said the same about the kernels of the projective covers which appear in the resolution. It was this example that motivated the present paper. Remark 5.1. In [6] Eilenberg gave a criterion for an N-graded ring A to be perfect. Namely, Proposition 15 of [6] says that if A 0 is semiprimary then A is graded perfect. Note that every semiprimary ring is perfect (p.318 [1]) and therefore this result can be deduced from Proposition 5.2 in this paper. Now we give an example which shows that the condition "Γ is artinian" in Proposition 5.2 is essential. Let Γ = (Z, +) and denote ( * , Γ) by C. Given a field K, define a C-graded K-algebra A by and multiplication a k a l = a k+l . In fact, A is just the polynomial algebra in one variable considered as a C-graded algebra. We define a C-graded A-module X by and the action of A on X is given by a k x l = x k+l . Proposition 5.3. The module X has no projective cover in A-mod. Proof. Suppose φ : P ։ X is a projective cover of X. Then, by Theorem 5.1 [19], P is a direct summand of the free module F A (X) and there is an idempotent e : F A (X) → F A (X) such that f e = f , where f : Note that for every k, the set { a k-l ⊗ x l | k ≥ l} is a basis of the vector space F A (X) k , so we can write e(a 0 ⊗ x k ) = l≤k λ k,l a k-l ⊗ x l , where λ k,l ∈ K. Now the coefficient of . Since e is an idempotent we get that λ 2 k,k = λ k,k . For every k there are two possibilities: either λ k,k = 1 or λ k,k = 0. Let I ⊂ Z be the set of k's such that λ k,k = 1. We will show that the set I contains infinitely many elements. Suppose k is the minimal element of I. Then either e(a 0 ⊗ x k-1 ) = 0 or e (a 0 ⊗ , where λ k-1,l = 0 and l < k. The first alternative is impossible as In the second case all the monomials different from λ k-1,l a k-1-l ⊗x l in e (a 0 ⊗ x k-1 ) are of the form λ k-1,m a k-1-m ⊗x m for m < l. Since the coefficient of a k-1-l ⊗x l in e (λ k-1,m a k-1-m ⊗ x m ) is zero, and e (a 0 ⊗ x k-1 ) = e 2 (a 0 ⊗ x k-1 ), it follows that the coefficient of a k-1-l ⊗ x l in e (a k-1-l ⊗ x l ) is one, or in other words, that l ∈ I. This gives a contradiction between assumptions that k is the minimal element of I and l ≤ k -1 < k. Let us fix k, l ∈ I, l < k. Denote a 0 ⊗ x ka k-l ⊗ x l by v. We have e(v) = a 0 ⊗ x k + . . . , where all other summands are of the form µa k-m ⊗ x m , m < k. This shows that e(v) = 0. Moreover, f e(v) = f (v) = x kx k = 0. Thus e(v) ∈ ker(f ) ∩ P = ker (φ). Next we show that ker (φ) is not a small subobject of P . For this we will find an A-submodule Q of P such that Ae(v) + Q = P and e (v) ∈ Q. Let and B = B ′ ∪ {e(v)}. We will prove that B generates P as an A-module. For this we have only to show that for every n > k, n ∈ I the element e (a 0 ⊗ x n ) of P is in the A-linear span of B. We have where λ n,s = 0. If s = k then we can rewrite the above sum in the form where µ n,l = λ n,l + λ n,k and µ n,t = λ n,t for t = l. Thus e (a 0 ⊗ x n ) belongs to the A-linear span of B. If s = k, then e (a 0 ⊗ x s ) ∈ B as s ∈ I. Now, for each t < s in (1), either t belongs to I, or e (a 0 ⊗ x t ) can be written as an A-linear combination of elements e (a 0 ⊗ x r ) with r < t. So we keep applying e to each of these until we get only e (a 0 ⊗ x r ) with either r ∈ I or r ≤ k (note that we are left with a finite number of indices to deal, since we are only concerned with those k < t ≤ s). We conclude then that e (a 0 ⊗ x n ) belongs to the A-linear span of B. Let us denote by Q the A-linear span of B ′ . We will show that the element e (v) of P k is not in Q k . Every element w of Q k can be written in the form where the sum is over t ∈ I, t < k and t ∈ I, t = k. Thus in fact w = t µ 2 > . . . stabilizes. Therefore α 1 , α 2 , . . . stabilizes as well. As we mentioned before, in this appendix we prove that the radical of an abelian category is unique and characterize it.