N-Koszul Algebras, A summary

E. L. Green E. N. Marcos R. Martínez-Villa  Pu Zhang
Dept. of Math. Dept. of Math. Instituto de Matemáticas Dept. of Math.
Virginia Tech USP  UNAM at Morelia  USTC
Blacksburg  São Paulo  Morelia  Hefei
VA, USA  S. Paulo, Brasil Mochoacán, MX  China

Abstract

In this paper we study N-koszul algebras which were introduced by R. Berger. We show that when n ≥ 3, these are classified by the Ext-algebra being generated in degrees 0, 1, and 2. We give a description of the Ext-algebra using the analogous of the Koszul complex and we also show that it is, is a Koszul algebra, after regrading. This notions can be generalized in various ways one was is to study categories of homogeneous algebras and this can be found in [2]

1 A general overview

In [1], Roland Berger introduced what he called “generalized” Koszul algebras. The generalized Koszul algebras are graded algebras A = K ⊕ A_{1} ⊕ A_{2} ⊕ ⋯ which are generated in degrees 0 and 1 such that there is a graded projective resolution of K for which the k^{th} projective in the resolution is generated in degree δ(k) where

δ(k) = \{ \begin{matrix}\frac{k}{2} n if k is even \\ \frac{k−1}{2}n + 1 if k is odd, \end{matrix} for some n.

We generalize this definition to the nonlocal case, i.e., K is replaced by a semisimple Kalgebra and we call this class of algebras N-koszul algebras.

We provide some general tools to study when the Yoneda product map

Ext_{A}^{k}(A_{0}, A_{0}) ⊗ Hom_{A}(M, A_{0}) → Ext^{k}(M, A_{0})

is surjective where A = A_{0} ⊕ A_{1} ⊕ ⋯ is a graded algebra and M is a graded left A-module.

We summarize the contents of the paper. In order to do that we need to introduce some notation:

2 Notation

Let K be a commutative ring and A = A_{0} ⊕ A_{1} ⊕ A_{2} ⊕ ⋯ be a graded K-algebra where the direct sum is as K-modules. Assume that A is generated in degrees 0 and 1; that is, A_{i} · A_{j} = A_{i+j} for all 0 ≤ i, j < ∞. Let Gr(A) denote the category of graded A-modules and degree 0 homomorphisms and Mod(A) denote the category of left A-modules. We denote by gr(A) and mod(A) the full subcategories of Gr(A) and Mod(A) respectively, consisting of finitely generated modules. Let F : Gr(A) → Mod(A) denote the forgetful functor and Gr_{0}(A) (respectively gr_{0}(A)) be the full subcategory of Gr(A) whose objects are the graded modules (respectively, finitely generated modules) generated in degree 0.

We assume A_{0} is a semisimple Artin algebra. The graded Jacobson radical of A, which we denote by r_{A}, or simply r, when no confusion can arise, is A_{1} ⊕ A_{2} ⊕ · · ·. Since A is generated in degrees 0 and 1, it follows that r^{i} = A_{i} ⊕ A_{i+1} ⊕ · · ·. We fix a minimal graded projective resolution of A_{0},

P^{•} : ⋯ → P^{k} → ⋯ → P^{1} → P^{0} → A_{0} → 0,

where A_{0} is viewed as a graded A-module generated in degree 0.

3 Summary

One of the fundamental results of the paper is the following:

Proposition 3.1 Let M ∈ Gr_{0}(A) be finitely generated with minimal graded projective resolution (Q^{•}). Assume that P^{n} is generated in degree s and Q^{n} is finitely generated. Then Q^{n} is generated in degree s if and only if Ext_{A}^{k}(A_{0}, A_{0}) ⊗_{K} Hom_{A}(M, A_{0}) → Ext_{A}^{k}(M, A_{0}) is surjective; that is, Ext_{A}^{k}(A_{0}, A_{0}) · Hom_{A}(M, A_{0}) = Ext_{A}^{k}(M, A_{0}).

One immediately consequence of this theorem is the following:

Corollary 3.2 Let M ∈ Gr_{0}(A) with minimal graded projective resolution Q^{•}. Assume that P^{k} is generated in degree s and Q^{k} is generated in degree t. Then t ≥ s.

If M_{1}, M_{2}, M_{3} are A-modules, we let

Y_{m,k} : Ext_{A}^{m}(M_{2}, M_{3}) ⊗_{K} Ext_{A}^{k}(M_{1}, M_{2}) → Ext_{A}^{m+k}(M_{1}, M_{3})

be the Yoneda product. We will usually write Y_{m,k} as Y when no confusion can arise. Furthermore, we will denote the image of Y in Ext_{A}^{m+k}(M_{1}, M_{3}) by Ext_{A}^{m}(M_{2}, M_{3})·Ext_{A}^{k}(M_{1}, M_{2}).

Lemma 3.3 Let M be a finitely generated graded A-module. The map Y : Ext_{A}^{k}(A_{0}, A_{0}) ⊗_{K} Hom_{A}(M, A_{0}) → Ext_{A}^{k}(M, A_{0}) factors through Ext_{A}^{k}(M/rM, A_{0}) → Ext_{A}^{k}(M, A_{0}), the map induced from the canonical surjection M → M/rM. Furthermore, the induced map Y′ : Ext_{A}^{k}(A_{0}, A_{0})⊗_{K}Hom_{A}(M, A_{0}) → Ext_{A}^{k}(M/rM, A_{0}) is surjective.

The fundamental proposition, 3.1, is used to prove the following:

Proposition 3.4 Suppose that P^{i} is finitely generated with generators in degree n_{i}, for i = α, β, α + β. Assume that

n_{α+β} = d_{α} + d_{β}.

Then the Yoneda maps Ext_{A}^{α}(A_{0}, A_{0})⊗_{K}Ext_{A}^{β}(A_{0}, A_{0}) → Ext_{A}^{α+β}(A_{0}, A_{0}) and Ext_{A}^{β}(A_{0}, A_{0})⊗_{K} Ext_{A}^{α}(A_{0}, A_{0}) → Ext_{A}^{α+β}(A_{0}, A_{0}) are both surjective. Thus,

Ext_{A}^{α+β}(A_{0}, A_{0}) = Ext_{A}^{α}(A_{0}, A_{0}) · Ext_{A}^{β}(A_{0}, A_{0})
= Ext_{A}^{β}(A_{0}, A_{0}) · Ext_{A}^{α}(A_{0}, A_{0}).

In section 4 of the paper appears the definition of N-koszul algebras.

Let A = A_{0} + A_{1} + A_{2} + ⋯ be a graded K-algebra generated in degrees 0 and 1, where A_{0} is a product of the field K, A_{1} is a finitely generated K-module and that P^{•} is a minimal graded A-projective resolution of A_{0}. We say that A is a N-koszul algebra if, for each n ≥ 0, P^{n} can be generated in exactly one degree, δ(n), and

δ(k) = \{ \begin{matrix}\frac{k}{2} n & if k is even \\ (\frac{k−1}{2}n) + 1 & if k is odd \end{matrix} .

We present the following characterization of N-koszul algebras.

Theorem 3.5 Let A = T_{A_{0}}(A_{1})/I where I can be generated by elements of ⊗_{A_{0}}^{n}A_{1} for some n ≥ 2. Then A is a N-koszul algebra if and only if the Ext-algebra E(A) can be generated in degrees 0, 1, and 2 in the ext-degree grading.

From the proof of the above theorem, we get the following important result.

Corollary 3.6 If A is a N-koszul algebra, with n > 2, then

Ext_{A}^{2m+1}(A_{0}, A_{0}) · Ext_{A}^{2k+1}(A_{0}, A_{0}) = (0),

for all k, m ≥ 0.

In section 5 we introduce the notion of N-koszul modules which is a natural generalization of the analogous notion of Koszul modules.

We say a left graded A-module M is a N-koszul module if there is a graded A-projective resolution ⋯ → Q^{2} → Q^{1} → Q^{0} → M → 0 such that Q^{n} is generated in degree δ(n) where

δ(k) = \{ \begin{matrix}\frac{k}{2} n & if k is even \\ (\frac{k−1}{2}n) + 1 & if k is odd \end{matrix} .

If M is a N-koszul module then M ∈ Gr_{0}(A) since Q^{0} is generated in degree 0. Note that if n = 2, then a module is N-koszul if and only if it has a linear projective resolution. Thus, in this case, being a N-koszul coincides with being a Koszul module.

We state in this summary two results which are proved in section 5 of the paper:

Proposition 3.7 Let A be a N-koszul algebra of type n and M a N-koszul module. Then Ω^{2}(M)[−n] and Ω^{1}(rM)[−n] are both N-koszul A-modules.

If M is a left A-module, let E(M) denote the left E(A)-module ⊕_{n≥0}Ext^{k}(M, A_{0}), where the module structure is given by the Yoneda product.

Theorem 3.8 Let A = A_{0} + A_{1} + ⋯ be a N-koszul algebra of type n with n ≥ 3 and let M be a left N-koszul A-module. Then E(M) can be generated in degree 0. Moreover, Ext^{2k+1}(A_{0}, A_{0}) · Ext^{2m+1}(M, A_{0}) = (0) for all k, m ≥ 0.

Section 6 on the paper studies the sum of the even extensions. The main result in this section is the following:

Let M be a left N-koszul A-module. We let

E^{ev}(A) = ⊕_{k≥0}Ext^{2k}(A_{0}, A_{0})

and

E^{ev}(M) = ⊕_{k≥0}Ext^{2k}(M, A_{0}).

We grade E^{ev}(A) by E^{ev}(A)_{k} = Ext^{2k}(A_{0}, A_{0}) and view E^{ev}(M) as a graded E^{ev}(A)- module where E^{ev}(M)_{k} = Ext_{A}^{2k}(M, A_{0}). We call this the even-grading.

The following result is the main result of this section.

Theorem 3.9 Let A = A_{0} + A_{1} + ⋯ be a N-koszul algebra of type n and M a N-koszul module. Then, in the even-grading, E^{ev}(A) is a Koszul algebra and E^{ev}(M) is a Koszul E^{ev}(A)-module.

In section 7 we study the Ext of a N-koszul algebra, the main result is the following: We denote by Ê(A) the Ext-algebra with a new grading where the homogeneous component of degree k is the sum of Ext^{2k−1} + Ext^{2k}. We show the following result.

Theorem 3.10 Let A = A_{0} + A_{1} + ⋯ be a N-koszul algebra with n ≥ 3. Let M be a N-koszul A-module. Then Ê(A) is a Koszul algebra and Ê(M) is a Koszul Ê(A)-module.

In section 8 we give a new proof of one of Berger’s result.

We introduced a N-koszul complex. This complex is defined for any algebra of the type A = T_{A_{0}}(A_{1})/I where I is generated by elements of degree n. It happens that this complex is a minimal resolution of A_{0} if and only if the algebra is a N-koszul algebra. We end this summary by describing this complex. We point out that some of the results which we got in the earlier sections can also be obtained by using this complex, in particular one can get a good description of the Ext algebra of A, by using it.

Let R = I ∩ (⊗_{A_{0}}^{n}A_{1}). Note that R is an A_{0}-A_{0}-submodule of ⊗_{A_{0}}^{n}A_{1}. We now assume K is a field and that A_{0} is not only semisimple, but, as a ring, A_{0} is K × K × ⋯ × K. Let T = T_{A_{0}}(A_{1}) and if x ∈ T, let x̄ denote π(x) where π : T → A is the canonical surjection. In this case, T is isomorphic to a path algebra KΓ for some quiver Γ. Let {v_{1}, …, v_{n}} be the arrows of Γ. Then the v_{i}’s are a full set of orthogonal idempotents. We say a nonzero element x ∈ T is left uniform if there exists a vertex v_{i} such that x = v_{i}x. If x is left uniform, we let o(x) = v_{i} if x = v_{i}x.

We define the generalized Koszul complex of R as follows. Let H_{0} = A_{0}, H_{1} = A_{1}, and, for k ≥ n,

H_{k} = ∩_{i+j+d=k}(⊗_{A_{0}}^{i}A_{1}) ⊗_{A_{0}} R ⊗_{A_{0}} (⊗_{A_{0}}^{j}A_{1}).

As usual, we let

δ(k) = \{ \begin{matrix}\frac{k}{2} n & if k is even \\ (\frac{k−1}{2}n) + 1 & if k is odd \end{matrix}

. We define Q^{k} = A ⊗_{A_{0}} H_{δ(k)} and note that Q^{k} is a projective left A-module for k ≥ 0.

We wish to define maps d^{k} : Q^{k} → Q^{k−1} for k ≥ 1. For this we need the following lemma which relates to the condition (ec) in Berger’s work. To simplify notation, we will denote ⊗_{A_{0}}^{i}A_{1} as simply A_{1}^{i} and write ⊗_{A_{0}} as simply ⊗.

Lemma 3.11 notation, if A is N-koszul then, for 2 ≤ i < n,

(R ⊗ A_{1}^{i}) ∩ (A_{1}^{i} ⊗ R) ⊆ A_{1}^{i−1} ⊗ R ⊗ A_{1}.

We define d^{m} : Q^{m} → Q^{m−1}. Recall that Q^{m} = A ⊗ H_{δ(m)}. From the definition and that R ⊂ A_{1}^{n}, we note that H_{δ(m)} ⊂ A_{1}^{δ(m)}. We write elements of H_{δ(m)} as x_{1} ⊗ ⋯ ⊗ x_{δ(m)} where the x_{i} are in A_{1}. If m = 2k, define

d^{m}(a ⊗ x_{1} ⊗ ⋯ ⊗ x_{kn}) = ax_{1}x_{2} ⋯ x_{n−1} ⊗ x_{n} ⊗ ⋯ ⊗ x_{kn}.

If m = 2k + 1, define

d^{m}(a ⊗ x_{1} ⊗ ⋯ ⊗ x_{kn+1}) = ax_{1} ⊗ x_{2} ⊗ ⊗ ⋯ ⊗ x_{kn+1}.

Using the description of H_{δ(k)} in the form of the corollary one shows that the maps are well-defined. We now can state one of Berger’s main results.

Theorem 3.12 [1, Thm 2.1] Let A = KΓ/I where I is an ideal generated in degree n. The following statements are equivalent.

(i) A is a N-koszul algebra.

(ii) {Q^{k}, d^{k}} is a minimal A-projective resolution of A_{0}.

In section 9 we provide a description of the Ext-algebra E(A) when A is a N-koszul algebra with n > 2 We need to introduce some notation in order to be able to state this description.

Recall that since A_{0} = ∏_{i=1}^{n} K, then the indecomposable A_{0}-A_{0}-bimodules are 1-dimen-sional over K and of the form e_{i}A_{0}e_{j}, where e_{i} = (0, …, 0, 1, 0, …, 0) with 1 occurring in the i^{th} component. Furthermore, A_{0}^{op} = A_{0} since A_{0} is a commutative ring. Since A_{0} ⊗_{K} A_{0} is a semisimple ring, it follows that every A_{0}-A_{0}-bimodule is a direct sum of copies of the 1-dimensional simple modules e_{i}A_{0}e_{j}, 1 ≤ i, j ≤ n.

Let V be a finitely generated A_{0}-A_{0}-bimodule. If W is an A_{0}-A_{0}-submodule of V, let W^{∗} = Hom_{A_{0}}(W, A_{0}) where the Hom is as left A_{0}-modules. The right A_{0}-module structure on W gives W^{∗} a left A_{0}-module structure. The right A_{0}-module structure on A_{0} gives W^{∗} an A_{0}-A_{0}-bimodule structure. Note that ^{∗} is a duality on A_{0}-A_{0}-bimodules and that if V is a finitely generated bimodule, then V^{∗∗} is naturally isomorphic to V as bimodules. Let W^{⊥} = {f ∈ V^{∗} | f(W) = 0}. We see that W^{⊥} is an A_{0}-A_{0}-bimodule if W is.

We identify A_{0} and A_{0}^{∗}. There is a natural isomorphism between (A_{1}^{i})^{∗} = (⊗^{i} A_{1})^{∗}

and ⊗^{i} A_{1}^{∗} = (A_{1}^{∗})^{i}, which we view as an identification. Let R^{⊥} = {f ∈ (A_{1}^{∗})^{n} | f(x) = 0 for all x ∈ R}. Let T^{∗} be the tensor algebra T_{A_{0}}(A_{1}^{∗}) = A_{0} ⊕ A_{1}^{∗} ⊕ (A_{1}^{∗})^{2} ⊕ · · ·. The dual algebra of A is defined to be A^{!} = T^{∗}/ < R^{⊥} >.

We see that A^{!} is a graded algebra since R^{⊥} is contained in (A_{1}^{∗})^{n}. Thus A^{!} = A_{0}^{!} ⊕ A_{1}^{!} ⊕ A_{2}^{!} ⊕ · · ·. Let B = B_{0} ⊕ B_{1} ⊕ B_{2} ⊕ ⋯ where B_{k} = A_{δ(k)}^{!} as vector spaces.

We main result in this section is the following:

Theorem 3.13 If A is a N-koszul algebra and n ≥ 2 then E(A) is isomorphic to B as graded algebras. In particular, Ext_{A}^{k}(A_{0}, A_{0}) is isomorphic to A_{δ(k)}^{!}.

In the final section we give some examples of N-koszul algebras with n > 2, in particular we characterize the monomial algebras which are N-koszul, this was done in [1], we show that his conclusions carry over quotient of quiver algebras.

References

[1] R. Berger Koszulity of nonquadratic algebras, J. of Algebra, 239, (2001), 705-734.

[2] R. Berger, M. Dubois-Violette and M. Warmbst Homogeneous Algebras To appear Journal of Algebra.( Available at math. QA).

[3] A.A. Beilinson, V. Ginsburg, and W. Soergel, Koszul duality patterns in representation theory, J. Am. Math. Soc. 9 (1996), 473-527.

[4] E.L. Green, D. Happel, D. Zacharia, Projective resolutions over Artin algebras with zero relations, Illinois J. Math. 29 (1985) 180-190.

[5] E.L. Green, R. Martínez-Villa, Koszul and Yoneda algebras Representation theory of algebras (Cocoyoc, 1994), CMS Conf. Proc., 18, Amer. Math. Soc., Providence, RI, (1996) 247-297.

[6] E.L. Green, R. Martínez-Villa, Koszul and Yoneda algebras II Algebras and modules, II (Geiranger, 1996), CMS Conf. Proc., 24, Amer. Math. Soc., Providence, RI, (1998) (227–244).

[7] E.L. Green, Ø. Solberg, D. Zacharia, Minimal projective resolutons, Trans. Am. Math. Soc., 353, (2001) 2915-2939.

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