Koszul and Gorenstein properties for homogeneous algebras. (English) Zbl 1125.16017
This paper studies various homological properties, such as Koszulity, AS-Gorenstein-ness and Poincaré duality for homogeneous algebras.
We begin by recapitulating three relevant definitions from the paper: A graded \(k\)-algebra \(A\) is ‘\(N\)-homogeneous’ (\(N\geqslant 2\)) if \(A=\text{Tens}(V)/I(R)\) where \(\text{Tens}(V)\) is the tensor algebra of a finite dimensional \(k\)-vector space \(V\), and \(I(R)\) is the two-sided ideal generated by a subspace \(R\subseteq V^{\otimes N}\). The first author [in J. Algebra 239, No. 2, 705-734 (2001; Zbl 1035.16023)] defined a notion of ‘(generalized) Koszulity’ for \(N\)-homogeneous algebras, which is also the definition used in this paper. Finally, if \(A\) is connected of finite global dimension \(D\), it is called ‘AS-Gorenstein’ provided that \(\text{Ext}^i_A(k,A)=0\) for \(i\neq D\) and \(\text{Ext}^D_A(k,A)\cong k\).
Sections 2, 3, and 4 are devoted to a satisfying discussion of various equivalent characterizations of Koszulity via (bimodule) Koszul \(N\)-complexes; and computations of some Yoneda products relevant for Section 5.
In Section 5 the authors give a criterion for AS-Gorenstein-ness which they use to prove that an \(N\)-homogeneous, generalized Koszul algebra \(A\) of finite global dimension is AS-Gorenstein if and only if its Yoneda algebra \(E(A)\) is Frobenius.
The final Section 6 deals with Poincaré duality, and the main result is a generalization of a theorem due to M. Van den Bergh [Proc. Am. Math. Soc. 126, No. 5, 1345-1348 (1998); erratum ibid. 130, No. 9, 2809-2810 (2002; Zbl 0894.16005)]: An \(N\)-homogeneous, generalized Koszul and AS-Gorenstein algebra \(A\) of finite global dimension \(D\) has Poincaré duality: \(\text{HH}^i(A,M)\cong\text{HH}_{D-i}(A,{_{\varepsilon^{D+1}\varphi}M})\). Here \(\text{HH}_*\) (\(\text{HH}^*\)) denotes Hochschild (co)homology, \(M\) is any \(A\)-\(A\)-bimodule, and \(\varepsilon,\varphi\) are appropriate automorphisms of \(A\).
We begin by recapitulating three relevant definitions from the paper: A graded \(k\)-algebra \(A\) is ‘\(N\)-homogeneous’ (\(N\geqslant 2\)) if \(A=\text{Tens}(V)/I(R)\) where \(\text{Tens}(V)\) is the tensor algebra of a finite dimensional \(k\)-vector space \(V\), and \(I(R)\) is the two-sided ideal generated by a subspace \(R\subseteq V^{\otimes N}\). The first author [in J. Algebra 239, No. 2, 705-734 (2001; Zbl 1035.16023)] defined a notion of ‘(generalized) Koszulity’ for \(N\)-homogeneous algebras, which is also the definition used in this paper. Finally, if \(A\) is connected of finite global dimension \(D\), it is called ‘AS-Gorenstein’ provided that \(\text{Ext}^i_A(k,A)=0\) for \(i\neq D\) and \(\text{Ext}^D_A(k,A)\cong k\).
Sections 2, 3, and 4 are devoted to a satisfying discussion of various equivalent characterizations of Koszulity via (bimodule) Koszul \(N\)-complexes; and computations of some Yoneda products relevant for Section 5.
In Section 5 the authors give a criterion for AS-Gorenstein-ness which they use to prove that an \(N\)-homogeneous, generalized Koszul algebra \(A\) of finite global dimension is AS-Gorenstein if and only if its Yoneda algebra \(E(A)\) is Frobenius.
The final Section 6 deals with Poincaré duality, and the main result is a generalization of a theorem due to M. Van den Bergh [Proc. Am. Math. Soc. 126, No. 5, 1345-1348 (1998); erratum ibid. 130, No. 9, 2809-2810 (2002; Zbl 0894.16005)]: An \(N\)-homogeneous, generalized Koszul and AS-Gorenstein algebra \(A\) of finite global dimension \(D\) has Poincaré duality: \(\text{HH}^i(A,M)\cong\text{HH}_{D-i}(A,{_{\varepsilon^{D+1}\varphi}M})\). Here \(\text{HH}_*\) (\(\text{HH}^*\)) denotes Hochschild (co)homology, \(M\) is any \(A\)-\(A\)-bimodule, and \(\varepsilon,\varphi\) are appropriate automorphisms of \(A\).
Reviewer: Henrik Holm (Copenhagen)
MSC:
| 16S37 | Quadratic and Koszul algebras |
| 16E40 | (Co)homology of rings and associative algebras (e.g., Hochschild, cyclic, dihedral, etc.) |
| 16E65 | Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) |
| 16S38 | Rings arising from noncommutative algebraic geometry |
Keywords:
Koszul algebras; Gorenstein algebras; complexes; Hochschild homology; Hochschild cohomologyReferences:
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