Koszul and Gorenstein Properties for Homogeneous Algebras

By R. Berger, N. Marconnet. 2003

92 citations recorded by the source corpus.

The Koszul property was generalized to homogeneous algebras of degree $$N>2$$ in [5], and related to $$N$$-complexes. We show that if the $$N$$-homogeneous algebra $$A$$ is generalized Koszul, AS-Gorenstein and of finite global dimension, then one can apply the Van den Bergh duality theorem to $$A$$ i.e., there is a Poincaré duality between Hochschild homology and cohomology of $$A$$ as for $$N = 2$$.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5522c70810e2113da53f60e910455d06baf6ad3d

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Citations in the corpus

Superpotentials and higher order derivations (2008, methodology): “= 0 for all φ∈( P lW ∗⊗l⊗R⊥ ⊗W∗⊗k−l−2). This is the same as to say that w∈ \ l W⊗l⊗R⊗W⊗k−l−2 = \ l W⊗l⊗W 2 ⊗W ⊗k−l−2. We conclude immediately that Wk⊂(A! k) ∗. Remark 6.6. According to theorem 5.4 in [6], the pairing between the Wkcan be extended to a pairing between the (A! k) ∗, but this pairing i”
Generalized Operads and Their Inner Cohomomorphisms (2006, background): “ N. Formulas (2.22)–(2.24) still hold in the new setup. Unit object for • is now k[ε]/(εN). As a consequence, Koszul complexes become N–complexes leading to an interesting new homological eﬀects: see [BerM] and references therein. c) Homogeneous algebras generated in degree one. This case is treated”
Skew Calabi-Yau triangulated categories and Frobenius Ext-algebras (2014, methodology): “1 = Ext1 A(k,k), we have µE|E1 = (−1)d+1(µ A|A1) ∗. In particular, µ A= 1 (that is, Ais a Calabi-Yau algebra) if and only if E is a graded-symmetric Frobenius algebra. The theorem above was proved in [BM] in the N-Koszul case, but many regular algebras of dimension 4 and higher are not N-Koszul. In ”
Poincaré-Birkhoff-Witt deformations of Calabi-Yau algebras (2006, background): “3) in [6]) (x, y) = 〈x⊗ g,W4〉, (y, x) = 〈g ⊗ x,W4〉, where g is any linear form on V ⊗3 extending y : R→ k, and consequently x⊗ g and g⊗ x are linear forms on V ⊗4.”
Calabi-Yau algebras (2006, unknown): “iangles N∈Q} λ(N)·N ∈F cyc. (4.4.4) Thus, A CQ Γ,Φ Γ is a quadratic algebra which is, moreover, a CY algebra of dimension 3, by Theorem 4.4.6 below. It follows that A CQ Γ,Φ Γ is a Koszul algebra, by [BM], [BT]. Remark 4.4.5. (i) The corresondence between edges of the quiver Q Γ and intertwiners xij”