Related papers: Koszul complexes and spectral sequences associated…

Let $R$ be a local commutative noetherian ring and $HKR$ the homology ring of the corresponding Koszul complex. We study the homological properties of $HKR$ in particular the so-called Avramov spectral sequence. When the embedding dimension…

We study the differential graded Lie algebra of endomorphisms of the Koszul resolution of a regular sequence on a unitary commutative $K$-algebra $R$ and we prove that it is homotopy abelian over $K$, while it is generally not formal over…

Let $n\ge 1$ and $A$ be a commutative algebra of the form $\boldsymbol k[x_1,x_2,\dots, x_n]/I$ where $\boldsymbol k$ is a field of characteristic $0$ and $I\subseteq \boldsymbol k[x_1,x_2,\dots, x_n]$ is an ideal. Assume that there is a…

It has been shown recently, in a joint work with Michel Dubois-Violette and Marc Wambst (see math.QA/0203035), that Koszul property of $N$-homogeneous algebras (as defined in the original paper) becomes natural in a $N$-complex setting. A…

Through the theory of Lie bi-algebroids and generalized complex structures, one could define a cohomology theory naturally associated to a holomorphic Poisson structure. It is known that it is the hypercohomology of a bi-complex such that…

This paper introduces the category of marked curved Lie algebras with curved morphisms, equipping it with a closed model category structure. This model structure is---when working over an algebraically closed field of characteristic…

The object of the paper is the dependence of Koszul complexes and dependence of dual Koszul complexes of two systems of non-homogeneous polynomials, when one system is a part of other system, in connection with the duality in a Koszul…

Let S=K[x_1,...,x_n] be a polynomial ring over a field K and I a homogeneous ideal in S generated by a regular sequence f_1,f_2,...,f_k of homogeneous forms of degree d. We study a generalization of a result of Conca, Herzog, Trung, and…

A holomorphic Poisson structure induces a deformation of the complex structure as Hitchin's generalized geometry. Its associated cohomology naturally appears as the limit of a spectral sequence of a double complex. The first sheet of this…

We classify nonabelian extensions of Lie algebroids in the holomorphic category. Moreover we study a spectral sequence associated to any such extension. This spectral sequence generalizes the Hochschild-Serre spectral sequence for Lie…

We study the degeneration relations on the varieties of associative and Lie algebra structures on a fixed finite dimensional vector space and give a description of them in terms of Gerstenhaber formal deformations. We use this result to…

Let $A$ be a Koszul (or more generally, $N$-Koszul) Calabi-Yau algebra. Inspired by the works of Kontsevich, Ginzburg and Van den Bergh, we show that there is a derived non-commutative Poisson structure on $A$, which induces a graded Lie…

We give an upper bound on the number of the page on which the spectral sequence corresponding to a locally free isometric action of an abelian Lie group degenerates. We give examples showing that these bounds are indeed sharp. Finally, we…

We study a spectral sequence approximating Lie algebroid cohomology associated to a Lie subalgebroid. This is a simultaneous generalisation of several classical constructions in differential geometry, including the Leray-Serre spectral…

This work concerns the Koszul complex $K$ of a commutative noetherian local ring $R$, with its natural structure as differential graded $R$-algebra. It is proved that under diverse conditions, involving the multiplicative structure of…

In this article we present a self-contained account of two important results in complex supergeometry: (1) Koszul's Splitting theorem and (2) Donagi and Witten's decomposition of the super Atiyah class. These results are related in the same…

Each connected graded, graded-commutative algebra $A$ of finite type over a field $\Bbbk$ of characteristic zero defines a complex of finitely generated, graded modules over a symmetric algebra, whose homology graded modules are called the…

We develop a Koszul-theoretic framework for comparing classical Alexander-type invariants with infinitesimal invariants arising from finite-type commutative differential graded algebra models. The central mechanism is Koszul linearization,…

Let $\check{C}_{\underline{x}}$ denote the \v{C}ech complex with respect to a system of elements $\underline{x} = x_1,\ldots,x_r$ of a commutative ring $R$. We construct a bounded complex $\mathcal{L}_{\underline{x}}$ of free $R$-modules…

We motivate and study the reduced Koszul map, relating the invariant bilinear maps on a Lie algebra and the third homology. We show that it is concentrated in degree 0 for any grading in a torsion-free abelian group, and in particular it…