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Koszul calculus
(Cambridge University Press, 2018-05)
We present a calculus that is well-adapted to homogeneous quadratic algebras. We define this calculus on Koszul cohomology - resp. homology - by cup products - resp. cap products. The Koszul homology and cohomology are ...
On a definition of multi-Koszul algebras
(Academic Press Inc Elsevier Science, 2013-02)
In this article we introduce the notion of multi-Koszul algebra for the case of a nonnegatively graded connected algebra with a finite number of generators of degree 1 and with a finite number of relations, as a generalization ...
PBW-deformations and deformations à la Gerstenhaber of N-Koszul algebras
(European Mathematical Society, 2014-07)
In this article we establish an explicit link between the classical theory of deformations à la Gerstenhaber (and a fortiori with the Hochschild cohomology) and (weak) PBW-deformations of homogeneous algebras. Our point ...
A case study in bigraded commutative algebra
(Chapman and Hall, 2007)
We study the commutative algebra of three bihomogeneous polynomials p_0,p_1,p_2 of degree (2,1) in variables x,y;z,w, assuming that they never vanish simultaneously on P^1 x P^1. Unlike the situation for P^2, the Koszul ...
Generating degrees for graded projective resolutions
(World Scientific, 2018-10)
We provide a framework connecting several well-known theories related to the linearity of graded modules over graded algebras. In the first part, we pay a particular attention to the tensor products of graded bimodules ...
Hochschild homology and cohomology of down–up algebras
(Academic Press Inc Elsevier Science, 2018-03)
We present a detailed computation of the cyclic and the Hochschild homology and cohomology of generic and 3-Calabi–Yau homogeneous down–up algebras. This family was defined by Benkart and Roby in [3] in their study of ...
Solving a sparse system using linear algebra
(Elsevier, 2015-04)
We give a new theoretical tool to solve sparse systems with finitely many solutions. It is based on toric varieties and basic linear algebra; eigenvalues, eigenvectors and coefficient matrices. We adapt Eigenvalue theorem ...
The simplest minimal free resolutions in P1×P1
(Springer, 2021)
We study the minimal bigraded free resolution of an ideal with three generators of the same bidegree, contained in the bihomogeneous maximal ideal 〈s, t〉∩〈u, v〉 of the bigraded ring K[s,t;u,v]. Our analysis involves tools ...