September 23, 2004
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March 13, 2021
The Koszul homology algebra of a commutative local (or graded) ring $R$ tends to reflect important information about the ring $R$ and its properties. In fact, certain classes of rings are characterized by the algebra structure on their Koszul homologies. In this paper we survey some classical results on the Koszul homology algebras of such rings and highlight some applications. We report on recent progress on the Koszul homology algebras of Koszul algebras and examine some op...
October 15, 2015
This text consists of five relatively systematic notes on Gr\"obner bases and free resolutions of modules over solvable polynomial algebras.
February 18, 2019
Differential graded algebra techniques have played a crucial role in the development of homological algebra, especially in the study of homological properties of commutative rings carried out by Serre, Tate, Gulliksen, Avramov, and others. In this article, we extend the construction of the Koszul complex and acyclic closure to a more general setting. As an application of our constructions, we shine some light on the structure of the Ext algebra of quotients of skew polynomial...
December 6, 2016
We investigate the standard generalized Gorenstein algebras of homological dimension three, giving a structure theorem for their resolutions. Moreover in many cases we are able to give a complete description of their graded Betti numbers.
December 11, 2022
We prove that in the polynomial ring $Q=\mathsf{k}[x,y,z,w]$, with $\mathsf{k}$ an algebraically closed field of characteristic zero, all Gorenstein homogeneous ideals $I$ such that $(x,y,z,w)^4\subseteq I \subseteq (x,y,z,w)^2$ can be obtained by \emph{doubling} from a grade three perfect ideal $J\subset I$ such that $Q/J$ is a locally Gorenstein ring. Moreover, a graded minimal free resolution of the $Q$-module $Q/I$ can be completely described in terms of a graded minimal ...
January 9, 2009
Let $A$ be a semigroup whose only invertible element is 0. For an $A$-homogeneous ideal we discuss the notions of simple $i$-syzygies and simple minimal free resolutions of $R/I$. When $I$ is a lattice ideal, the simple 0-syzygies of $R/I$ are the binomials in $I$. We show that for an appropriate choice of bases every $A$-homogeneous minimal free resolution of $R/I$ is simple. We introduce the gcd-complex $D_{gcd}(\bf b)$ for a degree $\mathbf{b}\in \A$. We show that the homo...
September 21, 2022
The question we address in this paper is: which monomial ideals have minimal cellular resolutions, that is, minimal resolutions obtained from homogenizing the chain maps of CW-complexes? Velasco gave families of examples of monomial ideals that do not have minimal cellular resolutions, but those examples have large minimal generating sets. In this paper, we show that if a monomial ideal has at most four generators, then the ideal and its (monomial) Artinian reductions have mi...
May 12, 2006
The Koszul homology of modules of the polynomial ring $R$ is a central object in commutative algebra.It is strongly related with the minimal free resolution of these modules, and thus with regularity, Hilbert functions, etc. Here we consider the case of modules of the form $R/I$ where $I$ is a monomial ideal. So far, some good algorithms have been given in the literature and implemented in different Computer Algebra Systems (e.g. CoCoa, Singular, Macaulay), which compute mini...
November 3, 2020
Differential graded (DG) algebras are powerful tools from rational homotopy theory. We survey some recent applications of these in the realm of homological commutative algebra.
May 24, 2019
The minimal free resolution of the coordinate ring of a complete intersection in projective space is a Koszul complex on a regular sequence. In the product of projective spaces $\mathbb{P}^1 \times \mathbb{P}^1$, we investigate which sets of points have a virtual resolution that is a Koszul complex on a regular sequence. This paper provides conditions on sets of points; some of which guarantee the points have this property, and some of which guarantee the points do not have t...