arXiv subjects

Oana Veliche

Publications and source records attributed to Oana Veliche.

22 records · Page 2Linked to original sources

arXiv2007

A test complex for Gorensteinness

Let $R$ be a commutative noetherian ring with a dualizing complex. By recent work of Iyengar and Krause, the difference between the category of acyclic complexes and its subcategory of totally acyclic complexes measures how far $R$ is from being Gorenstein. In particular, $R$ is Gorenstein if and only if every acyclic complex is totally acyclic. In this note we exhibit a specific acyclic complex with the property that it is totally acyclic if and only if $R$ is Gorenstein.

math.AC↗

arXiv2007

Stable cohomology over local rings

The focus of this paper is on a poorly understood invariant of a commutative noetherian local ring $R$ with residue field $k$: the stable cohomology modules $\hat{Ext}^{n}_R(k,k)$, defined for each $n\in\mathbb{Z}$ by Benson and Carlson, Mislin, and Vogel; it coincides with Tate cohomology when $R$ is Gorenstein. It is proved that important properties of $R$, such as being regular, complete intersection, or Gorenstein, are detected by the $k$-rank of $\hat{Ext}^{n}_R(k,k)$ for an arbitrary $n\in\mathbb{Z}$. Such numerical characterizations are made possible by results on the structure of $\mathbb{Z}$-graded $k$-algebra carried by $\hat{Ext}^{n}_R(k,k)$. It is proved that in many cases this algebra is determined by the absolute cohomology algebra through a canonical homomorphism ${Ext}^{n}_R(k,k)\to\hat{Ext}^{n}_R(k,k)$.

math.AC↗

arXiv2006

Acyclicity over local rings with radical cube zero

This paper studies infinite acyclic complexes of finitely generated free modules over a commutative noetherian local ring $(R,m)$ with $m^3=0$. Conclusive results are obtained on the growth of the ranks of the modules in acyclic complexes, and new sufficient conditions are given for total acyclicity. Results are also obtained on the structure of rings that admit acyclic complexes; part of this structure is exhibited by every ring $R$ that admits a non-free finitely generated non-zero module $M$ with $Ext^n(M,R)=0$ for a few $n>0$.

math.AC↗

arXiv2004

Gorenstein projective dimension for complexes

We define and study a notion of Gorenstein projective dimension for complexes of left modules over associative rings. For complexes of finite Gorenstein projective dimension we define and study a Tate cohomology theory. Tate cohomology groups have a natural transformation to classical Ext groups. In the case of module arguments, we show that these maps fit into a long exact sequence, where every third term is a relative cohomology group defined for left modules of finite Gorenstein projective dimension.

math.AC↗