Liana M. Şega

Free resolutions over short local rings

Let R be a local ring with maximal ideal m admitting a non-zero element a ∈ m for which the ideal (0 : a) is isomorphic to R/aR. We study minimal free resolutions of finitely generated R-modules M , with particular attention to the case when m = 0. Let e denote the minimal number of generators of m. If R is Gorenstein with m = 0 and e ≥ 3, we show that P M (t) is rational with denominator HR(−t) = 1 − et + et 2 − t, for each finitely generated R-module M . In particular, this conclusion applies to generic Gorenstein algebras of socle degree 3.

Vanishing of cohomology over Gorenstein rings of small codimension

We prove that if M, N are finite modules over a Gorenstein local ring R of codimension at most 4, then the vanishing of Ext n R(M, N) for n >> 0 is equivalent to the vanishing of Ext n R(N, M) for n >> 0. Furthermore, if R has no embedded deformation, then such vanishing occurs if and only if M or N has finite projective dimension.

Asymmetric complete resolutions and vanishing of ext overGorenstein rings

We construct a class of Gorenstein local rings

Short Koszul modules

R

Poincaré series of compressed local Artinian rings with odd top socle degree

which admit minimal complete

Cohomology of finite modules over short Gorenstein rings

R

Nonvanishing cohomology and classes of Gorenstein rings

-free resolutions

Independence of the total reflexivity conditions for modules

\bd C

Free resolutions over short Gorenstein local rings

such that the sequence

Extensions of a dualizing complex by its ring: commutative versions of a conjecture of Tachikawa

\{\rank_R C_i\}