19 research outputs found
Nonvanishing cohomology and classes of Gorenstein rings
We give counterexamples to the following conjecture of Auslander: given a
finitely generated module over an Artin algebra , there exists a
positive integer such that for all finitely generated -modules
, if \Ext_{\Lambda}^i(M,N)=0 for all , then
\Ext_{\Lambda}^i(M,N)=0 for all . Some of our examples moreover
yield homologically defined classes of commutative local rings strictly between
the class of local complete intersections and the class of local Gorenstein
rings.Comment: 16 page
Free resolutions over short local rings
The structure of minimal free resolutions of finite modules M over
commutative local rings (R,m,k) with m^3=0 and rank_k(m^2) < rank_k(m/m^2)is
studied. It is proved that over generic R every M has a Koszul syzygy module.
Explicit families of Koszul modules are identified. When R is Gorenstein the
non-Koszul modules are classified. Structure theorems are established for the
graded k-algebra Ext_R(k,k) and its graded module Ext_R(M,k).Comment: 17 pages; number of minor changes. This article will appear in the
Journal of the London Math. So
The Scarf complex and betti numbers of powers of extremal ideals
This paper is concerned with finding bounds on betti numbers and describing
combinatorially and topologically (minimal) free resolutions of powers of
ideals generated by a fixed number of square-free monomials. Among such
ideals, we focus on a specific ideal , which we call {\it
extremal}, and which has the property that for each the betti numbers
of are an upper bound for the betti numbers of for
any ideal generated by square-free monomials (in any number of
variables). We study the Scarf complex of the ideals and
use this simplicial complex to extract information on minimal free resolutions.
In particular, we show that has a minimal free resolution
supported on its Scarf complex when or when , and we
describe explicitly this complex. For any and , we also show that
is the smallest possible, or in other words equal
to the number of edges in the Scarf complex. These results lead to effective
bounds on the betti numbers of , with as above. For example, we obtain
that pd for all ideals generated by square-free monomials
and any