16 research outputs found
Strong Artin-Rees Property in Rings of dimension one and two
In this note we give a simple proof of the fact that local rings of dimension
one have the strong uniform Artin-Rees property. Moreover, we give two examples
of rings of dimension two where the property fails.Comment: 7 page
Constructing Big Indecomposable modules
Let be local Noetherian ring of depth at least two. We prove that there
are indecomposable -modules which are free on the punctured spectrum of
constant, arbitrarily large, rank.Comment: 8 pages, Minor modifications
Uniform Artin-Rees Bounds for Syzygies
Let be a local Noetherian ring, let be a finitely generated
-module and let be a free resolution of
. We find a uniform bound such that the Artin-Rees containment holds for
all integers , for all integers , and for all ideals of
. In fact, we show that a considerably stronger statement holds. The uniform
bound holds for all ideals and all resolutions of th syzygy modules. In
order to prove our statements, we introduce the concept of Koszul annihilating
sequences.Comment: 14 page
Finite Gorenstein representation type implies simple singularity
Let R be a commutative noetherian local ring and consider the set of
isomorphism classes of indecomposable totally reflexive R-modules. We prove
that if this set is finite, then either it has exactly one element, represented
by the rank 1 free module, or R is Gorenstein and an isolated singularity (if R
is complete, then it is even a simple hypersurface singularity). The crux of
our proof is to argue that if the residue field has a totally reflexive cover,
then R is Gorenstein or every totally reflexive R-module is free.Comment: Final version, to appear in Adv. Math. 14 p
Brauer-Thrall for totally reflexive modules
Let R be a commutative noetherian local ring that is not Gorenstein. It is
known that the category of totally reflexive modules over R is representation
infinite, provided that it contains a non-free module. The main goal of this
paper is to understand how complex the category of totally reflexive modules
can be in this situation.
Local rings (R,m) with m^3=0 are commonly regarded as the structurally
simplest rings to admit diverse categorical and homological characteristics.
For such rings we obtain conclusive results about the category of totally
reflexive modules, modeled on the Brauer-Thrall conjectures. Starting from a
non-free cyclic totally reflexive module, we construct a family of
indecomposable totally reflexive R-modules that contains, for every n in N, a
module that is minimally generated by n elements. Moreover, if the residue
field R/m is algebraically closed, then we construct for every n in N an
infinite family of indecomposable and pairwise non-isomorphic totally reflexive
R-modules, that are all minimally generated by n elements. The modules in both
families have periodic minimal free resolutions of period at most 2.Comment: Final version; 34 pp. To appear in J. Algebr
Exact pairs of homogeneous zero divisors
Let S be a standard graded Artinian algebra over a field k. We identify
constraints on the Hilbert function of S which are imposed by the hypothesis
that S contains an exact pair of homogeneous zero divisors. As a consequence,
we prove that if S is a compressed level algebra, then S does not contain any
homogeneous zero divisors
A Construction of Totally Reflexive Modules
We construct infinite families of pairwise non-isomorphic indecomposable totally reflexive modules of high multiplicity. Under suitable conditions on the totally reflexive modules M and N, we find infinitely many non-isomorphic indecomposable modules arising as extensions of M by N. The construction uses the bimodule structure of Ext1R((M,N) over the endomorphism rings of N and M. Our results compare with a recent theorem of Celikbas, Gheibi and Takahashi, and broaden the scope of that theorem