10,085 research outputs found
Simplicial presheaves of coalgebras
The category of simplicial R-coalgebras over a presheaf of commutative unital
rings on a small Grothendieck site is endowed with a left proper, simplicial,
cofibrantly generated model category structure where the weak equivalences are
the local weak equivalences of the underlying simplicial presheaves. This model
category is naturally linked to the R-local homotopy theory of simplicial
presheaves and the homotopy theory of simplicial R-modules by Quillen
adjunctions. We study the comparison with the R-local homotopy category of
simplicial presheaves in the special case where R is a presheaf of
algebraically closed (or perfect) fields. If R is a presheaf of algebraically
closed fields, we show that the R-local homotopy category of simplicial
presheaves embeds fully faithfully in the homotopy category of simplicial
R-coalgebras.Comment: 24 page
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
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