342 research outputs found
Liaison classes of modules
We propose a concept of module liaison that extends Gorenstein liaison of
ideals and provides an equivalence relation among unmixed modules over a
commutative Gorenstein ring. Analyzing the resulting equivalence classes we
show that several results known for Gorenstein liaison are still true in the
more general case of module liaison. In particular, we construct two maps from
the set of even liaison classes of modules of fixed codimension into stable
equivalence classes of certain reflexive modules. As a consequence, we show
that the intermediate cohomology modules and properties like being perfect,
Cohen-Macaulay, Buchsbaum, or surjective-Buchsbaum are preserved in even module
liaison classes. Furthermore, we prove that the module liaison class of a
complete intersection of codimension one consists of precisely all perfect
modules of codimension one
Gorenstein algebras presented by quadrics
We establish restrictions on the Hilbert function of standard graded
Gorenstein algebras with only quadratic relations. Furthermore, we pose some
intriguing conjectures and provide evidence for them by proving them in some
cases using a number of different techniques, including liaison theory and
generic initial ideals
Glicci simplicial complexes
One of the main open questions in liaison theory is whether every homogeneous
Cohen-Macaulay ideal in a polynomial ring is glicci, i.e. if it is in the
G-liaison class of a complete intersection. We give an affirmative answer to
this question for Stanley-Reisner ideals defined by simplicial complexes that
are weakly vertex-decomposable. This class of complexes includes matroid,
shifted and Gorenstein complexes respectively. Moreover, we construct a
simplicial complex which shows that the property of being glicci depends on the
characteristic of the base field. As an application of our methods we establish
new evidence for two conjectures of Stanley on partitionable complexes and on
Stanley decompositions
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