L. Szpiro's conjecture on Gorenstein algebras in codimension 2

Fe b 20 04 L . Szpiro ’ s conjecture on Gorenstein algebras in codimension 2

A Gorenstein A−algebra R of codimension 2 is a perfect finite A−algebra such that R ∼= ExtA(R,A) holds as R−modules, A being a Cohen-Macaulay local ring with dimA− dimA R = 2. I prove a structure theorem for these algebras improving on an old theorem of M. Grassi [Gra]. Special attention is paid to the question how the ring structure of R is encoded in its Hilbert resolution. It is shown that R...

An Improved Multiplicity Conjecture for Codimension Three Gorenstein Algebras

The Multiplicity Conjecture is a deep problem relating the multiplicity (or degree) of a Cohen-Macaulay standard graded algebra with certain extremal graded Betti numbers in its minimal free resolution. In the case of level algebras of codimension three, Zanello has proposed a stronger conjecture. We prove this conjecture for the case of codimension three graded Gorenstein algebras.

Canonical surfaces in P^4 and Gorenstein algebras in codimension 2

On the Weak Lefschetz Property for Artinian Gorenstein Algebras of Codimension Three

We study the problem of whether an arbitrary codimension three graded artinian Gorenstein algebra has the Weak Lefschetz Property. We reduce this problem to checking whether it holds for all compressed Gorenstein algebras of odd socle degree. In the first open case, namely Hilbert function (1, 3, 6, 6, 3, 1), we give a complete answer in every characteristic by translating the problem to one of...

On Cm-finite Gorenstein Artin Algebras

An artin algebra A is said to be CM-finite if there are only finitely many, up to isomorphisms, indecomposable finitely-generated Gorenstein-projective A-modules. We prove that for a Gorenstein artin algebra, it is CM-finite if and only if every its Gorenstein-projective module is a direct sum of finitely-generated Gorenstein-projective modules.