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

الأصل · EN

A Gorenstein A-algebra R of codimension 2 is a perfect finite A-algebra such that R=Ext²(R,A) holds as R-modules, A being a Cohen-Macaulay local ring with dim(A)-dimₐ(R)=2. I prove a structure theorem for these algebras improving on an old theorem of M. Grassi. Special attention is paid to the question how the ring structure of R is encoded in its Hilbert resolution. It is shown that R is automatically a ring once one imposes a weak depth condition on a determinantal ideal derived from a presentation matrix of R over A. The interplay of Gorenstein algebras and Koszul modules as introduced by M. Grassi is clarified. Questions of applicability to canonical surfaces in P⁴ have served as a guideline in these investigations.