L. Szpiroʹs conjecture on Gorenstein algebras in codimension 2

A Gorenstein A-algebra R of codimension 2 is a perfect finite A-algebra such that holds as R-modules, A being a Cohen–Macaulay local ring with dimA−dimAR=2. The aim of this article is to prove a structure theorem for these algebras improving on an old theorem of M. Grassi [Koszul modules and Gorenstein algebras, J. Algebra 180 (1996) 918–953]. 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 very weak depth condition on a determinantal ideal derived from a presentation matrix of R over A. Graded analogues of the aforementioned results are also included. Questions of applicability to the theory of surfaces of general type (namely, canonical surfaces in ) have served as a guideline in these commutative algebra investigations.