Fe b 20 04 L . Szpiro ’ s conjecture on Gorenstein algebras in codimension 2
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چکیده
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 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. Furthermore, the interplay of Gorenstein algebras and Koszul modules as introduced by M. Grassi is clarified. I include graded analogues of the afore-mentioned results when possible. Questions of applicability to the theory of surfaces of general type (namely, canonical surfaces in P) have served as a guideline in these commutative algebra investigations. 0 Introduction and statement of results Motivation for investigating the types of questions treated in the present article sprang from two different, though closely related sources, the first algebro-geometric, the second one purely algebraic in spirit. Though in this paper I will restrict myself to touching upon the latter only, both themes can, I think, be best understood in conjunction, so I will briefly discuss them jointly in the introduction. From the point of view of algebraic geometry, the perhaps earliest traces of the story may be located in the book [En] by F. Enriques where he treats the structure of canonical surfaces in P4 with q = 0, pg = 5, K 2 = 8 and 9 (cf. loc. cit., p. 284ff. ; they are the complete intersections of type (2, 4) and (3, 3); the word “canonical” means that for those surfaces the 1−canonical
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تاریخ انتشار 2004