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Gorenstein Algebras in Codimension 2 and Koszul Modules
Abstract
The notion of Koszul module introduced in Grassi (19969.
Grassi , M. ( 1996 ). Koszul modules and Gorenstein algebras . J. Alg. 180 : 918 – 953 . View all references) is a generalization of the concept of complete intersection ring to arbitrary finite modules over some base ring A (cf. Def. 1.3). A Gorenstein A-algebra R of codimension 2 is a perfect finite A-algebra such that holds as R-modules, where A is a Cohen–Macaulay local ring with dim A − dim A R = 2.Here we prove (Thm. 1.4) that every Gorenstein algebra R of codimension 2 has a resolution over A (assuming 2 is invertible in A) that is simultaneously Gorenstein-symmetric and of Koszul module type; this is a generalization to algebras of higher rank of the easy fact that, if R is a quotient of A, it has a resolution over A given by a length two Koszul complex.
... Also, Grassi defines Koszul Modules [9] which are quasi-Gorenstein modules with certain types of free resolutions. Furthermore, a Gorenstein algebra is a quasi-Gorenstein module and there is a collection of work about such algebras [10,11,12,13]. ...
Let R be a commutative ring. A quasi-Gorenstein R-module is an R-module such that the grade of the module and the projective dimension of the module are equal and the canonical module of the module is isomorphic to the module itself. After discussing properties of finitely generated quasi-Gorenstein modules, it is shown that this definition allows for a characterization of diagonal matrices of maximal rank over a Noetherian factorial domain R extending a theorem of Frobenius and Stickelberger to modules of projective dimension 1 over a commutative Noetherian factorial domain.
In [Böh] a structure theorem for Gorenstein algebras in codimension 2 was obtained. In the first part of this article we give a geometric application and prove a structure theorem for good birational canonical projections of regular surfaces of general type with pg = 5 to P 4 (theorem 1.6). In the second part we show how this can be used to analyze the moduli space of canonical surfaces with q = 0, pg = 5 and K 2 = 11 (theorem 2.4).
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.
We show that a Gorenstein subcanonical codimension 3 subscheme Z in X = P^N, N > 3, can be realized as the locus along which two Lagrangian subbundles of a twisted orthogonal bundle meet degenerately, and conversely. We extend this result to singular Z and all quasiprojective ambient schemes X under the necessary hypothesis that Z is strongly subcanonical in a sense defined below. A central point is that a pair of Lagrangian subbundles can be transformed locally into an alternating map. In the local case our structure theorem reduces to that of Buchsbaum-Eisenbud and says that Z is Pfaffian. We also prove codimension one symmetric and skew-symmetric analogues of our structure theorems.
A good canonical projection of a surface S of general type is a morphism to the 3-dimensional projective space P^3 given by 4 sections of the canonical line bundle. To such a projection one associates the direct image sheaf F of the structure sheaf of S, which is shown to admit a certain length 1 symmetrical locally free resolution. The structure theorem gives necessary and sufficient conditions for such a length 1 locally free symmetrical resolution of a sheaf F on P^3 in order that yield a canonically projected surface of general type. The result was found in 1983 by the first author in the regular case, and the main ingredient here for the irregular (= non Cohen-Macaulay) case is to replace the use of Hilbert resolutions with the use of Beilinson's monads. Most of the paper is devoted then to the analysis of irregular canonical surfaces with p_g=4 and low degree (the analysis in the regular case had been initiated by F. Enriques). For q=1 we classify completely the (irreducible) moduli space for the minimal value K^2=12. We also study other moduli spaces, q=2, K^2=18, and q=3, K^2=12. The last family is provided by the divisors in a (1,1,2) polarization on some Abelian 3-fold. A remarkable subfamily is given by the pull backs, under a 2-isogeny, of the theta divisor of a p.p.Abelian 3-fold: for those surfaces the canonical map is a double cover of a canonical surface (i.e., such that its canonical map is birational). The canonical image is a sextic surface with a plane cubic as double curve, and moreover with an even set of 32 nodes as remaining singular locus. Our method allows to write the equations of these transcendental objects, but not yet to construct new ones by computer algebra.
In the last two decades Cohen-Macaulay rings and modules have been central topics in commutative algebra. This book meets the need for a thorough, self-contained introduction to the homological and combinatorial aspects of the theory of Cohen-Macaulay rings, Gorenstein rings, local cohomology, and canonical modules. A separate chapter is devoted to Hilbert functions (including Macaulay's theorem) and numerical invariants derived from them. The authors emphasize the study of explicit, specific rings, making the presentation as concrete as possible. So the general theory is applied to Stanley-Reisner rings, semigroup rings, determinantal rings, and rings of invariants. Their connections with combinatorics are highlighted, e.g. Stanley's upper bound theorem or Ehrhart's reciprocity law for rational polytopes. The final chapters are devoted to Hochster's theorem on big Cohen-Macaulay modules and its applications, including Peskine-Szpiro's intersection theorem, the Evans-Griffith syzygy theorem, bounds for Bass numbers, and tight closure. Throughout each chapter the authors have supplied many examples and exercises which, combined with the expository style, will make the book very useful for graduate courses in algebra. As the only modern, broad account of the subject it will be essential reading for researchers in commutative algebra.
I first define Koszul modules, which are a generalization to arbitrary rank of complete intersections. After a study of some of their properties, it is proved that Gorenstein algebras of codimension one or two over a local or graded CM ring are Koszul modules, thus generalizing a well known statement for rank one modules. The general techniques used to describe Koszul modules are then used to obtain a structure theorem for Gorenstein algebras in codimension one and two, over a local or graded CM ring.
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 P4) have served as a guideline in these commutative algebra investigations.
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