×

Cohen-Macaulay modules on quadrics. With an appendix by Ragnar-Olaf Buchweitz: The comparison theorem (p. 96- 116). (English) Zbl 0633.13008

Singularities, representation of algebras, and vector bundles, Proc. Symp., Lambrecht/Pfalz/FRG 1985, Lect. Notes Math. 1273, 58-95; 96-116 (1987).
[For the entire collection see Zbl 0619.00007.]
From the abstract: “This paper analyzes the graded \(MCM=\max imal\) Cohen-Macaulay) modules over rings of the form \(R=k[x_ 1,...,x_ r]/Q\), when Q is a quadratic form defining a regular projective hypersurface and k is an arbitrary field.”
Let Q be a quadratic form on a vector space V of dimension r over a field k. Q is considered as an element of \(S_ 2(V^*)\subset k[V^*]=S\). Let \(R=S/Q\) and \({\mathfrak m}\) the maximal ideal generated by \(V^*\). A graded R-module M is said to be linear if it is an MCM module admitting a graded free presentation of the form \(P^ n(-1)\to^{f}R^ m\to M\to 0\) with f a matrix of linear forms. The main result is stated as follows: The functor \(M\to M/{\mathfrak m}M\) is an equivalence of categories from the category of linear R-modules without free summands and maps of degrees zero to the category of modules over \(C_ 0(Q)\), the even Clifford algebra of Q. Under this equivalence, modules which correspond to the first syzygy module and the dual module are given. If Q is regular, then every MCM \(R_{{\mathfrak m}}\)- (or \(\hat R_{{\mathfrak m}}\)-)module is the localization (or completion) of a linear module. R has either 1 or 2 isomorphism classes of nonfree indecomposable MCM modules. It is described by properties of Q which case occurs. The rank of the sum of the distinct indecomposable nonfree MCM modules is given.
The appendix by the first author is to categorically analyze the equivalence stated above. The main result is the existence of a natural exact equivalence between \(D^ b(R)\) and \(D^ b(E)\), where \(D^ b(\quad)\) denotes the derived category of complexes with bounded cohomology of graded, finitely generated modules and \(E=Ext_ R(k,k)\), the Ext algebra over R of the ground field k. This equivalence provides some informations about MCM(R), the category of graded MCM R-modules modulo projectives.
Reviewer: Y.Aoyama

MSC:

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13C05 Structure, classification theorems for modules and ideals in commutative rings
14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal)
14J99 Surfaces and higher-dimensional varieties
11E04 Quadratic forms over general fields

Citations:

Zbl 0619.00007