Topic
Complete intersection ring
About: Complete intersection ring is a research topic. Over the lifetime, 70 publications have been published within this topic receiving 2510 citations.
Papers published on a yearly basis
Papers
TL;DR: In this article, it was shown that the maximal Cohen-Macaulay modules with periodic resolutions are the maximal 4-modules without free direct summands, and the maximal 5-modules with periodic resolution are maximal 3-modules.
Abstract: Let R be a regular local ring, and let A = R/(x), where x is any nonunit of R. We prove that every minimal free resolution of a finitely generated A -module becomes periodic of period 1 or 2 after at most dim A steps, and we examine generalizations and extensions of this for complete intersections. Our theorems follow from the properties of certain universally defined endomorphisms of complexes over such rings. Let A be a commutative ring, and let x G A be a nonzero divisor. How does homological algebra over A/(x) = B differ from that over AI In this paper we will study a certain natural endomorphism / of complexes of free A / (x)-modu\es which seems to reflect some of the difference. For example, the (homotopic) triviality of t is an obstruction (closely related to the usual one in Ext2,) to the lifting of a complex of free 5-modules to a complex of free /I-modules. More generally, if x,, . . . , xn is an A -sequence, we study « natural endomorphisms /,,..., tn of complexes of free A/(xx, . . . , x")-modules, and try to use them to explain the way in which free resolutions over A/(xx, . . . , x") differ from free resolutions over A (the construction and elementary properties of these endomorphisms is given in §1). In this paper, we will study the case in which A is a regular local ring and B = A/(xx, . . . , x") is not regular. (It would also be very interesting to understand the case in which both A and A/(x) = B were regular-with, say, A of mixed characteristic and B ramified or of characteristic p.) In this case, the homological algebra over A is dominated, roughly speaking, by the fact that minimal /I-free resolutions are finite; we seek to understand the eventual behavior of minimal 5-free resolutions in terms of the tt. For example, if « = 1, so that B = A/(x), we prove that / is eventually an isomorphism, so that every minimal 5-free resolution becomes periodic of period 2 after at most 1 + dim B steps (§6). We also show that the 5-modules with periodic resolutions are the maximal Cohen-Macaulay modules without free direct summands. Since the periodic part of a periodic resolution over A/(x) (or more generally, over A/(xx, . . . , xn), if x,, . . . , x" is an A -sequence) is easy to describe explicitly (§5), this yields information on maximal Cohen- Macaulay modules.
863 citations
1 Jan 1998
TL;DR: The notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matematica, Institut d'Estudis Catalans, July 15-26, 1996.
Abstract: This text is based on the notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matematica, Institut d’Estudis Catalans, July 15–26, 1996
450 citations
TL;DR: In this article, a homological invariant for a finite module over a commutative noetherian ring, called its CI-dimension, is introduced, which provides a rich structure theory of free resolutions.
Abstract: A new homological invariant is introduced for a finite module over a commutative noetherian ring: its CI-dimension. In the local case, sharp quantitative and structural data are obtained for modules of finite CI-dimension, providing the first class of modules of (possibly) infinite projective dimension with a rich structure theory of free resolutions.
313 citations
TL;DR: In this paper, the authors developed geometric methods for the study of finite modules over a local complete intersection R. They showed that the least number of equations needed to cut out R from a regular local ring equals the codimension of R, where codim R = νR(m)− dim R and m denotes the minimal number of generators of an R-module M.
Abstract: Quillen’s geometric approach to the cohomology of finite groups [30] has revolutionized modular representation theory. The ideology and techniques involved have been generalized and extended to representations of various Hopf algebras, culminating in the recent work of Friedlander and Suslin [17] on finite group schemes. In this paper we develop geometric methods for the study of finite modules over a local complete intersection R. If R is such a ring and m is its maximal ideal, then the m-adic completion R has the form Q/(f), where f is a regular sequence and Q is a regular local ring that can be taken to be a ring of formal power series over a field or a discrete valuation ring. The least number of equations needed to cut out R from a regular local ring equals the codimension of R, where codim R = νR(m)− dim R and νR(M) denotes the minimal number of generators of an R-module M. In [5] a cone, that is, a homogeneous algebraic set VR(M) is attached to each finite R-module M and used to study its minimal free resolution. Here we prove
306 citations
197 citations
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Performance Metrics
| Year | Papers |
|---|---|
| 2021 | 4 |
| 2020 | 5 |
| 2019 | 7 |
| 2018 | 3 |
| 2017 | 3 |
| 2016 | 3 |