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If I is an ideal in a Gorenstein ring S, and S/I is Cohen-Macaulay, then the same is true for any linked ideal I ; but such statements hold for residual intersections of higher codimension only under restrictive hypotheses, not satisfied even by ideals as simple as the ideal Ln of minors of a generic 2 × n matrix when n > 3. In this paper we initiate the study of a different sort of Cohen-Macaulay property that holds for certain general residual intersections of the maximal (interesting) codimension, one less than the analytic spread of I. For example, suppose that K is the residual intersection of Ln by 2n − 4 general quadratic forms in Ln. In this situation we analyze S/K and show that In−3(S/K) is a self-dual maximal Cohen-Macaulay S/K-module with linear free resolution over S. The technical heart of the paper is a result about ideals of analytic spread 1 whose high powers are linearly presented.
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Tate resolutions and {MCM} approximations
Let M be a finitely generated Cohen-Macaulay module of codimension m over a Gorenstein Ring R=S/I, where S is a regular ring. We show how to form a quasi-isomorphism ϕ from an R-free resolution of M to the dual of an R-free resolution of M∨:=ExtmR(M,R) using the S-free resolutions of R and M. The mapping cone of ϕ is then a Tate resolution of M, allowing us to compute the maximal Cohen-Macaulay approximation of M. "In the case when R is 0-dimensional local, and M is the residue field, the formula for ϕ becomes a formula for the socle of R generalizing a well-known formula for the socle of a zero-dimensional complete intersection. "When I⊂J⊂S are ideals generated by regular sequences, the R-module M=S/J is called a quasi-complete intersection, and ϕ was studied in detail by Kustin and Şega. We relate their construction to the sequence of `EagonNorthcott'-like complexes originally introduced by Buchsbaum and Eisenbud.
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- Award ID(s):
- 2001649
- PAR ID:
- 10349531
- Date Published:
- Journal Name:
- Contemporary mathematics
- Volume:
- 773
- ISSN:
- 0271-4132
- Page Range / eLocation ID:
- 37-43
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1090/conm/773/15531
