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A Family of Complexes Associated to an Almost Alternating Map, with Applications to Residual Intersections
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A family {Dq} of complexes of free R-modules is constructed for each almost alternating matrix p = (XY) of shape (g,f) over a commutative noetherian ring R. (In p,X is a g x g alternating matrix and Y is an arbitrary g x f matrix). The complexes Dq are in many ways analogous to the Buchsbaum-Rim and Eagon-Northcott complexes which are associated to a generic matrix. For example, the complex Dq is isomorphic to the (shifted) dual of the complex Df-2-q ; and the complex Dq can be obtained as the mapping cone of two of the complexes which correspond to an almost alternating matrix of shape (g, f ...
A family {Dq} of complexes of free R-modules is constructed for each almost alternating matrix p = (XY) of shape (g,f) over a commutative noetherian ring R. (In p,X is a g x g alternating matrix and Y is an arbitrary g x f matrix). The complexes Dq are in many ways analogous to the Buchsbaum-Rim and Eagon-Northcott complexes which are associated to a generic matrix. For example, the complex Dq is isomorphic to the (shifted) dual of the complex Df-2-q ; and the complex Dq can be obtained as the mapping cone of two of the complexes which correspond to an almost alternating matrix of shape (g, f - 1). Roughly speaking, the complex Dq is obtained by pasting a graded strand of algebra K/J (where K is the Koszul algebra associated to p and J is a two generated ideal of K), together with a different graded strand of the same algebra. The position in Dq where the two strands are patched together involves pfaffians, of various sizes, of the alternating map which corresponds to p. If p is sufficiently general, then Dq is acyclic for all g 5 - 1. If the maximal order pfaffians of X generate a grade three Gorenstein ideal I of R and f 5 3, then Do resolves R/J, where J is an f-residual intersection of I. In the generic case, the divisor class group of R/J is the infinite cyclic group generated by the cokernel of p, and Dq resolves a representative of the class q(cockerp) from CM(R/J) for all q 5-1. When f = O, then Dq resolves the qth poser Iq of the grade three Gorenstein ideal I.
Table of contents:
Preliminary concepts; The definition of the family {Dq}; Each Dq is a complex; Elementary facts about the complexes {Dq}; Grade calculations; Acyclicity in the case f = O; Acyclicity in the generic case; Acyclicity in the non-generic case; The residual intersection of a grade three Gorenstein Ideal.
Table of contents:
Preliminary concepts; The definition of the family {Dq}; Each Dq is a complex; Elementary facts about the complexes {Dq}; Grade calculations; Acyclicity in the case f = O; Acyclicity in the generic case; Acyclicity in the non-generic case; The residual intersection of a grade three Gorenstein Ideal.