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Abstract The Burch index is a new invariant of a local ringRwhose positivity implies a kind of linearity in resolutions ofR-modules. We show that ifRhas depth zero and Burch index at least 2, then any non-free 7thR-syzygy contains the residue field as a direct summand. We compute the Burch index in various cases of interest. 

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. 

Francis Sowerby Macaulay began his career working on Brill and Noether’s theory of algebraic plane curves and their interpretation of the Riemann–Roch and Cayley–Bacharach theorems; in fact it is Macaulay who first stated and proved the modern form of the Cayley–Bacharach theorem. Later in his career Macaulay developed ideas and results that have become important in modern commutative algebra, such as the notions of unmixedness, perfection (the Cohen–Macaulay property), and super-perfection (the Gorenstein property), work that was appreciated by Emmy Noether and the people around her. He also discovered results that are now fundamental in the theory of linkage, but this work was forgotten and independently rediscovered much later. The name of a computer algebra program (now Macaulay2) recognizes that much of his work is based on examples created by refined computation. Though he never spoke of the connection, the threads of Macaulay’s work lead directly from the problems on plane curves to his later theorems. In this paper we will explain what Macaulay did, and how his results are connected. 

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. 

Abstract We prove duality results for residual intersections that unify and complete results of van Straten,Huneke–Ulrich and Ulrich, and settle conjectures of van Straten and Warmt. Suppose that I is an ideal of codimension g in a Gorenstein ring,and {J\subset I} is an ideal with {s=g+t} generators such that {K:=J:I} has codimension s . Let {{\overline{I}}} be the image of I in {{\overline{R}}:=R/K} . In the first part of the paper we prove, among other things, that under suitable hypotheses on I , the truncated Rees ring {{\overline{R}}\oplus{\overline{I}}\oplus\cdots\oplus{\overline{I}}{}^{t+1}} is a Gorenstein ring, and that the modules {{\overline{I}}{}^{u}} and {{\overline{I}}{}^{t+1-u}} are dualto one another via the multiplication pairing into {{{\overline{I}}{}^{t+1}}\cong{\omega_{\overline{R}}}} . In the second part of the paper we study the analogue of residue theory, and prove that, when {R/K} is a finite-dimensional algebra over a field of characteristic 0 and certain other hypotheses are satisfied, the socle of {I^{t+1}/JI^{t}\cong{\omega_{R/K}}} is generated by a Jacobian determinant.