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1. Residual intersections and linear powers

   https://doi.org/10.1090/btran/127  

   Eisenbud, David; Huneke, Craig; Ulrich, Bernd (October 2023, Transactions of the American Mathematical Society, Series B)

   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 $I^{\prime }$; but such statements hold for residual intersections of higher codimension only under restrictive hypotheses, not satisfied even by ideals as simple as the ideal $L_n$of minors of a generic $2\times <\#comment/>n$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 $L_n$by $2n-<\#comment/>4$general quadratic forms in $L_n$. In this situation we analyze $S/K$and show that $I^{n-<\#comment/>3}\left(S/K\right)$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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2. Tate resolutions and {MCM} approximations

   https://doi.org/10.1090/conm/773/15531  

   David Eisenbud and Frank-Olaf Schreyer (January 2021, Contemporary mathematics)

   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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3. The family of perfect ideals of codimension 3, of type 2 with 5 generators

   https://doi.org/10.1090/proc/14646  

   Celikbas, Ela; Jai, Laxmi; Kraskiewicz, Witold; Weyman, Jerzy (March 2020, Proceedings of the American Mathematical Society)

   In this paper we define an interesting family of perfect ideals of codimension three, with five generators, of Cohen-Macaulay type two with trivial multiplication on the $$ \operatorname {Tor}$$ algebra. This family is likely to play a key role in classifying perfect ideals with five generators of type two. 

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4. Duality and socle generators for residual intersections

   https://doi.org/10.1515/crelle-2017-0045  

   Eisenbud, David; Ulrich, Bernd (November 2019, Journal für die reine und angewandte Mathematik (Crelles Journal))

   null (Ed.)

   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. 

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5. Schemes Supported on the Singular Locus of a Hyperplane Arrangement in ℙ n

   https://doi.org/10.1093/imrn/rnaa113  

   Migliore, Juan; Nagel, Uwe; Schenck, Henry (May 2020, International Mathematics Research Notices)

   Abstract A hyperplane arrangement in $$\mathbb P^n$$ is free if $R/J$ is Cohen–Macaulay (CM), where $$R = k[x_0,\dots ,x_n]$$ and $$J$$ is the Jacobian ideal. We study the CM-ness of two related unmixed ideals: $$ J^{un}$$, the intersection of height two primary components, and $$\sqrt{J}$$, the radical. Under a mild hypothesis, we show these ideals are CM. Suppose the hypothesis fails. For equidimensional curves in $$\mathbb P^3$$, the Hartshorne–Rao module measures the failure of CM-ness and determines the even liaison class of the curve. We show that for any positive integer $$r$$, there is an arrangement for which $$R/J^{un}$$ (resp. $$R/\sqrt{J}$$) fails to be CM in only one degree, and this failure is by $$r$$. We draw consequences for the even liaison class of $$J^{un}$$ or $$\sqrt{J}$$. 

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