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1. 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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2. Stable behavior of Frobenius powers over a general hypersurface

   https://doi.org/10.1090/btran/199  

   Miller, Claudia; Rahmati, Hamidreza; R_G, Rebecca (November 2024, Transactions of the American Mathematical Society, Series B)

   The main goal of this paper is to prove, in positive characteristic $p$, stability behavior for the graded Betti numbers in the periodic tails of the minimal resolutions of Frobenius powers of the homogeneous maximal ideals for very general choices of hypersurface in three variables whose degree has the opposite parity to that of $p$. We also find some of the structure of the matrix factorization giving the resolution. We achieve this by developing a method for obtaining the degrees of the generators of the defining ideal of an $\mathfrak{c}$-compressed Gorenstein Artinian graded algebra from its socle degree, where $\mathfrak{c}$is a Frobenius power of the homogeneous maximal ideal. As an application, we also obtain the Hilbert–Kunz function of the hypersurface ring, as well as the Castelnuovo–Mumford regularity of the quotients by Frobenius powers of the homogeneous maximal ideal. 

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3. Residual Intersections and linear powers

   EISENBUD, David; HUNEKE, CRAIG; ULRICH, BERND (July 2024, Transactions of the American Mathematical Society)

   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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4. A Jacobian criterion for nonsingularity in mixed characteristic

   https://doi.org/10.1353/ajm.2024.a944363  

   Hochster, Melvin; Jeffries, Jack (December 2024, American Journal of Mathematics)

   abstract: We give a version of the usual Jacobian characterization of the defining ideal of the singular locus in the equal characteristic case: the new theorem is valid for essentially affine algebras over a complete local algebra over a mixed characteristic discrete valuation ring. The result makes use of the minors of a matrix that includes a row coming from the values of a $$p$$-derivation. To study the analogue of modules of differentials associated with the mixed Jacobian matrices that arise in our context, we introduce and investigate the notion of a {\it perivation}, which may be thought of, roughly, as a linearization of the notion of $$p$$-derivation. We also develop a mixed characteristic analogue of the positive characteristic $$\Gamma$$-construction, and apply this to give additional nonsingularity criteria. 

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5. Betti numbers for connected sums of graded Gorenstein Artinian algebras

   https://doi.org/10.1090/tran/9286  

   Altafi, Nasrin; Di_Gennaro, Roberta; Galetto, Federico; Grate, Sean; Miró-Roig, Rosa; Nagel, Uwe; Seceleanu, Alexandra; Watanabe, Junzo (October 2024, Transactions of the American Mathematical Society)

   The connected sum construction, which takes as input Gorenstein rings and produces new Gorenstein rings, can be considered as an algebraic analogue for the topological construction having the same name. We determine the graded Betti numbers for connected sums of graded Artinian Gorenstein algebras. Along the way, we find the graded Betti numbers for fiber products of graded rings; an analogous result was obtained in the local case by Geller [Proc. Amer. Math. Soc. 150 (2022), pp. 4159–4172]. We relate the connected sum construction to the doubling construction, which also produces Gorenstein rings. Specifically, we show that, for any number of summands, a connected sum of doublings is the doubling of a fiber product ring. 

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