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Cohomological jump loci and duality in local algebra

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Abstract

In this article a higher order support theory, called the cohomological jump loci, is introduced and studied for dg modules over a Koszul extension of a local dg algebra. The generality of this setting applies to dg modules over local complete intersection rings, exterior algebras and certain group algebras in prime characteristic. This family of varieties generalizes the well-studied support varieties in each of these contexts. We show that cohomological jump loci satisfy several interesting properties, including being closed under (Grothendieck) duality. The main application of this support theory is that over a local ring the homological invariants of Betti degree and complexity are preserved under duality for finitely generated modules having finite complete intersection dimension.

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Authors and Affiliations

  1. Department of Mathematics, University of Utah, Salt Lake City, UT, 84112, USA

    Benjamin Briggs, Daniel McCormick & Josh Pollitz

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  1. Benjamin Briggs
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  2. Daniel McCormick
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  3. Josh Pollitz
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Correspondence to Josh Pollitz.

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Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

For part of this work the Benjamin Briggs was hosted by the Mathematical Sciences Research Institute in Berkeley, California, supported by the National Science Foundation under Grant No. 1928930. The Daniel McCormick and Josh Pollitz worked on this project while supported by the RTG grant from the National Science Foundation No. 1840190, as well as being partly supported by the National Science Foundation under Grants No. 2001368 and 2002173, respectively

This work has benefited significantly from numerous comments of Srikanth Iyengar, for which we are grateful. We also thank a referee for helpful comments on the manuscript.

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Briggs, B., McCormick, D. & Pollitz, J. Cohomological jump loci and duality in local algebra. Math. Z. 304, 30 (2023). https://doi.org/10.1007/s00209-023-03276-9

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