Cohomological jump loci and duality in local algebra

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...

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Bibliographic Details
Published in:Mathematische Zeitschrift 2023-06, Vol.304 (2), Article 30
Main Authors: Briggs, Benjamin, McCormick, Daniel, Pollitz, Josh
Format: Article
Language:English
Subjects:
Algebra
Completeness
Intersections
Mathematics
Mathematics and Statistics
Modules
Rings (mathematics)
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Summary: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.
ISSN:0025-5874
1432-1823
DOI:10.1007/s00209-023-03276-9