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Cohomological supports over derived complete intersections and local rings

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Abstract

A theory of cohomological support for pairs of DG modules over a Koszul complex is investigated. These specialize to the support varieties of Avramov and Buchweitz defined over a complete intersection ring, as well as support varieties over an exterior algebra. The main objects of study are certain DG modules over a polynomial ring; these determine the aforementioned cohomological supports and are shown to encode (co)homological information about pairs of DG modules over a Koszul complex. The perspective in this article leads to new proofs of well-known results for pairs of complexes over a complete intersection. Furthermore, these cohomological supports are used to define a support theory for pairs of objects in the derived category of an arbitrary commutative noetherian local ring. Finally, we calculate several examples; one of which answers a question of D. Jorgensen in the negative.

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Notes

  1. This last deduction is essentially the same argument from [5, 5.7] for complexity over a complete intersection ring.

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Acknowledgements

This paper was partially completed at the University of Nebraska-Lincoln while I was finishing my thesis under the supervision of Luchezar Avramov and Mark Walker. It is a pleasure to thank both of them for innumerable conversations relating to this work. I would also like to thank Srikanth Iyengar and the referee for helpful comments on this article.

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

    Josh Pollitz

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Correspondence to Josh Pollitz.

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The author was partly supported through National Science Foundation Grants DMS 1103176, DMS 1840190, and DMS 2002173.

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Pollitz, J. Cohomological supports over derived complete intersections and local rings. Math. Z. 299, 2063–2101 (2021). https://doi.org/10.1007/s00209-021-02738-2

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