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\(\textbf{GL}\)-algebras in positive characteristic I: the exterior algebra

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Abstract

We study the category of \(\textbf{GL}\)-equivariant modules over the infinite exterior algebra in positive characteristic. Our main structural result is a shift theorem à la Nagpal. Using this, we obtain a Church–Ellenberg type bound for the Castelnuovo–Mumford regularity. We also prove finiteness results for local cohomology.

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Abbreviations

k :

The base field, always algebraically closed of characteristic \(p > 0\)

\(\textbf{V}\) :

A fixed infinite dimensional k-vector space with basis \(\{e_i\}_{i \ge 1}\)

\(\textbf{GL}\) :

The group of automorphisms of \(\textbf{V}\) fixing all but finitely many of the basis vectors \(e_i\)

\({{\,\textrm{Vec}\,}}\) :

The category of k-vector spaces

\({{\,\textrm{Rep}\,}}^\textrm{pol}(\textbf{GL})\) :

The category of polynomial representations of \(\textbf{GL}\)

\(\textbf{Pol}\) :

The category of strict polynomial functors \({{\,\textrm{Vec}\,}}\rightarrow {{\,\textrm{Vec}\,}}\)

R :

The exterior algebra of \(\textbf{V}\)

\(L_{\lambda }\) :

The irreducible polynomial representation of \(\textbf{GL}\) with highest weight \(\lambda \)

\(\varvec{\Sigma }\) :

The Schur derivative functor, which we also call the shift functor

\(\varvec{\Delta }\) :

The difference functor

\(-^{<n}\) :

The submodule generated by all elements of degree less than n

\(-^{(r)}\) :

The r-th Frobenius twist of a \(\textbf{GL}\)-representation

\(t_i(-)\) :

The generation degree of \({{\,\textrm{Tor}\,}}_i^R(-, k)\)

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Acknowledgements

I am grateful to Andrew Snowden for being generous with his ideas and providing numerous comments on a draft which greatly improved the exposition. Thanks are also due to Rohit Nagpal for pointing out helpful references, and Nate Harman for helpful discussions. The work was supported in part by NSF grants DMS #1453893 and DMS #1840234.

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  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, USA

    Karthik Ganapathy

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Ganapathy, K. \(\textbf{GL}\)-algebras in positive characteristic I: the exterior algebra. Sel. Math. New Ser. 30, 63 (2024). https://doi.org/10.1007/s00029-024-00960-4

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