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Constructing equivariant vector bundles via the BGG correspondence. (English) Zbl 1439.14063

Summary: We describe a strategy for the construction of finitely generated \(G\)-equivariant \(\mathbb{Z}\)-graded modules \(M\) over the exterior algebra for a finite group \(G\). By an equivariant version of the BGG correspondence, \(M\) defines an object \(\mathcal{F}\) in the bounded derived category of \(G\)-equivariant coherent sheaves on projective space. We develop a necessary condition for \(\mathcal{F}\) being isomorphic to a vector bundle that can be simply read off from the Hilbert series of \(M\). Combining this necessary condition with the computation of finite excerpts of the cohomology table of \(\mathcal{F}\) makes it possible to enlist a class of equivariant vector bundles on \(\mathbb{P}^4\) that we call strongly determined in the case where \(G\) is the alternating group on 5 points.

MSC:

14F08 Derived categories of sheaves, dg categories, and related constructions in algebraic geometry
16E05 Syzygies, resolutions, complexes in associative algebras
16E20 Grothendieck groups, \(K\)-theory, etc.

Software:

Nemo; CAP; GAP; Magma
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References:

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