Equivariant Modules

A module $N$ over a ring $A$ is a $G$-equivariant module if $N$ is also a representation of $G$ in a way compatible with the module structure. The lattice of an equivariant module is a convenient way to describe an equivariant module. We introduce an explicit elementary technique for understanding the lattice of equivariant modules. Then we apply this technique to two questions related to equivariant modules. \nIn Chapter 2 we work with equivariant modules for $\\GL(V)$ acting on the polynomial ring $R=\\Sym V$. We introduce for every partition $\\lambda$ the elementary equivariant module $M_{\\lambda}$. Then we prove that any finitely generated equivariant module admits a filtration with associated graded being the direct sum of modules of only two kinds: either $M_{\\lambda}$ or truncations of $M_{\\lambda}$. We use our technique to show that each $M_{\\lambda}$ has a linear resolution and describe also the resolution of its truncations.\nIn Chapter 3 we look at a family of equivariant complexes. One can find this family in the appendix of the famous book by D. Eisenbud "Commutative Algebra with a View Towards Algebraic Geometry". This family includes the Eagon-Northcott and Buschsbaum-Rim complexes. Our objective is to study this family, and, in particular, refine the knowledge of its cohomology.\nFirst, we obtain these complexes from the derived images of twists of the Koszul complex on the projective space. This idea apparently goes back to Kempf [1970]. Taking this "geometric" point of view, we interpret the cohomology of these complexes as the cohomology of certain vector bundles on projective space, and proceed with calculations. Our technique allows us to describe the lattice of cohomology as an equivariant module.\nFinally, we put the above complexes in the realm of tilting theory: non-exactness of this family in certain regions can be seen as a failure of the exceptional sequence of line bundles on the projective space to lift to an exceptional sequence on a certain vector bundle. This observation creates a curious contrast with the results of Buchweitz-Leushke-Van den Bergh, stating that the exceptional sequence of twisted differential forms does lift to an exceptional sequence on the same vector bundle.

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