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The goal of this paper is to study irreducible families of codimension 4, arithmetically Gorenstein schemes defined by the submaximal minors of a homogeneous matrix whose entries are homogeneous forms of degree . Under some numerical assumption on and , we prove that the closure of is an irreducible component of , show that is generically smooth along , and compute the dimension of in terms of and . To achieve these results we first prove that is determined by a regular section of where and is a codimension-2, arithmetically Cohen–Macaulay scheme defined by the maximal minors of the matrix obtained deleting a suitable row of .
Following Soulé’s ideas from 1979, we give a presentation of the abstract group for any semisimple (connected) simply connected absolutely almost simple -group . As an application, we give a description of in terms of direct limits, and show that the Whitehead group and the naive group of connected components of coincide.
We show that the universal unitary completion of certain locally algebraic representation of with is nonzero, topologically irreducible, admissible and corresponds to a -dimensional crystalline representation with nonsemisimple Frobenius via the -adic Langlands correspondence for .
We consider a finite group acting on a graded module and define an equivariant degree that generalizes the usual nonequivariant degree. The value of this degree is a module for the group, up to a rational multiple. We investigate how this behaves when the module is a ring and apply our results to reprove some results of Kuhn on the cohomology of groups.