Extensions of symmetric tensors by alternating tensors
Let F be a free module of rank d over a commutative ring R. Let \(\bigwedge^ nF\) and \(D_ nF\) be the n-th homogeneous components of the exterior algebra \(\bigwedge F=\sum \bigwedge^ iF \) and the divided power algebra \(DF=\sum D_ jF \), respectively. The author shows that the extension group \(Ext^ m(\bigwedge^ mF,D_ nF)\) in the category of the polynomial representations of the algebraic group scheme GL(F) over R is isomorphic as an R-module to the homogeneous component in degree \( n\) of the homology module \(H_{n-m}(DG_ a)\) of the hyperalgebra \(DG_ a\) of the algebraic group scheme \(G_ a\) over R. This paper makes an important contribution to representation theory.
polynomial representations of the algebraic group scheme
scientific article; zbMATH DE number 4116710