Equivariant resolutions over Veronese rings

Abstract

Working in a polynomial ring $S=\mathbf{k}[x_1,\dots ,x_n]$ where $\mathbf{k}$ is an arbitrary commutative ring with $1$, we consider the $d^{th}$ Veronese subalgebras $R=S^{(d)}$, as well as natural $R$-submodules $M=S^{(\ge r,d)}$ inside $S$. We develop and use characteristic-free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple $G{L}_n(\mathbf{k})$-equivariant minimal free $R$-resolutions for the quotient ring $\mathbf{k}=R/R_{+}$ and for these modules $M$. These also lead to elegant descriptions of ${\mathrm{Tor}}_i^R(M,M^{\prime })$ for all $i$ and ${\mathrm{Hom}}_R(M,M^{\prime })$ for any pair of these modules $M,M^{\prime }$.

37 pages. Further minor edits. Version to appear in J. London Math. Soc

Cited by in corpus (1)

• Ribbon Schur Functors