Equivariant resolutions over Veronese rings

Abstract

Working in a polynomial ring $S=\mathbf{k}[x_1,\ldots,x_n]$ 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 $M=S^{(\geq r, d)}$ inside $S$. We develop and use characteristic-free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple $GL_n(\mathbf{k})$-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 $\mathrm{Tor}^R_i(M,M')$ for all $i$ and $\mathrm{Hom}_R(M,M')$ for any pair of these modules $M,M'$.