Equivariant resolutions over Veronese rings

Working in a polynomial ring S=k[x1,…,xn]$S={\mathbf {k}}[x_1,\ldots ,x_n]$, where k${\mathbf {k}}$ is an arbitrary commutative ring with 1, we consider the d$d$th Veronese subalgebras R=S(d)$R={S^{(d)}}$, as well as natural R$R$‐submodules M=S(⩾r,d)$M={S^{({\geqslant r},{d})}}$ inside S$S$. We develop and use characteristic‐free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple GLn(k)$GL_n({\mathbf {k}})$‐equivariant minimal free R$R$‐resolutions for the quotient ring k=R/R+${\mathbf {k}}=R/R_+$ and for these modules M$M$. These also lead to elegant descriptions of ToriR(M,M′)$\operatorname{Tor}^R_i(M,M^{\prime})$ for all i$i$ and HomR(M,M′)$\operatorname{Hom}_R(M,M^{\prime})$ for any pair of these modules M,M′$M,M^{\prime}$.