Let X be a quotient surface singularity, and define $G^{def}(X,r)$ as the directed graph of maximal Cohen-Macaulay (MCM) modules with edges corresponding to deformation incidences. We conjecture that the number of connected components of $G^{def}(X,r)$ is equal to the order of the divisor class group of X, and when X is a rational double point (RDP), we observe that this follows from a result of A. Ishii. We view this as an enrichment of the McKay correspondence. For a general quotient singularity X, we prove the conjecture...

Let $A=K[x_1,\cdots ,x_m]$ be a polynomial ring in $m$ variables and let $\mathbf{d}=(d_0\<\cdots \<d_m)$ be a strictly increasing sequence of $m+1$ integers. Boij and Söderberg conjectured the existence of graded $A$-modules $M$ of finite length having pure free resolution of type $\mathbf{d}$ in the sense that for $i=0,\cdots ,m$ the $i$-th syzygy module of $M$ has generators only in degree $d_i$.This paper provides a construction, in characteristic zero, of modules with this property that are also $GL\left(m\right)$-equivariant. Moreover, the construction works over rings of the form $A{\otimes }_{K}B$ where $A$ is a polynomial...

Let $H$ denote a finite index subgroup of the modular group $\Gamma$ and let $\rho$ denote a finite-dimensional complex representation of $H.$ Let $M\left(\rho \right)$ denote the collection of holomorphic vector-valued modular forms for $\rho$ and let $M\left(H\right)$ denote the collection of modular forms on $H$. Then $M\left(\rho \right)$ is a $\mathbb{Z}$-graded $M\left(H\right)$-module. It has been proven that $M\left(\rho \right)$ may not be projective as a $M\left(H\right)$-module. We prove that $M\left(\rho \right)$ is Cohen-Macaulay as a $M\left(H\right)$-module. We also explain how to apply this result to prove that if $M\left(H\right)$ is a polynomial ring, then $M\left(\rho \right)$ is a free...