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...

The moduli space of rank-$n$ commutative algebras equipped with an ordered basis is an affine scheme ${𝔅}_n$ of finite type over $\mathbb{Z}$, with geometrically connected fibers. It is smooth if and only if $n\le 3$. It is reducible if $n\ge 8$ (and the converse holds, at least if we remove the fibers above $2$ and $3$). The relative dimension of ${𝔅}_n$ is $\frac{2}{27}{n}^3+O\left(n^{8/3}\right)$. The subscheme parameterizing étale algebras is isomorphic to ${GL}_n/S_n$, which is of dimension only $n^2$. For $n\ge 8$, there exist algebras that are not limits of étale algebras. The dimension calculations...

Let $G$ be a finite simple graph with the vertex set $V$ and let $I_G$ be its edge ideal in the polynomial ring $S=𝕂\left[V\right]$. We compute the depth and the Castelnuovo-Mumford regularity of $S/I_G$ when $G=G_1\circ G_2$ or $G=G_1*G_2$ is a graph obtained from Cohen-Macaulay bipartite graphs $G_1$, $G_2$ by the $\circ$ operation or $*$ operation, respectively.

Let $R$ be a commutative ring with unity and $U\left(R\right)$ be the set of unit elements of $R$. In this paper, we introduce and investigate some properties of a new kind of graph on the ring $R$, namely, the prime ideals intersection graph of $R$, denoted by $G_p\left(R\right)$. The $G_p\left(R\right)$ is a graph with vertex set $R^{*}-U\left(R\right)$ and two distinct vertices $a$ and $b$ are adjacent if and only if there exists a prime ideal $𝔭$ of $R$ such that $a,b\in 𝔭$. We obtain necessary and sufficient conditions on $R$ such that $G_p\left(R\right)$ is disconnected. We find the diameter and girth of $G_p\left(R\right)$....