The problem of finding a characterization of Cohen--Macaulay simplicial complexes has been studied intensively by many authors. There are several attempts at this problem available for some special classes of simplicial complexes satisfying some technical conditions. This paper is a survey, with some new results, of some of these developments. The new results about simplicial complexes with Serre's condition are an analogue of the known results for Cohen--Macaulay simplicial complexes.

Given a finitely generated module $M$ over a commutative local ring (or a standard graded $k$-algebra) $(R,\m,k)$, we detect its complexity in terms of numerical invariants coming from suitable $\m$-stable filtrations $\mathbb{M}$ on $M.$ We study the Castelnuovo-Mumford regularity of $gr_{\mathbb{M}}(M) $ and the linearity defect of $M, $ denoted $\ld_R(M), $ through a deep investigation based on the theory of standard bases. If $M$ is a graded $R$-module, then $\reg_R(gr_{\mathbb{M}}(M)) \lt \infty $ implies $\reg_R(M)\lt \infty$ and the converse holds provided $M$ is of homogenous type. An analogous result can be proved in the local case in terms of the linearity defect. Motivated by a positive answer in the graded case, we present for local rings a partial answer to a question raised by Herzog and Iyengar of whether $\ld_R(k)\lt \infty$ implies $R$ is Koszul.

In this paper, we introduce and study $\mathbb P$-radical and $\mathbb{M}$-radical modules over commutative rings. We say that an $R$-module $M$ is \textit{$\mathbb P$-radical} whenever $M$ satisfies the equality $(\sqrt[p]{{\mathcal{P}}M}:M)=\sqrt{{\mathcal{P}}}$ for every prime ideal ${\mathcal{P}}\supseteq Ann({\mathcal{P}}M)$, where $\sqrt[p]{{\mathcal{P}}M}$ is the intersection of all prime submodules of $M$ containing ${\mathcal{P}}M$. Among other results, we show that the class of $\mathbb P$-radical modules is wider than the class of primeful modules (introduced by Lu \cite{Lu4}). Also, we prove that any projective module over a Noetherian ring is $\mathbb P$-radical. This also holds for any arbitrary module over an Artinian ring. Furthermore, we call an $R$-module $M$ by $\mathbb{M}$-\textit{radical} if $(\sqrt[p]{{\mathcal{M}}M}:M)={\mathcal{M}}$, for every maximal ideal ${\mathcal{M}}$ containing ${\rm Ann\,}(M)$. We show that the conditions $\mathbb P$-radical and $\mathbb{M}$-radical are equivalent for all $R$-modules if and only if $R$ is a Hilbert ring. Also, two conditions primeful and $\mathbb{M}$-radical are equivalent for all $R$-modules if and only if $\mbox{dim\,}(R)=0$. Finally, we remark that the results of this paper will be applied in a subsequent work of the authors to construct a structure sheaf on the spectrum of $\mathbb P$-radical modules in the point of algebraic geometry view.

We study lowest-weight irreducible representations of rational Cherednik algebras attached to the complex reflection groups $G(m, r, n)$ in characteristic~$p$. Our approach is mostly from the perspective of commutative algebra. By studying the kernel of the contravariant bilinear form on Verma modules, we obtain formulas for a Hilbert series of irreducible representations in a number of cases, and present conjectures in other cases. We observe that the form of the Hilbert series of irreducible representations and the generators of the kernel tend to be determined by the value of $n$ modulo~$p$ and are related to special classes of subspace arrangements. Perhaps the most novel (conjectural) discovery from the commutative algebra perspective is that the generators of the kernel can be given the structure of a ``matrix regular sequence'' in some instances, which we prove in some small cases.

Given two determinantal rings over a field $k$, we consider the Rees algebra of the diagonal ideal, the kernel of the multiplication map. The special fiber ring of the diagonal ideal is the homogeneous coordinate ring of the secant variety. When the Rees algebra and the symmetric algebra coincide, we show that the Rees algebra is Cohen-Macaulay.

In this paper a realization of all classical and most exceptional finite groups of Lie type defined over a field $\mathbb{F}_{q}$ (where $q=p^r$ is a prime power) as Galois groups over rational function fields over the prime field $\mathbb{F}_p$ is provided. Our approach runs by restricting the ground field of the groups and using criteria for bounds for Galois groups, derived from the theory of Frobenius modules.