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In this paper we extend a result of Dubouloz on the Cancellation Problem in higher dimensions ($\geqslant 2$) over the field of complex numbers to fields of arbitrary characteristic. We then apply the generalised result to describe the…

Let $k$ be a field, $m$ a positive integer, $\mathbb{V}$ an affine subvariety of $\mathbb{A}^{m+3}$ defined by a linear relation of the form $x_{1}^{r_{1}}\cdots x_{m}^{r_{m}}y=F(x_{1}, \ldots , x_{m},z,t)$, $A$ the coordinate ring of…

A semidomain is an additive submonoid of an integral domain that is closed under multiplication and contains the identity element. Although atomicity and divisibility in integral domains have been systematically investigated for more than…

This is a short report on the discussions of appearance of tensors in algebraic statistics and rigidity theory, during the semester ``AGATES: Algebraic Geometry with Applications to TEnsors and Secants". We briefly survey some of the…

We show that for a square-free monomial ideal, the regularity of its symbolic (second) power and of the integral closure of of its (second) power can differ from the regularity of its ordinary (second) power by an arbitrarily large integer.

We study commutative Cohen-Macaulay rings whose Cohen-Macaulay representation theory are controlled by representations of quivers, which we call hereditary representation type. Based on tilting theory and cluster tilting theory, we…

We show that the small and large restricted injective dimensions coincide for Cohen-Macaulay rings of finite Krull dimension. Based on this, and inspired by the recent work of Sather-Wagstaff and Totushek, we suggest a new definition of…

For a given family $(G_i)_{i \in \N}$ of finitely generated abelian groups, we construct a Dedekind domain $D$ having the following properties. \begin{enumerate} \item $\Pic(D) \cong \bigoplus_{i \in \N}G_i$. \item For each $i \in \N$,…

Let $\mathbb{K}$ be a field and $n$ be a positive integer. Let $\Gamma =([n], E)$ be a simple graph, where $[n]=\{1,\ldots, n\}$. If $S=\mathbb{K}[x_1, \ldots, x_n, y_1, \ldots, y_n]$ is a polynomial ring, then the graded ideal \[…

A degree of a module $M$ is a numerical measure of information carried by $M$. We highlight some of Vasconcelos' outstanding contributions to the theory of degrees, bridging commutative algebra and computational algebra. We present several…

Let $k$ be a perfect field of prime characteristic $p$, $G$ a finite group scheme over $k$, and $V$ a finite-dimensional $G$-module. Let $S=\mathop{\mathrm{Sym}}V$ be the symmetric algebra with the standard grading. Let $M$ be a $\Bbb…

The shedding vertices of simplicial complexes are studied from an algebraic point of view. Based on this perspective, we introduce the class of ass-decomposable monomial ideals which is a generalization of the class of Stanley-Reisner…

Let $R$ be a commutative ring. An $R$-module $M$ is said to be $w$-split if Ext$_{R}^1(M,N)$ is a GV-torsion $R$-module for all $R$-modules $N$. It is known that every projective module is $w$-split, but the converse is not true in general.…

We extend the epsilon multiplicity of ideals defined by Ulrich and Validashti to epsilon multiplicity of filtrations, and show that under mild assumptions this multiplicity exists as a limit. We show that in rather general rings, the…

This is a revised version of the previous version with a new appendix consisting of characteristic two case. We define quasi-quadratic modules in a commutative ring generalizing the notion of quadratic modules. The main theorem is a…

Let D be a Euclidean domain, with fraction field K. Let R(D) be the subring of K generated by the reciprocals of the nonzero elements of D. The main theorem states that if R(D) is not equal to K, then R(D) is a rank 1 discrete valuation…

We give a short proof that any smooth (means formally smooth and finitely presented) homomorphism of rings can be obtained by base change from a smooth homomorphism of noetherian rings. Together with the elegant short proof by J. Conde-Lago…

Let $S$ be a numerical semigroup. We will say that $h\in {\mathbb{N}} \backslash S$ is an {\it isolated gap }of $S$ if $\{h-1,h+1\}\subseteq S.$ A numerical semigroup without isolated gaps is called perfect numerical semigroup. Denote by…

In this paper, we suggest a new method for a given tensor to find CP decompositions using a less number of rank $1$ tensors. The main ingredient is the Least Absolute Shrinkage and Selection Operator (LASSO) by considering the decomposition…

In this work we will show that if $F$ is a positive integer, then ${\mathrm{Sat}}(F)=\{S\mid S \mbox{ is a saturated numerical semigroup with Frobenius number } F\}$ is a covariety. As a consequence, we present two algorithms: one that…