交换代数

We study the semiring $\mathbb{N}_0[\alpha]$ as an additive monoid where $\alpha$ is a positive real algebraic number. In the atomic case, the atoms of $\mathbb{N}_0[\alpha]$ are precisely the powers $\alpha^n$ up to a certain nonnegative…

We construct examples of local cohomology modules of ramified regular local rings with infinitely many associated primes and infinite Bass numbers.

In this note we rectify the proof of Theorem 3.11 in [arXiv:2403.02876]. We also present a set of examples at the end discussing various cases.

Exploiting an iterative formula already introduced in a previous manuscript to count the number $O_d$ of finite $O$-sequences of multiplicity $d$, we obtain some new information about $O_d$. Letting $A_d$ be the number of the finite…

The starting point is the class of the following simplicial complexes $\Delta$ with 2-linear resolutions. The facets of $\Delta$ are $F_1,\ldots,F_n$, and we demand that for each $i$ $F_i\cap (F_1\cup \cdots\cup F_{i-1}\cup…

Let $G$ be a finite group acting on a finite dimensional complex vector space $V$ via linear transformations. Let $\mathbb{C}[V]^G$ be the algebra of polynomials that are invariant under the induced $G$-action on the polynomial ring…

Let $k$ be a field of positive characteristic and $R = k[x_0,\dots, x_n]$. We consider ideals $I\subseteq R$ generated by homogeneous polynomials of degree $d$. Takagi and Watanabe proved that $\mathrm{fpt}(I)\geq \mathrm{height}(I)/d$; we…

Generalizing previous work of the first author, we introduce and study a characteristic free analog of the $F$-threshold for non-principal ideals, BCM-thresholds. We show that this coincides with the classical $F$-threshold for weakly…

Let $K$ be a field of characteristic $p>0$, $A=K[[Y]]$ be a power series ring in one variable and $Q(A)$ be the field of fraction of $A$. Suppose that $R=A[X_1,\ldots,X_n]$ is a standard $\mathbb{N}^n$-graded polynomial ring over $A$, i.e.,…

We introduce a weighted version of the module of logarithmic derivations of a divisor in weighted projective space, and provide a generalization of Saito's criterion for freeness in terms of weighted multiple eigenschemes (wME-schemes).…

We study the number $O_d$ of finite $O$-sequences of a given multiplicity $d$, with particular attention to the computation of $O_d$. We show that the sequence $(O_d)_d$ is sub-Fibonacci, and that if the sequence $(O_d / O_{d-1})_d$…

It is known that for a monomial ideal $I$, the number of minimal generators, $\mu(I^n)$, eventually follows a polynomial pattern for increasing $n$. In general, little is known about the power at which this pattern emerges. Even less is…

Let $O$ be a discrete valuation ring and $A := O[X_{m \times n}]/I_{m}(X)$ the determinantal ring of maximal minors. We consider algebra maps $\lambda \colon A \to O$, which is tantamount to choosing rank-deficient matrices $a \in O^{m…

This note develops some fundamental properties of resultants and related notions. It represents my own personal exploration of this domain, which I found more instructive than seeking answers in the standard literature. Consequently,…

Let $R$ be a commutative noetherian ring, and let $C$ be a semidualizing $R$-module. In this paper, we study levels of bounded complexes of finitely generated $R$-modules with respect to the full subcategory $\mathsf{G}_{C}(R)$ consisting…

The conjecture that every modular lattice is integral is disproved.

We present a Lean 4 Mathlib formalization of Nagata's factoriality theorem: if R is a noetherian domain and S <= R is a prime-generated submonoid such that S^{-1}R is a UFD, then R itself is a UFD. The prime-generated hypothesis -- every…

In this paper, we introduce and study the class of generalized square-difference factor absorbing (gsdf-absorbing) submodules of modules over commutative rings. We provide various characterizations and properties of gsdf-absorbing…

Let $G$ be a finite connected simple graph, and let $\mathcal{J}_{K_m,G}$ denote its generalized binomial edge ideal. By investigating the colon ideals of $\mathcal{J}_{K_m,G}$, we derive a formula for the local $\mathrm{v}$-number of…

In this work, we investigate the connections between dominating sets, private neighbors, and perfect codes in graphs, and their relationships with commutative algebra. In particular, we estimate the $\mathtt{v}$-number of closed…