Using the concept of vector partition functions, we investigate the asymptotic behavior of graded Betti numbers of powers of homogeneous ideals in a polynomial ring over a field. Our main results state that if the polynomial ring is equipped with a positive ${\mathbb{Z}}^d$-grading, then the Betti numbers of powers of ideals are encoded by finitely many polynomials.

Specially, in the case of $\mathbb{Z}$-grading, for each homological degree $i$ we can split ${\mathbb{Z}}^2=\left\{\left(\mu ,t\right)\mid t,\mu \in \mathbb{Z}\right\}$ in a finite number of regions such that for each region there is a polynomial in $\mu$ and $t$ that computes ${dim}_k\left({Tor}_i^S{\left(I^t,k\right)}_{\mu }\right)$. This refines, in a graded situation, the result of Kodiyalam on Betti numbers of powers of ideals.

Our main statement treats the case of a power products of homogeneous ideals in a ${\mathbb{Z}}^d$-graded algebra, for a positive grading.

In this note, we show that in a commutative ring $R$ with unity, for any $n>0$, if $I$ is an $n$-absorbing ideal of $R$, then ${\left(\sqrt{I}\right)}^n\subseteq I$.

Among other results, we prove the following:

1. A locally Archimedean stable domain satisfies accp.

2. A stable domain $R$ is Archimedean if and only if every nonunit of $R$ belongs to a height-one prime ideal of the integral closure $R^{\prime }$ of $R$ in its quotient field (this result is related to Ohm’s theorem for Prüfer domains).

3. An Archimedean stable domain $R$ is one-dimensional if and only if $R^{\prime }$ is equidimensional (generally, an Archimedean stable local domain is not necessarily one-dimensional).

4. An Archimedean finitely stable semilocal domain with stable maximal ideals is locally Archimedean, but generally, neither Archimedean stable domains, nor Archimedean semilocal domains are necessarily locally Archimedean.

An explicit construction is given of a minimal free resolution of the ideal generated by all squarefree monomials of a given degree. The construction relies upon and exhibits the natural action of the symmetric group on the syzygy modules. The resolution is obtained over an arbitrary coefficient ring; in particular, it is characteristic free. Two applications are given: an equivariant resolution of De Concini–Procesi rings indexed by hook partitions, and a resolution of FI-modules.

We compute the resultants for quadratic binomial complete intersections. As an application we show that any quadratic binomial complete intersection can have the set of square-free monomials as a vector space basis if the generators are put in a normal form.

A quaternion rational surface is a rational surface generated by two rational space curves via quaternion multiplication. In general, the structure of the graded minimal free resolution of a rational surface is unknown. The goal of this paper is to construct the graded minimal free resolution of a quaternion rational surface generated by two rational space curves. We will provide the explicit formulas for the maps of these graded minimal free resolutions. The approach we take is to utilize the information of the $\mu$-bases of the generating rational curves, and create the generating sets for the first and second syzygy modules in the graded minimal free resolutions. In addition, we show that the ideal generated by the first syzygy module expressed in terms of moving planes is exactly the same as the ideal generated by the parametrization in the affine ring.

Dress and Siebeneicher gave a significant generalization of the construction of Witt vectors, by producing for any profinite group $G$, a ring-valued functor ${\mathbf{W}}_G$. This paper gives the first concrete interpretation of any Witt–Burnside rings outside the procyclic cases in terms of known rings. In particular, the rings ${\mathbf{W}}_{{\mathbf{Z}}_p^2}\left(k\right)$, where $k$ is a field of characteristic $p>0$ have a quotient realized as rings of Lipschitz continuous functions on the $p$-adic upper half plane ${\mathbf{P}}^1\left({\mathbf{Q}}_p\right)$. As a consequence we show that the Krull dimensions of the rings ${\mathbf{W}}_{{\mathbf{Z}}_p^d}\left(k\right)$ are infinite for $d\ge 2$ and we show the Teichmüller representatives form an analogue of the van der Put basis for continuous functions on ${\mathbf{Z}}_p$.

Support and rank varieties of modules over a group algebra of an elementary abelian $p$-group have been well studied. In particular, Avrunin and Scott showed that in this setting, the rank and support varieties are equivalent. Avramov and Buchweitz proved an analogous result for pairs of modules over arbitrary commutative local complete intersection rings. In this paper we study support and rank varieties in the triangulated category of totally acyclic chain complexes over a complete intersection ring and show that these varieties are also equivalent.