Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2210.16342

Equivariant resolutions over Veronese rings

Ayah Almousa·Michael Perlman·Alexandra Pevzner·Victor Reiner·Keller VandeBogert
View original source ↗Read source document ↗Export metadata ↓

Abstract

Working in a polynomial ring $S=\mathbf{k}[x_1,\ldots,x_n]$ where $\mathbf{k}$ is an arbitrary commutative ring with $1$, we consider the $d^{th}$ Veronese subalgebras $R=S^{(d)}$, as well as natural $R$-submodules $M=S^{(\geq r, d)}$ inside $S$. We develop and use characteristic-free theory of Schur functors associated to ribbon skew diagrams as a tool to construct simple $GL_n(\mathbf{k})$-equivariant minimal free $R$-resolutions for the quotient ring $\mathbf{k}=R/R_+$ and for these modules $M$. These also lead to elegant descriptions of $\mathrm{Tor}^R_i(M,M')$ for all $i$ and $\mathrm{Hom}_R(M,M')$ for any pair of these modules $M,M'$.

Explore related subjects

math.ACmath.CO

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ayah Almousa, Michael Perlman, Alexandra Pevzner, Victor Reiner, Keller VandeBogert. 2023-11-23. Equivariant resolutions over Veronese rings. https://doi.org/10.1112/jlms.12848

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Asymptotic $\mathrm{v}$-number of graded families of ideals and the Newton-Okounkov region

In this paper, we prove that for Noetherian graded families $\mathcal{I} = \{I_k\}_{k \ge 0}$ of homogeneous ideals in a Noetherian $\mathbb{N}$-graded Noetherian domain, $\lim\limits_{k \to \infty} \frac{\mathrm{v}(I_k)}{k}$ exists, and is given by $\frac{α(I_r)}{r}$ for some $r \ge 1$, where $α(I)$ denotes the initial degree. Extending these results to integral closures, we show that $ \lim\limits_{k\to\infty}\frac{\mathrm{v}(\overline{I_k})}{k} = \lim\limits_{k\to\infty}\frac{α(\overline{I_k})}{k}=\lim\limits_{k\to\infty}\frac{\mathrm{v}(I_k)}{k}=\lim\limits_{k\to\infty}\frac{α(I_k)}{k} $. For a graded family of monomial ideals in a polynomial ring, we provide a combinatorial interpretation of these limits via Newton--Okounkov regions $Δ(\mathcal{I})$. This connection is further generalized to arbitrary homogeneous ideals using good valuations. We also establish that both $\operatorname{reg}(I_k)$ and $\mathrm{v}(I_k)$ are eventually quasi-linear functions of $k$ for any Noetherian graded family. For a stable monomial ideal $I$ we show that $\mathrm{v}(I) < \operatorname{reg}(I)$. Finally, for a zero-dimensional homogeneous ideal $I$ in a polynomial ring $S$, we prove that $\mathrm{v}(I) < e(S/I)$, where $e(S/I)$ denotes the multiplicity.

math.AC↗
arXiv2026

Border Bases and Border Basis Schemes

This survey invites the readers on a journey spanning more than twenty years of research through the landscape of border basis schemes. During most of this period, I had the pleasure of working with Martin Kreuzer, and more recently with Le Ngoc Long. Along this journey, one encounters border bases, which are characterized by the remarkable property that their associated multiplication matrices commute pairwise. This property, in turn, provides a natural foundation for defining border basis schemes (BBS). These are beautiful schemes, elegantly defined by simple quadratic equations. However, the number of indeterminates in their coordinate rings can be huge. This necessitates a suitable re-embedding into a polynomial ring with fewer indeterminates. This task is accomplished in a more general setting, and the reward is reaped in the BBS scenario, where the notions of cotangent equivalence and exposed indeterminates play a fundamental role, for instance, in showing that planar Box BBS are affine cells. The center stage is taken by positive $P_0$-algebras and the unimodular matrix problem, which allows us to prove that regular algebras of this kind are free. Finally, we turn our attention to special BBS and interesting subschemes of BBS. Are there no more open problems? Fortunately, many remain, and a selection of these challenges marks not a final destination, but a new horizon, suggesting that this journey may continue in the near future. on the border of the soul bases of unseen photographs, schemes of ancient thoughts, slowly return into poems and theorems L. Robbiano, 2026}

math.AC↗
arXiv2026

On a question of Nori: The high-codimension conductor case

Let $R$ be an affine domain of dimension $d$ over an infinite field, with normalisation $\tilde{R}$ smooth, and let $\mathfrak{c}$ be the conductor. We answer a question of Nori in the affirmative when $R$ is singular only in high codimension.

math.AC↗