arXiv subjects

Luigi Ferraro

Publications and source records attributed to Luigi Ferraro.

At least 19 recordsLinked to original sources

arXiv2026

Intersection theorems over DG-rings revisited

In this work we generalize two recently proved intersection theorems for DG-rings. The Derived Improved New Intersection Theorem concerns the length of semi-free DG-modules over DG-rings and it was recently proved by the second author. We show that it holds under weaker hypotheses. Foxby's Intersection Theorem was generalized to DG-rings by Yang and we improve the inequality that they provided. As an application we prove a DG version of the classic result that finite length modules of finite projective dimension only exist over Cohen-Macaulay rings, generalizing another result of Yang.

math.AC↗

arXiv2026

Amplitude inequalities for local (co)homology

Peskine and Szpiro's Intersection Theorem for finitely generated modules was generalized by Foxby to all modules and bounded complexes of such. We strengthen Foxby's result and prove a dual result, which for complexes with finitely generated homology recovers Iversen's Amplitude Inequality but also applies to derived complete complexes.

math.AC↗

arXiv2026

The derived depth formula for modules of finite quasi-projective dimension

Let $R$ be a commutative Noetherian local ring. We prove a variety of new formulae for modules of finite quasi-projective or finite quasi-injective dimension. These include the Derived Depth Formula, itself an extension of Auslander famous depth formula, a variation of the Derived Depth Formula for width, an extended version of Ischebeck's Formula, and a Dependency formula in the vein of Jorgensen. Several special cases of our main results are new even under stronger assumptions on the vanishing of various complete intersection dimensions.

math.AC↗

arXiv2025

Quasi-homological dimensions with respect to semidualizing modules

Gheibi, Jorgensen and Takahashi recently introduced the quasi-projective dimension of a module over commutative Noetherian rings, a homological invariant extending the classic projective dimension of a module, and Gheibi later developed the dual notion of quasi-injective dimension. Takahashi and White in 2010 introduced the projective and injective dimension of a module with respect to a semidualizing module, which likewise generalize their classic counterparts. In this paper we unify and extend these theories by defining and studying the quasi-projective and quasi-injective dimension of a module with respect to a semidualizing module. We establish several results generalizing classic formulae such as the Auslander-Buchsbaum formula, Bass' formula, Ischebeck's formula, Auslander's depth formula and Jorgensen's dependency formula. Furthermore, we prove a special case of the Auslander-Reiten conjecture and investigate rigidity properties of Ext and Tor.

math.AC↗

arXiv2025

A DG $\Gamma$-structure on the Generalized Taylor Resolution

Free resolutions of ideals in commutative rings provide valuable insights into the complexity of these ideals. In 1966, Taylor constructed a free resolution for monomial ideals in polynomial rings, which Gemeda later showed admits a differential graded (DG) algebra structure. In 2002, Avramov proved that the Taylor resolution admits a DG algebra structure with divided powers. In 2007, Herzog introduced the generalized Taylor resolution, which is usually smaller than the original Taylor resolution. Recently, in 2023, VandeBogert showed that the generalized Taylor resolution also admits a DG algebra structure. In this paper, we extend Avramov's result by proving that the generalized Taylor resolution admits a DG algebra structure with divided powers. Along the way, we correct a sign error in VandeBogert's product formula. We provide applications concerning homotopy Lie algebras and resolutions of squarefree monomial ideals.

math.AC↗

arXiv2024

The Herzog-Takayama resolution over a skew polynomial ring

Let $\Bbbk$ be a field, and let $I$ be a monomial ideal in the polynomial ring $R=\Bbbk[x_1,\ldots,x_n]$. In her thesis, Taylor introduced a complex that provides a finite free resolution of $R/I$ as an $R$-module. Building on this, Ferraro, Martin and Moore extended this construction to monomial ideals in skew polynomial rings. Since the Taylor resolution is generally not minimal, significant effort has been devoted to identifying classes of ideals with minimal free resolutions that are relatively straightforward to construct. In a 1987 paper, Eliahou and Kervaire developed a minimal free resolution for a class of monomial ideals in $R$ known as stable ideals. This result was later generalized to stable ideals in skew polynomial rings by Ferraro and Hardesty. In a 2002 paper, Herzog and Takayama constructed a minimal free resolution for monomial ideals with linear quotients, a broader class of ideals containing stable ideals. Their resolution reduces to the Eliahou-Kervaire resolution in the stable case. In this paper, we generalize the Herzog-Takayama resolution to skew polynomial rings.

math.RA↗

arXiv2024

Trimming Five Generated Gorenstein Ideals

Let $(R,\mathfrak{m},\Bbbk)$ be a regular local ring of dimension 3. Let $I$ be a Gorenstein ideal of $R$ of grade 3. It follows from a result of Buchsbaum and Eisenbud that there is a skew-symmetric matrix of odd size such that $I$ is generated by the sub-maximal pfaffians of this matrix. Let $J$ be the ideal obtained by multiplying some of the pfaffian generators of $I$ by $\mathfrak{m}$; we say that $J$ is a trimming of $I$. In a previous work, the first author and A. Hardesty constructed an explicit free resolution of $R/J$ and computed a DG algebra structure on this resolution. They utilized these products to analyze the Tor algebra of such trimmed ideals. Missing from their result was the case where $I$ is five generated. In this paper we address this case.

math.AC↗

arXiv2022

The Improved New Intersection Theorem revisited

We prove a generalized version of Evans and Griffith's Improved New Intersection Theorem: Let I be an ideal in a local ring R. If a finite free R-complex, concentrated in nonnegative degrees, has I-torsion homology in positive degrees, and the homology in degree 0 has an I-torsion minimal generator, then the length of the complex is at least dim R - dim R/I. This improves the bound ht I obtained by Avramov, Iyengar, and Neeman in 2018.

math.AC↗

arXiv2022

The homotopy Lie algebra of a Tor-independent tensor product

In this article we investigate a pair of surjective local ring maps $S_1\leftarrow R\to S_2$ and their relation to the canonical projection $R\to S_1\otimes_R S_2$, where $S_1,S_2$ are Tor-independent over $R$. Our main result asserts a structural connection between the homotopy Lie algebra of $S:=S_1\otimes_R S_2$, denoted $π(S)$, in terms of those of $R,S_1$ and $S_2$. Namely, $π(S)$ is the pullback of (adjusted) Lie algebras along the maps $π(S_i)\to π(R)$ in various cases, including when the maps above have residual characteristic zero. Consequences to the main theorem include structural results on André-Quillen cohomology, stable cohomology, and Tor algebras, as well as an equality relating the Poincaré series of the common residue field of $R,S_1,S_2$ and $S$.

math.AC↗

arXiv2022

The Tor algebra of trimmings of Gorenstein ideals

Let $(R,\mathfrak{m},\Bbbk)$ be a regular local ring of dimension 3. Let $I$ be a Gorenstein ideal of $R$ of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that $I$ is generated by the sub-maximal pfaffians of this matrix. Let $J$ be the ideal obtained by multiplying some of the pfaffian generators of $I$ by $\mathfrak{m}$; we say that $J$ is a trimming of $I$. Building on a recent paper of Vandebogert, we construct an explicit free resolution of $R/J$ and compute a partial DG algebra structure on this resolution. We provide the full DG algebra structure in the appendix. We use the products on this resolution to study the Tor algebra of such trimmed ideals and we use the information obtained to prove that recent conjectures of Christensen, Veliche and Weyman on ideals of class $\mathbf{G}$ hold true in our context. Furthermore, we address the realizability question for ideals of class $\mathbf{G}$.

math.AC↗

arXiv2021

Rigidity of Ext and Tor via flat-cotorsion theory

Let p be a prime ideal in a commutative noetherian ring R and denote by k(p) the residue field of the local ring R_p. We prove that if an R-module M satisfies Ext_R^n(k(p),M) = 0 for some n >= dim R, then Ext_R^i(k(p),M) = 0 holds for all i >= n. This improves a result of Christensen, Iyengar, and Marley by lowering the bound on n. We also improve existing results on Tor-rigidity. This progress is driven by the existence of minimal semi-flat-cotorsion replacements in the derived category as recently proved by Nakamura and Thompson.

math.AC↗

arXiv2021

The Taylor resolution over a skew polynomial ring

Let $\Bbbk$ be a field and let $I$ be a monomial ideal in the polynomial ring $Q=\Bbbk[x_1,\ldots,x_n]$. In her thesis, Taylor introduced a complex which provides a finite free resolution for $Q/I$ as a $Q$-module. Later, Gemeda constructed a differential graded structure on the Taylor resolution. More recently, Avramov showed that this differential graded algebra admits divided powers. We generalize each of these results to monomial ideals in a skew polynomial ring $R$. Under the hypothesis that the skew commuting parameters defining $R$ are roots of unity, we prove as an application that as $I$ varies among all ideals generated by a fixed number of monomials of degree at least two in $R$, there is only a finite number of possibilities for the Poincaré series of $\Bbbk$ over $R/I$ and for the isomorphism classes of the homotopy Lie algebra of $R/I$ in cohomological degree larger or equal to two.

math.RA↗

arXiv2021

The Eliahou-Kervaire resolution over a skew polynomial ring

In a 1987 paper, Eliahou and Kervaire constructed a minimal resolution of a class of monomial ideals in a polynomial ring, called stable ideals. As a consequence of their construction they deduced several homological properties of stable ideals. Furthermore they showed that this resolution admits an associative, graded commutative product that satisfies the Leibniz rule. In this paper we show that their construction can be extended to stable ideals in skew polynomial rings. As a consequence we show that the homological properties of stable ideals proved by Eliahou and Kervaire hold also for stable ideals in skew polynomial rings. Finally we show that the resolution we construct admits a product generalizing the one given by Eliahou and Kervaire in the commutative case.

math.RA↗

arXiv2021

Support varieties over skew complete intersections via derived braided Hochschild cohomology

In this article we study a theory of support varieties over a skew complete intersection $R$, i.e. a skew polynomial ring modulo an ideal generated by a sequence of regular normal elements. We compute the derived braided Hochschild cohomology of $R$ relative to the skew polynomial ring and show its action on $\mathrm{Ext}_R(M,N)$ is noetherian for finitely generated $R$-modules $M$ and $N$ respecting the braiding of $R$. When the parameters defining the skew polynomial ring are roots of unity we use this action to define a support theory. In this setting applications include a proof of the Generalized Auslander-Reiten Conjecture and that $R$ possesses symmetric complexity.

math.RA↗

arXiv2021

Semisimple Reflection Hopf Algebras of Dimension Sixteen

For each nontrivial semisimple Hopf algebra $H$ of dimension sixteen over $\mathbb{C}$, the smallest dimension inner-faithful representation of $H$ acting on a quadratic AS regular algebra $A$ of dimension 2 or 3, homogeneously and preserving the grading, is determined. Each invariant subring $A^H$ is determined. When $A^H$ is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that $H$ is a reflection Hopf algebra for $A$.

math.RA↗

arXiv2020

The InvariantRing package for Macaulay2

We describe a significant update to the existing InvariantRing package for Macaulay2. In addition to expanding and improving the methods of the existing package for actions of finite groups, the updated package adds functionality for computing invariants of diagonal actions of tori and finite abelian groups as well as invariants of arbitrary linearly reductive group actions. The implementation of the package has been completely overhauled with the aim of serving as a unified resource for invariant theory computations in Macaulay2.

math.AC↗

arXiv2020

Simple $\mathbb{Z}$-graded domains of Gelfand-Kirillov dimension two

Let $k$ be an algebraically closed field and $A$ a $\mathbb{Z}$-graded finitely generated simple $k$-algebra which is a domain of Gelfand-Kirillov dimension 2. We show that the category of $\mathbb{Z}$-graded right $A$-modules is equivalent to the category of quasicoherent sheaves on a certain quotient stack. The theory of these simple algebras is closely related to that of a class of generalized Weyl algebras (GWAs). We prove a translation principle for the noncommutative schemes of these GWAs, shedding new light on the classical translation principle for the infinite-dimensional primitive quotients of $U(\mathfrak{sl}_2)$.

math.RA↗

arXiv2019

Three infinite families of reflection Hopf algebras

Let $H$ be a semisimple Hopf algebra acting on an Artin-Schelter regular algebra $A$, homogeneously, inner-faithfully, preserving the grading on $A$, and so that $A$ is an $H$-module algebra. When the fixed subring $A^H$ is also AS regular, thus providing a generalization of the Chevalley-Shephard-Todd Theorem, we say that $H$ is a reflection Hopf algebra for $A$. We show that each of the semisimple Hopf algebras $H_{2n^2}$ of Pansera, and $\mathcal{A}_{4m}$ and $\mathcal{B}_{4m}$ of Masuoka is a reflection Hopf algebra for an AS regular algebra of dimension 2 or 3.

math.RA↗