arXiv subjects
Publications and source records attributed to Andrew J. Soto Levins.
Let R be a commutative noetherian local ring and M and N be R-complexes with finitely generated homology. We prove that formulas for the homological dimensions of the derived Hom complex RHom(M,N), in terms of the homological dimensions of M and N, hold without the a priori assumptions on the dimensions of M or N present in classic results.
Foundational work by Avramov and Buchweitz, as well as by Huneke and Jorgensen, established symmetry results for the vanishing of Ext over local complete intersections and AB-rings, respectively. Later, several authors studied symmetry in the vanishing of Ext over local Gorenstein rings under additional homological assumptions on the modules involved. In this work, we provide a generalized symmetry in the vanishing of Ext for homologically finite complexes over a Noetherian ring with a dualizing complex, under additional homological conditions on the complexes involved. The results in this paper extend and unify several symmetry results on the vanishing of Ext previously proved over local Gorenstein rings and show that the dualizing complex has been hidden in these results.
We introduce and study a notion of large homomorphisms on the homotopy lie coalgebra; these homomorphisms are a variant of the large homomorphisms of Levin. As a consequence of our work, we establish new cases of a homotopy lie coalgebra analog of a conjecture of Quillen as proposed by Briggs.
We introduce quasi-Gorenstein morphisms of commutative local dg-algebras and use a Gorenstein version of the virtually small property to characterize them, a result which is new even for homomorphisms of local rings. In a different direction, we characterize exact sequences in a noetherian local ring, in the sense of Avramov, Henriques, and Şega, in terms of quasi-Gorenstein morphisms involving Koszul complexes.
We prove that a commutative noetherian ring $R$ is Gorenstein of dimension at most $d$ if $d+1$ is an upper bound on the G-levels of perfect $R$-complexes. For $R$ local, we prove a formula for levels, with respect to injective or Gorenstein injective $R$-modules, of $R$-complexes with finitely generated homology; it mimics Bass' classic formula for injective dimension of finitely generated $R$-modules.
This paper is concerned with lifting modules along a surjective map of noetherian local rings, say $φ\colon R \twoheadrightarrow S$. A finitely generated $R$-module $L$ is a naive lift of an $S$-module $M$ if $L \otimes_R S \cong M$. We are concerned with the maximum depth and dimension among all naive lifts of $M$, which we call the liftable depth and liftable dimension, respectively, of $M$ along $φ$. We approach this via a notion of lifting systems that we introduce in this paper. We then provide a necessary and sufficient condition for a module of finite length to lift and Serre lift to a regular local ring in terms of lifting systems.
If a module $M$ has finite projective dimension, then the Ext modules of $M$ against any other module eventually vanish and the projective dimension of $M$ gives a uniform bound for this vanishing. For modules of infinite projective dimension there can still exist a bound. Such a bound is called the Auslander bound. In this paper we define a similar bound for complexes and give applications.
We describe cohomological conditions that are necessary and sufficient for the existence of balanced dualizing dg-modules, generalizing a theorem of Van den Bergh for balanced dualizing complexes over graded algebras. As a consequence, we show that a dg-algebra satisfying certain finiteness conditions admits a balanced dualizing dg-module if and only if its zeroth cohomology algebra admits a balanced dualizing complex. Additionally, we obtain a host of new examples of dg-algebras whose associated noncommutative spaces satisfy Serre duality.
The Euler form is an Ext analog of the Euler characteristic, and in this paper we study the Euler form and give some applications. The first being a question of Jorgensen, which bounds the projective dimension of a module over a complete intersection by using the vanishing of self extensions. Our second application uses the Euler form to yield a new result involving the vanishing of the higher Herbrand difference. Along the way we translate some of our results to the graded setting.
We introduce and study a notion of Serre liftable modules; these are modules that are liftable to modules of the maximal possible dimension over a regular local ring. We establish new cases of Serre's positivity conjecture over ramified regular local rings by proving it for Serre liftable modules. Furthermore, we show that the length of a nonzero Serre liftable module is bounded below by the Hilbert-Samuel multiplicity of the local ring. This establishes special cases of the Length Conjecture of Iyengar-Ma-Walker.
Let $R$ be a regular local ring containing a field, let $I$ be an ideal with $d=\text{ht}{I}$, and assume $\text{ht}{p}=d$ for every minimal prime $p$ of $I$. We compute the Bass numbers $μ^{0}(q,H_{I}^{d}(R))$ and $μ^{1}(q,H_{I}^{d}(R))$ for all primes $q$. We then study $μ^{2}(q,H_{I}^{d}(R))$ by considering the associated primes of $H_{I}^{d+1}(R)$.