Related papers: The ascent-descent property for $2$-term silting c…
In this paper, we develop 2-dimensional algebraic theory which closely follows the classical theory of modules. The main results are giving definitions of 2-module and the representation of 2-ring. Moreover, for a 2-ring $\cR$, we prove…
For any commutative ring $A$ we introduce a generalization of $S$-noetherian rings using a hereditary torsion theory $\sigma$ instead of a multiplicatively closed subset $S\subseteq{A}$. It is proved that if $A$ is a totally…
We find a necessary and sufficient condition for the existence of the tensor product of modules over a Lie conformal algebra. We provide two algebraic constructions of the tensor product. We show the relation between tensor product and…
We prove a Decomposition Theorem for the direct image of an irreducible local system on a smooth complex projective variety under a morphism with values in another smooth complex projective variety. For this purpose, we construct a category…
Let $X$ be a smooth projective complex curve. We prove that a Torelli type theorem holds, under certain conditions, for the moduli space of $\alpha$-polystable quadratic pairs on $X$ of rank 2.
In this paper we consider birational properties of ramification in excellent local rings. We extend earlier results of the author with Olivier Piltant to show that strong local monomialization is true along a valuation dominating a…
We investigate scalar restriction, scalar extension, and scalar coextension functors for graded modules, including their interplay with coarsening functors, graded tensor products, and graded Hom functors. This leads to several…
Given a non-unit, non-zero-divisor, central element $x$ of a ring $\Lambda$, it is well known that many properties or invariants of $\Lambda$ determine, and are determined by, those of $\Lambda / x \Lambda$ and $\Lambda_x$. In the present…
In this work we introduce a new concept, namely, $\tau_{s}$-extending modules (rings) which is torsion-theoretic analogues of extending modules and then we extend many results from extending modules to this new concept. For instance we show…
Persistence modules are a central algebraic object arising in topological data analysis. The notion of interleaving provides a natural way to measure distances between persistence modules. We consider various classes of persistence modules,…
We study the behavior of modules of $m$-integrable derivations of a commutative finitely generated algebra in the sense of Hasse-Schmidt under base change. We focus on the case of separable ring extensions over a field of positive…
We extend Latimer and MacDuffee's theorem to a general commutative domain and apply this result to study similarity of matrices over integral rings of number fields. We also conjecture similarity over discrete valuation rings can be descent…
In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties…
We prove a modularity lifting theorem for potentially Barostti-Tate representations over totally real fields, generalising recent results of Kisin. Unfortunately, there was an error in the original version of this paper, meaning that we can…
Given a multiplicative subset $S$ in a commutative ring $R$, we consider $S$-weakly cotorsion and $S$-strongly flat $R$-modules, and show that all $R$-modules have $S$-strongly flat covers if and only if all flat $R$-modules are…
We prove a tight connection between reflexive modules over a one-dimensional ring $R$ and its birational extensions that are self-dual as $R$-modules. Consequently, we show that a complete local reduced Arf ring has finitely many…
We compare the obstruction classes defined in arXiv:1101.4069 to those defined by Illusie. We also give sheaf theoretic proofs of some of the standard properties of the cotangent complex.
Motivated by recent progress in topological data analysis, we establish a Matlis duality between injective hulls and flat covers of persistence modules. This extends to a duality between minimal flat and minimal injective resolutions. We…
We characterize Gorenstein modules over those local rings that admit a finite contracting endomorphism.
We give a characteristic free proof of the main result of our previous paper (math.AC/0509697) concerning toroidalization of generating sequences of valuations in dimension two function fields. We show that when an extension of two…