Related papers: The direct limit closure of perfect complexes
Enochs' conjecture asserts that each covering class of modules (over any fixed ring) has to be closed under direct limits. Although various special cases of the conjecture have been verified, the conjecture remains open in its full…
Projectively coresolved Gorenstein flat modules were introduced recently by Saroch and Stovicek and were shown to be Gorenstein projective. While the relation between Gorenstein projective and Gorenstein flat modules is not well understood,…
Firstly, we compare the bounded derived categories with respect to the pure-exact and the usual exact structures, and describe bounded derived category by pure-projective modules, under a fairly strong assumption on the ring. Then, we study…
We define a filtration indexed by the integers on the tensor product of an integrable highest weight module and a loop module for a quantum affine algebra. We prove that the filtration is either trivial or strictly decreasing and give…
In this note we show that finitely generated unit $O_X[\sigma]$--modules for $X$ regular and $F$--finite have a minimal root (in the sense of [Lyubeznik, F-modules] Definition~3.6). This problem was posed by Lyubeznik and answered by…
It is proved that every commutative ring whose RD-injective modules are $\Sigma$-RD-injective is the product of a pure semi-simple ring and a finite ring. A complete characterization of commutative rings for which each artinian…
We describe Koszul type complexes associated with a linear map from any module to a free module, and vice versa with a linear map from a free module to an arbitrary module, generalizing the classical Koszul complexes. Given a short complex…
We study the classical K\"othe's problem, concerning the structure of non-commutative rings with the property that: ``every left module is a direct sum of cyclic modules". In 1934, K\"othe showed that left modules over Artinian principal…
We define a notion of complexity for modules over infinite groups. We show that if $M$ is a module over the group ring $kG$, and $M$ has complexity $\leq f$ (where $f$ is some complexity function) over some set of finite index subgroups of…
We prove the pro-supersolvable closure of a finitely generated subgroup of the free group is finitely generated. It extends similar results for pro-$p$ closures proved by Ribes-Zalesskii and pro-Nilpotent closures proved by…
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…
It has recently been proved that the class of unital exact C*-algebras is closed under taking reduced amalgamated free products. In particular, this implied that the class of exact discrete groups is closed under taking amalgamated free…
We show that the "profinite direct sum" is a good notion of infinite direct sums for profinite modules having properties similar to direct sums of abstract modules. For example, the profinite direct sum of projective modules is projective,…
Labourie and the author independently showed that a convex real projective structure on an oriented surface of genus at least 2 is equivalent to a conformal structure plus a holomorphic cubic differential U. We analyze the behavior of the…
We show that the virtual second Betti number of a finitely generated, residually free group $G$ is finite if and only if $G$ is either free, free abelian or the fundamental group of a closed surface. We also prove a similar statement in…
The higher direct image complex of a coherent sheaf (or finite complex of coherent sheaves) under a projective morphism is a fundamental construction that can be defined via a Cech complex or an injective resolution, both inherently…
It is shown that, if $R$ is either an Artin algebra or a commutative noetherian domain of Krull dimension $1$, then infinite direct products of $R$-modules resist direct sum decomposition as follows: If $(M_n)_{n \in \Bbb N}$ is a family of…
We prove a degree-one saving bound for the dimension of the space of cohomological automorphic forms of fixed level and growing weight on $\mathrm{SL}_2$ over any number field that is not totally real. In particular, we establish a sharp…
In 2006, Gao and Zeng \cite{GZ} gave the free field realizations of highest weight modules over a class of extended affine Lie algebras. In the present paper, applying the technique of localization to those free field realizations, we…
We show that every finitely generated group admits weak analogues of an invariant expectation, whose existence characterizes exact groups. This fact has a number of applications. We show that Hopf $G$-modules are relatively injective, which…