Finite flat and projective dimension

张量环上的（投射余分解）Gorenstein平坦模

(projectively coresolved) Gorenstein flat modules over tensor rings

本文针对张量环结构，刻画了其上的投射余分解Gorenstein平坦模的等价判定条件，明确描述了一类相关平坦模，并给出了其在两类环结构上的应用。

(n,m)-Strongly Gorenstein Projective Modules

This paper is a continuation of the papers J. Pure Appl. Algebra, 210 (2007), 437--445 and J. Algebra Appl., 8 (2009), 219--227. Namely, we introduce and study a doubly filtered set of classes of modules of finite Gorenstein projective dimension, which are called $(n,m)$-strongly Gorenstein projective ($(n,m)$-SG-projective for short) for integers $n\geq 1$ and $m\geq 0$. We are mainly interested in studying syzygies of these modules. As consequences, we show that a module $M$ has Gorenstein projective dimension at most $m$ if and only if $M\oplus G$ is $(1,m)$-SG-projective for some Gorenstei…

Rings over which every module has a flat $δ$-cover

Let $M$ be a module. A {\em $δ$-cover} of $M$ is an epimorphism from a module $F$ onto $M$ with a $δ$-small kernel. A $δ$-cover is said to be a {\em flat $δ$-cover} in case $F$ is a flat module. In the present paper, we investigate some properties of (flat) $δ$-covers and flat modules having a projective $δ$-cover. Moreover, we study rings over which every module has a flat $δ$-cover and call them {\em right generalized $δ$-perfect} rings. We also give some characterizations of $δ$-semiperfect and $δ$-perfect rings in terms of locally (finitely, quasi-, direct-) projective $δ$-covers and flat …

Stability of strongly Gorenstein flat modules

A left $R$-module $M$ is called two-degree Ding projective if there exists an exact sequence $...\longrightarrow D_{1}\longrightarrow D_{0}\longrightarrow D_{-1}\longrightarrow D_{-2}\longrightarrow...$ of Ding projective left $R$-modules such that $M\cong\ker (D_{0}\longrightarrow D_{-1})$ and $\Hom_{R} (-, F)$ leaves the sequence exact for any flat (or Gorenstein flat) left $R$-module $F$. In this paper, we show that the two-degree Ding projective modules are nothing more than the Ding projective modules.

当每个S-平坦模都是（平坦）投射模时

When every S-flat module is (flat) projective

本文研究含单位元交换环中S-平坦模均为平坦或投射模的环类，给出两类环的刻画，定义S-完全环，举例区分三类环并研究S-完全性的环构造传递性。

大族无限群上的Gorenstein模与维数

Gorenstein modules and dimension over large families of infinite groups

该研究刻画了大族无限群的群代数上的三类Gorenstein模，证明弱Gorenstein模与对应Gorenstein模的等价性，得到有限Gorenstein弱整体维数交换环上的相关结论，并确定了特定群的Gorenstein同调维数。

The cotorsion pair generated by the class of flat Mittag-Leffler modules

Let $R$ be a ring and denote by $\mathcal{FM}$ the class of all flat and Mittag-Leffler left $R$-modules. In \cite{BazzoniStovicek2} it is proved that, if $R$ is countable, the orthogonal class of $\mathcal{FM}$ consists of all cotorsion modules. In this note we extend this result to the class of all rings $R$ satisfying that each flat left $R$-module is filtered by totally ordered limits of projective modules. This class of rings contains all countable, left perfect and discrete valuation domains. Moreover, assuming that there do not exist inaccessible cardinals, we obtain that, over these ri…

Neat-Flat Modules

Let $R$ be a ring and $M$ be a right $R$-module. $M$ is called neat-flat if any short exact sequence of the form $0\to K\to N\to M\to 0$ is neat-exact i.e. any homomorphism from a simple right $R$-module $S$ to $M$ can be lifted to $N$. We prove that, a module is neat-flat if and only if it is simple-projective. Neat-flat right $R$-modules are projective if and only if $R$ is a right $\sum$-$CS$ ring. Every finitely generated neat-flat right $R$-module is projective if and only if $R$ is a right $C$-ring and every finitely generated free right $R$-module is extending. Every cyclic neat-flat ri…

环的经典与Gorenstein同调不变量

On classical and Gorenstein homological invariants of rings

本文证明任意环R上平坦左R-模的投射维数上确界等于Gorenstein平坦左R-模的Gorenstein投射维数上确界，由此得到左n-完全环的新刻画，并引入经典同调不变量的Gorenstein类比，研究其关系，最后给出左弱n-Σ-余挠环的Gorenstein类刻画。

Projective covers of flat contramodules

We show that a direct limit of projective contramodules (over a right linear topological ring) is projective if it has a projective cover. A similar result is obtained for $\infty$-strictly flat contramodules of projective dimension not exceeding $1$, using an argument based on the notion of the topological Jacobson radical. Covers and precovers of direct limits of more general classes of objects, both in abelian categories with exact and with nonexact direct limits, are also discussed, with an eye towards the Enochs conjecture about covers and direct limits, using locally split (mono)morphism…

Graded comodule categories with enough projectives

It is well-known that the category of comodules over a flat Hopf algebroid is abelian but typically fails to have enough projectives, and more generally, the category of graded comodules over a graded flat Hopf algebroid is abelian but typically fails to have enough projectives. In this short paper we prove that the category of connective graded comodules over a connective, graded, flat, finite-type Hopf algebroid has enough projectives. Applications to algebraic topology are given: the Hopf algebroids of stable co-operations in complex bordism, Brown-Peterson homology, and classical mod $p$ h…

On Gorenstein Projective, Injective and Flat Dimensions - A Functorial Description with Applications

Gorenstein homological dimensions are refinements of the classical homological dimensions, and finiteness singles out modules with amenable properties reflecting those of modules over Gorenstein rings. As opposed to their classical counterparts, these dimensions do not immediately come with practical and robust criteria for finiteness, not even over commutative noetherian local rings. In this paper we enlarge the class of rings known to admit good criteria for finiteness of Gorenstein dimensions: It now includes, for instance, the rings encountered in commutative algebraic geometry and, in the…

Neeman 平坦模与投射模定理的反模推广

A contramodule generalization of Neeman's flat and projective module theorem

本文将 Neeman 及 Bazzoni 等人的平坦模与投射模定理推广到完备分离拓扑环上的反模范畴，证明了投射反模同伦范畴、平坦反模导出范畴与平坦余挠反模同伦范畴三者等价，并给出反零调性的若干等价刻画。

Homological dimension based on a class of Gorenstein flat modules

In this paper, we study the relative homological dimension based on the class of projectively coresolved Gorenstein flat modules (PGF-modules), that were introduced by Saroch and Stovicek. The resulting PGF-dimension of modules has several properties in common with the Gorenstein projective dimension, the relative homological theory based on the class of Gorenstein projective modules. In particular, there is a hereditary Hovey triple in the category of modules of finite PGF-dimension, whose associated homotopy category is triangulated equivalent to the stable category of PGF-modules. Studying …

项目性忠实平坦降级在Lean中的形式化

Formalization in Lean of faithfully flat descent of projectivity

本文在Lean中形式化了项目性忠实平坦降级的理论，修正了Raynaud和Gruson的经典结果，并为研究交换诺特环的有限制维度提供了关键工具。

Almost free modules and Mittag--Leffler conditions

Drinfeld recently suggested to replace projective modules by the flat Mittag--Leffler ones in the definition of an infinite dimensional vector bundle on a scheme $X$. Two questions arise: (1) What is the structure of the class $\mathcal D$ of all flat Mittag--Leffler modules over a general ring? (2) Can flat Mittag--Leffler modules be used to build a Quillen model category structure on the category of all chain complexes of quasi--coherent sheaves on $X$? We answer (1) by showing that a module $M$ is flat Mittag--Leffler, if and only if $M$ is $\aleph_1$--projective in the sense of Eklof and M…

Homology theories and Gorenstein dimensions for complexes

In this paper, we first study the Gorenstein projective/flat dimension of complexes of modules. The relation between the Gorenstein projective/flat dimension for complexes and that for modules are investigated. Then we study Tate, stable and unbounded homology for complexes of modules. In the case of module arguments, we get some results that improve the known results.