RD-flatness and RD-injectivity

AB5范畴中大量注入性缺失

Injectivity paucity in AB5 categories of oversize chains

本文构造了无非零注入（或射影）对象的AB5范畴例子，结合Rickard的例子与2-函子构造方法，填补了文献中的空白。

关于单项式代数内射性的一个注记

A note on injectivity of monomial algebras

本文研究单项式代数的自内射性，通过正则模的 socle 映射可扩张条件给出等价刻画与完整分类，并证明此类代数均属于 Nakayama 代数。

交换半环上内射性的转置对称性

Transpose Symmetry of Injectivity over Commutative Semirings

该研究证明交换半环上矩阵映射的内射性、可消性及满射性与转置操作相关，还得到单位交换半环矩阵的稳定有限性定理，揭示了半环的乘法单位元与满射方阵存在性的关联。

On (co)pure Baer injective modules

For a given class of R-modules Q, a module M is called Q-copure Baer injective if any map from a Q-copure left ideal of R into M can be extended to a map from R into M. Depending on the class Q, this concept is both a dualization and a generalization of pure Baer injectivity. We show that every module can be embedded as Q-copure submodule of a Q-copure Baer injective module. Certain types of rings are characterized using properties of Q-copure Baer injective modules. For example a ring R is Q-coregular if and only if every Q-copure Baer injective R-module is injective.

Steenrod代数是自内射的，且Steenrod代数不是自内射的

The Steenrod algebra is self-injective, and the Steenrod algebra is not self-injective

本文通过证明Faith定理的分次推广，完整刻画了分次环上哪些分次内射模的余积仍为分次内射，并据此说明Steenrod代数在改变分次或作为非分次环时不再是自内射的。

On subinjectivity domains of pure-injective modules

As an alternative perspective on the injectivity of a pure-injective module, a pure-injective module M is said to be pi-indigent if its subinjectivity domain is smallest possible, namely, consisting of exactly the absolutely pure modules. A module M is called subinjective relative to a module N if for every extension K of N, every homomorphism N \to M can be extended to a homomorphism K \to M. The subinjectivity domain of the module M is defined to be the class of modules N such that M is N-subinjective. Basic properties of the subinjectivity domains of pure-injective modules and of pi-indigen…

局部域上曲线函数域上经典型群的Rost不变量单射性

Rost injectivity for classical groups over function fields of curves over local fields

本文证明了在剩余域为整体域或无实序局部域的完备离散赋值域上，外型A_n的绝对单连通单群的Rost不变量映射是单射，并由此推出局部域上曲线函数域上经典型群的Rost不变量也是单射。

$Σ$-pure injectivity and Brown representability

We prove that a right $R$-module $M$ is $Σ$-pure injective if and only if $\mathrm{Add}(M)\subseteq \mathrm{Prod}(M)$. Consequently, if $R$ is a unital ring, the homotopy category $\mathbf{K}({\mathrm{Mod}\text{-} R})$ satisfies the Brown Representability Theorem if and only if the dual category has the same property. We also apply the main result to provide new characterizations for right pure-semisimple rings or to give a partial positive answer to a question of G. Bergman.

On cotilting cotorsion and torsion pairs

In this paper we study cotorsion and torsion pairs induced by cotilting modules. We prove the existence of a strong relationship between the $Σ$-pure injectivity of the cotilting module and the property of the induced cotorsion pair to be of finite type. In particular for cotilting modules of injective dimension at most 1, or for noetherian rings, the two notions are equivalent. On the other hand we prove that a torsion pair is cogenerated by a $Σ$-pure injective cotilting module if and only if its heart is a locally noetherian Grothendieck category. Moreover we prove that any ring admitting a…

The stable module category of a general ring

For any ring R we construct two triangulated categories, each admitting a functor from R-modules that sends projective and injective modules to 0. When R is a quasi-Frobenius or Gorenstein ring, these triangulated categories agree with each other and with the usual stable module category. Our stable module categories are homotopy categories of Quillen model structures on the category of R-modules. These model categories involve generalizations of Gorenstein projective and injective modules that we derive by replacing finitely presented modules by modules of type FP-infinity. Along the way, we …