完备模的同伦理论

投射模之间单同态的同伦范畴

The homotopy category of monomorphisms between projective modules

本文研究交换诺特局部环上由投射模间余核被非零因子零化的单同态构成的范畴，引入其三角同伦范畴并证明可嵌入商环奇点范畴，进而证明几乎分裂序列存在性，刻画Auslander-Reiten平移并发现相关不变性。

Homotopical Aspects of Commutative Algebras II: Freeness Conditions for Quadratic Modules

This article investigates the homotopy theory of simplicial commutative algebras with a view to homological applications.

Compactly generated homotopy categories

Over an associative ring we consider a class $\mathbb{X}$ of left modules which is closed under set-indexed coproducts and direct summands. We investigate when the triangulated homotopy category $\mathsf{K}(\mathbb{X})$ is compactly generated, and give a number of examples.

Some perspective on Homotopy obstructions

Throughout $A$ will denote commutative noetherian ring, with $\dim A=d\geq 2$, and $P$ denote a projective $A$-module with $rank(P)=n$. In \cite{MM1} we considered the Homotopy obstruction sets $π_0\left({\mathcal LO}(P)\right)$, which has a structure of an abelian monoid, under suitable regularity and other conditions. In this article, we provide some perspective on these sets $π_0\left({\mathcal LO}(P)\right)$. Under similar regularity and other conditions, we prove if $P, Q$ are two projective $A$-modules, with $rank(P)=rank(Q)=d$ and $\det(P) \cong \det Q$, then $π_0\left({\mathcal LO}(Q)\…

同伦与交换原理

Homotopy and Commutativity Principle

本文证明了线性、辛群和传递群的交换原理，基于Quillen-Suslin局部全局原理，并在非对称应用下证明了各类群的结构特性。

Vasconcelos' conjecture on the conormal module

For any ideal $I$ of finite projective dimension in a commutative noetherian local ring $R$, we prove that if the conormal module $I/I^2$ has finite projective dimension over $R/I$, then $I$ must be generated by a regular sequence. This resolves a conjecture of Vasconcelos. We prove a similar result for the first Koszul homology module of $I$. When $R$ is a localisation of a polynomial ring over a field $K$ of characteristic zero, Vasconcelos conjectured that $R/I$ is a reduced complete intersection if the module $Ω_{(R/I)/K}$ of differentials has finite projective dimension; we prove this con…

复形的同伦极限

Homotopy limits of complexes

本文在阿贝尔范畴带乘积的复形范畴中提出同伦极限概念，证明其可计算导出极限复形、与极限拟同构，且是同伦上极限的对偶，余局部化子范畴对其稳定。

Complete intersections and equivalences with categories of matrix factorizations

We prove that one can realize certain triangulated subcategories of the singularity category of a complete intersection as homotopy categories of matrix factorizations. Moreover, we prove that for any commutative ring and non-zerodivisor, the homotopy category of matrix factorizations embeds into the homotopy category of totally acyclic complexes of finitely generated projective modules over the factor ring.

Matrix factorizations for self-orthogonal categories of modules

For a commutative ring $S$ and self-orthogonal subcategory $\mathsf{C}$ of $\mathsf{Mod}(S)$, we consider matrix factorizations whose modules belong to $\mathsf{C}$. Let $f\in S$ be a regular element. If $f$ is $M$-regular for every $M\in \mathsf{C}$, we show there is a natural embedding of the homotopy category of $\mathsf{C}$-factorizations of $f$ into a corresponding homotopy category of totally acyclic complexes. Moreover, we prove this is an equivalence if $\mathsf{C}$ is the category of projective or flat-cotorsion $S$-modules. Dually, using divisibility in place of regularity, we observ…