Presentations of algebras in chain complexes

Degreewise n-projective and n-flat model structures on chain complexes

In the paper "Cotorsion Pairs in C(R-Mod)", the authors construct an abelian model structure on the category of chain complexes Ch(R), where the class of cofibrant objects is given by the class of degreewise projective chain complexes. Using a generalization of a well known theorem by I. Kaplansky, we generalize the method used in that paper in order to obtain, for each integer n>1, a new abelian model structure on Ch(R), where the class of cofibrant objects is the class of chain complexes whose terms have projective dimension smaller or equal than n (dw-n-projective complexes), provided the r…

Simple omega-categories and chain complexes

The category of strict omega-categories has an important full subcategory whose objects are the simple omega-categories freely generated by planar trees or by globular cardinals. We give a simple description of this subcategory in terms of chain complexes, and we give a similar description of the opposite category, the category of finite discs, in terms of cochain complexes. Berger has shown that the category of simple omega-categories has a filtration by iterated wreath products of the simplex category. We generalise his result by considering wreath products of categories of chain complexes o…

Homological dimensions and abelian model structures on chain complexes

We construct Abelian model structures on the category of chain complexes over a ring $R$, from the notion homological dimensions of modules. Given an integer $n > 0$, we prove that the left modules over a ringoid $\mathfrak{R}$ with projective dimension at most $n$ form the left half of a complete cotorsion pair. Using this result we prove that there is a unique Abelian model structure on the category of chain complexes where the exact complexes are the trivial objects and the complexes with projective dimension at most $n$ form the class of trivially cofibrant objects. In the paper "Cotorsion…

T-model structures on chain complexes of presheaves

A tensor model structure is constructed on the category of chain complexes of presheaves of R-modules for a sheaf of rings R in a Grothendieck topos. If the topos has enough points, then the homotopy category is equivalent to the derived category. Some t-structures, generalizing the perverse t-structures, are constructed on the derived category.

Simplicial Structure on Complexes

While chain complexes are equipped with a differential $d$ satisfying $d^2 = 0$, their generalizations called $N$-complexes have a differential $d$ satisfying $d^N = 0$. In this paper we show that the lax nerve of the category of chain complexes is pointwise adjoint equivalent to the décalage of the simplicial category of $N$-complexes. This reveals additional simplicial structure on the lax nerve of the category of chain complexes which provides a categorfication of the triangulated homotopy category of chain complexes. We study this phenomena in general and present evidence that the axioms o…

Adjointness properties for relative extensions of disk and sphere chain complexes

We study the subgroup $\mathcal{E}xt^i_{\mathcal{C}}(\mathcal{F}; C, D)$ of $\mathcal{E}xt^i_{\mathcal{C}}(C,D)$ formed by those $i$-extensions of $C$ by $D$ in an Abelian category $\mathcal{C}$ which are ${\rm Hom}_{\mathcal{C}}(\mathcal{F},-)$-exact, and present a Baer-like description of this subgroup in terms of certain right derived functors of ${\rm Hom}_{\mathcal{C}}(-,-)$. We also study adjointness properties of these subgroups and the disk and sphere chain complex functors $\mathcal{C} \longrightarrow \textbf{Ch}(\mathcal{C})$, given by a collection of natural isomorphisms which gener…

K-flat complexes and derived categories

Let $R$ be a ring with identity. Inspired by recent work of Emmanouil, we show that the derived category of $R$ is equivalent to the chain homotopy category of all K-flat complexes with pure-injective components. This is implicitly related to a recollement we exhibit. It expresses $\mathcal{D}_{pur}(R)$, the pure derived category of $R$, as an attachment of the usual derived category $\mathcal{D}(R)$ with Emmanouil's quotient category $\mathcal{D}_{K-flat}(R):=K(R)/K-Flat$, which here we call the K-flat derived category. It follows that this Verdier quotient is a compactly generated triangulat…

具有支撑的链复形的导出等价

Derived equivalences for chain complexes with support

本文研究了具有有限A-维度支撑的链复形的导出等价，并探讨了其在K理论中的同伦纤维描述。

N-复形范畴上的结构

Structures on the Category of N-Complexes

该研究将链复形和分次对象的张量积推广到$N$-复形情形，探究了$N$-复形导出范畴的多种实现方式，丰富了$N$-复形理论的范畴结构研究。

Simplicial Descent Categories

Much of the homotopical and homological structure of the categories of chain complexes and topological spaces can be deduced from the existence and properties of the 'simple' functors Tot : {double chain complexes} -> {chain complexes} and geometric realization : {sSets} -> {Top}, or similarly, Tot : {simplicial chain complexes} -> {chain complexes} and | | : {sTop} -> {Top}. The purpose of this thesis is to abstract this situation, and to this end we introduce the notion of '(co)simplicial descent category'. It is inspired by Guillen-Navarros's '(cubical) descent categories'. The key ingredie…

相对无环复形的链同伦范畴的模型

Models for chain homotopy category of relative acyclic complexes

本文利用 Abel 范畴中的平衡对，在温和条件下把两类相对无环复形的链同伦范畴实现为模型范畴同伦范畴，并导出相对重配及 Gorenstein 模应用。

Cotorsion Pairs and Quillen Adjunctions

Let $F:\mathcal{A}\to \mathcal{B}$ be a left adjoint between abelian categories and let $Ch(F)$ be the induced left adjoint on chain complexes. If the abelian categories $\mathcal{A}$ and $\mathcal{B}$ are equipped with sufficiently nice cotorsion pairs, then we find model structures on their categories of chain complexes. This paper gives novel criteria under which the induced adjunction becomes a Quillen adjunction. In particular, our criteria can be checked on the level of the underlying abelian categories instead of chain complexes. We do the same for adjunctions of $n$ variables and we al…

Model structures and relative Gorenstein flat modules and chain complexes

A recent result by J. Šaroch and J. Šťov\'ıček asserts that there is a unique abelian model structure on the category of left $R$-modules, for any associative ring $R$ with identity, whose (trivially) cofibrant and (trivially) fibrant objects are given by the classes of Gorenstein flat (resp., flat) and cotorsion (resp., Gorenstein cotorsion) modules. In this paper, we generalise this result to a certain relativisation of Gorenstein flat modules, which we call Gorenstein $\mathcal{B}$-flat modules, where $\mathcal{B}$ is a class of right $R$-modules. Using some of the techniques considered by …