DG quotients of DG categories

On the structure of Calabi-Yau categories with a cluster tilting subcategory

We prove that for $d \geq 2$, an algebraic $d$-Calabi-Yau triangulated category endowed with a $d$-cluster tilting subcategory is the stable category of a DG category which is perfectly $(d+1)$-Calabi-Yau and carries a non degenerate $t$-structure whose heart has enough projectives.

Quotients of exact categories by pseudo-cluster tilting subcategories

We introduce the concept of a pseudo-cluster tilting subcategory from the viewpoint of the fact that the quotient of an exact category by a cluster tilting subcategory is an abelian category. We prove that the quotients in the case of pseudo-cluster tilting are always semi-abelian. In addition, it is abelian if and only if some self-orthogonal conditions are satisfied. We revisit the abelian quotient category of conflations by splitting ones, and get that there exists a unique exact substructure such that it is a cluster quotient.

Refining thick subcategory theorems

We use a $K$-theory recipe of Thomason to obtain classifications of triangulated subcategories via refining some standard thick subcategory theorems. We apply this recipe to the full subcategories of finite objects in the derived categories of rings and the stable homotopy category of spectra. This gives, in the derived categories, a complete classification of the triangulated subcategories of perfect complexes over some commutative rings. In the stable homotoy category of spectra we obtain only a partial classification of the triangulated subcategories of the finite $p$-local spectra. We use …

Galois理论与拟阿贝尔函子范畴中的同调

Galois theory and homology in quasi-abelian functor categories

本文研究有限范畴T上的函子范畴[T,A]（A为拟阿贝尔范畴），定义并证明其子范畴F为无挠Birkhoff子范畴，从而用于研究中心扩张和同调的广义Hopf公式。

Preresolving categories and derived equivalences

It is well known that a resolving subcategory $\mathcal{A}$ of an abelian subcategory $\mathcal{E}$ induces several derived equivalences: a triangle equivalence $\mathbf{D}^-(\mathcal{A})\to \mathbf{D}^-(\mathcal{E})$ exists in general and furthermore restricts to a triangle equivalence $\mathbf{D}^{\mathsf{b}}(\mathcal{A})\to \mathbf{D}^{\mathsf{b}}(\mathcal{E})$ if $\operatorname{res.dim}_{\mathcal{A}}(E)<\infty$ for any object $E\in \mathcal{E}$. If the category $\mathcal{E}$ is uniformly bounded, i.e. $\operatorname{res.dim}_{\mathcal{A}}(\mathcal{E})<\infty$, one obtains a triangle equiva…

Triangulated categories without models

We exhibit examples of triangulated categories which are neither the stable category of a Frobenius category nor a full triangulated subcategory of the homotopy category of a stable model category. Even more drastically, our examples do not admit any non-trivial exact functors to or from these algebraic respectively topological triangulated categories.