Related papers: Cohomological quotients and smashing localizations

We investigate abelian quotients arising from extriangulated categories via morphism categories, which is a unified treatment for both exact categories and triangulated categories. Let $(\mathcal{C},\mathbb{E},\mathfrak{s})$ be an…

Let $(\mathfrak{C},\mathbb{E},\mathfrak{s})$ be an Ext-finite, Krull-Schmidt and $k$-linear extriangulated category with $k$ a commutative artinian ring. We define an additive subcategory $\mathfrak{C}_r$ (respectively, $\mathfrak{C}_l$) of…

Given a commutative noetherian ring $R$ and a finite acyclic quiver $Q$, we study the tensor triangulated category $\mathcal{D}(RQ)$ endowed with the vertexwise tensor product. We find a description of the internal hom functor and show that…

The operation of tensor product of Cohomological Field Theories (or algebras over genus zero moduli operad) introduced in an earlier paper by the authors is described in full detail, and the proof of a theorem on additive relations between…

Let $R$ be a commutative Noetherian ring. We introduce the notion of localization functors $\lambda^W$ with cosupports in arbitrary subsets $W$ of $\text{Spec}\, R$; it is a common generalization of localizations with respect to…

We give a method to realize Verdier quotients as triangulated subfactors of an arbitrary triangulated category. We show that Iyama-Yoshino triangulated subfactors are Verdier quotients under suitable conditions.

A model structure on a category is a formal way of introducing a homotopy theory on that category, and if the model structure is abelian and hereditary, its homotopy category is known to be triangulated. So a good way to both build and…

Given an additive category $\mathcal{C}$ and an integer $n\geqslant 2$. We form a new additive category $\mathcal{C}[\epsilon]^n$ consisting of objects $X$ in $\mathcal{C}$ equipped with an endomorphism $\epsilon_X$ satisfying…

We introduce the notions of Koszul $N$-complex, $\check{\mathrm{C}}$ech $N$-complex and telescope $N$-complex, explicit derived torsion and derived completion functors in the derived category $\mathbf{D}_N(R)$ of $N$-complexes using the…

Consider a monad on an idempotent complete triangulated category with the property that its Eilenberg-Moore category of modules inherits a triangulation. We show that any other triangulated adjunction realizing this monad is 'essentially…

Building on the recent computation of the cohomology rings of smooth toric varieties and partial quotients of moment-angle complexes, we investigate the naturality properties of the resulting isomorphism between the cohomology of such a…

Derived categories were invented by Grothendieck and Verdier around 1960, not very long after the "old" homological algebra (of derived functors between abelian categories) was established. This "new" homological algebra, of derived…

For a connected reductive group $G$ and an affine smooth $G$-variety $X$ over the complex numbers, the localization functor takes $\mathfrak{g}$-modules to $D_X$-modules. We extend this construction to an equivariant and derived setting…

Let K be a comonad on a model category M. We provide conditions under which the associated category of K-coalgebras admits a model category structure such that the forgetful functor to M creates both cofibrations and weak equivalences. We…

Twisted complex $K$-theory can be defined for a space $X$ equipped with a bundle of complex projective spaces, or, equivalently, with a bundle of C$^*$-algebras. Up to equivalence, the twisting corresponds to an element of $H^3(X;\Z)$. We…

This work deals with relations between a bounded cohomological invariant and the geometry of Hermitian symmetric spaces of noncompact type. The invariant, obtained from the K\"ahler class, is used to define and characterize a special class…

We introduce a new algebraic concept of an algebra which is "almost" commutative (more precisely "quasi-commutative differential graded algebra" or ADGQ, in French). We associate to any simplicial set X an ADGQ - called D(X) - and show how…

Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C|=BC. We study the ring homomorphism HH*(kC) --> H*(|C|,k) and prove it is split surjective. This…

Let $R$ be a commutative noetherian local ring with residue field $k$. Denote by $\mathsf{D^b}(R)$ the bounded derived category of finitely generated $R$-modules. In this paper, we study the structure of the Verdier quotient…

We introduce the notion of being cohomologically complete for objects of the derived category of sheaves of $Z[\hbar]$-modules on a topological space. Then we consider a $Z[\hbar]$-algebra satisfying some suitable conditions and prove…