Related papers: Stratifying derived categories of cochains on cert…
For a commutative Noetherian ring $R$ with finite Krull dimension, we study the nullity classes in $D^c_{fg}(R)$, the full triangulated subcategory $D^c_{fg}(R)$ of the derived category $D(R)$ consisting of objects which can be represented…
We develop a theory of cosupport and costratification in tensor triangular geometry. We study the geometric relationship between support and cosupport, provide a conceptual foundation for cosupport as categorically dual to support, and…
We introduce a new topological invariant of a rigidly-compactly generated tensor-triangulated category and two new notions of support. The first is based on smashing subcategories: it is unknown whether the frame of smashing subcategories…
We establish conditions on a family of coproduct-preserving tt-functors $f_i\colon \mathcal{T}\to \mathcal{T}_i$ between tt-categories with small coproducts, ensuring that the localizing tensor ideal generated by an object $x \in…
For any compactly generated triangulated category we introduce two topological spaces, the shift-spectrum and the shift-homological spectrum. We use them to parametrise a family of thick subcategories of the compact objects, which we call…
We construct a spectral sequence associated to a stratified space, which computes the compactly supported cohomology groups of an open stratum in terms of the compactly supported cohomology groups of closed strata and the reduced cohomology…
We define the Chow ring of the classifying space of a linear algebraic group. In all the examples where we can compute it, such as the symmetric groups and the orthogonal groups, it is isomorphic to a natural quotient of the complex…
By a theorem of A.Bj\"orner, for every interval $[u,v]$ in the Bruhat order of a Coxeter group $W$, there exists a stratified space whose strata are labeled by the elements of $[u,v]$, adjacency is described by the Bruhat order, and each…
In this paper we show that, besides the usual calculus involving K\"ahler differentials, it is also possible to define conical calculus on schemes and perfectoid spaces; this can be done via a stratification process. Following some ideas…
We systematically develop a theory of stratification in the context of tensor triangular geometry and apply it to classify the localizing tensor-ideals of certain categories of spectral $G$-Mackey functors for all finite groups $G$. Our…
We classify localising subcategories of the stable module category of a finite group that are closed under tensor product with simple (or, equivalently all) modules. One application is a proof of the telescope conjecture in this context.…
This paper systematically develops a notion of regular sequences in the context of $R$-linear triangulated categories for a graded-commutative ring $R$. The notion has equivalent characterizations involving Koszul objects and local…
We construct a pairing, which we call factorization homology, between framed manifolds and higher categories. The essential geometric notion is that of a vari-framing of a stratified manifold, which is a framing on each stratum together…
Recollements of triangulated categories may be seen as exact sequences of such categories. Iterated recollements of triangulated categories are analogues of geometric or topological stratifications and of composition series of algebraic…
We obtain a theorem which allows to prove compact generation of derived categories of Grothendieck categories, based upon certain coverings by localizations. This theorem follows from an application of Rouquier's cocovering theorem in the…
Mutations occur in multiple algebraic contexts, often enjoying good combinatorial properties. In this paper we study mutations of pure-injective cosilting objects in compactly generated triangulated categories from a topological point of…
We classify complactly generated t-structures on the derived category of modules over a commutative Noetherian ring R in terms of decreasing filtrations by supports on Spec(R). A decreasing filtration by supports \phi : Z -> Spec(R)…
In this paper, we prove a version of Freyd's generating hypothesis for triangulated categories: if D is a cocomplete triangulated category and S is an object in D whose endomorphism ring is graded commutative and concentrated in degree…
We construct some new cohomology theories for topological groups and Lie groups and study some of its basic properties. For example, we introduce a cohomology theory based on measurable cochains which are continuous in a neighbourhood of…
This work concerns representations of a finite flat group scheme $G$, defined over a noetherian commutative ring $R$. The focus is on lattices, namely, finitely generated $G$-modules that are projective as $R$-modules, and on the full…