In this note, we discuss the derived functors of infinite products and homotopy limits. QC(X), the category of quasi-coherent sheaves on a Deligne-Mumford stack X , usually has the property that the derived functors of product vanish after a finite stage. We use this fact to study the convergence of certain homotopy limits and apply it compare the derived category of QC(X) with certain other cl...

Let D be a triangulated category with a cluster tilting subcategory U . The quotient category D/U is abelian; suppose that it has finite global dimension. We show that projection from D to D/U sends cluster tilting subcategories of D to support tilting subcategories of D/U , and that, in turn, support tilting subcategories of D/U can be lifted uniquely to weak cluster tilting subcategories of D.

This survey article on bivariant Kasparov theory and E-theory is mainly intended for readers with a background in homotopical algebra and category theory. We approach both bivariant K-theories via their universal properties and equip them with extra structure such as a tensor product and a triangulated category structure. We discuss the construction of the Baum– Connes assembly map via localisa...

For a weighted projective line, the stable category of its vector bundles modulo line bundles has a natural triangulated structure. We prove that, for any positive integers p, q, r and r′ with r′ r, there is an explicit recollement of the stable category of vector bundles on a weighted projective line of weight type (p, q, r) relative to the ones on weighted projective lines of weight types (p,...

For any natural number $n \geq 2$, we construct a triangulated monoidal category whose Grothendieck ring is isomorphic to the of cyclotomic integers $\mathbb{O}_n$.

For a finite group scheme G, we continue our investigation of those finite dimensional kG-modules which are of constant Jordan type. We introduce a Quillen exact category structure C(kG) on these modules and investigate K0(C(kG)). We study which Jordan types can be realized as the Jordan types of (virtual) modules of constant Jordan type. We also briefly consider thickenings of C(kG) inside the...

The discovery in July 2020 of fully-charm tetraquark led to the need for its theoretical explanation. For investigation such complex four-quark formation, modern mathematical apparatus theory derived categories is used. By representing diquarks as solitonic objects terms sheaves, one can explain measured mass broad resonance tetraquarks consisting di-charmonia.

We prove a modified version of Ravenel’s telescope conjecture. It is shown that every smashing subcategory of the stable homotopy category is generated by a set of maps between finite spectra. This result is based on a new characterization of smashing subcategories, which leads in addition to a classification of these subcategories in terms of the category of finite spectra. The approach presen...

Auslander’s formula shows that any abelian category C is equivalent to the category of coherent functors on C modulo the Serre subcategory of all effaceable functors. We establish a derived version of this equivalence. This amounts to showing that the homotopy category of injective objects of some appropriate Grothendieck abelian category (the category of ind-objects of C) is compactly generate...

A special case of a theorem due to Orlov states that for a hypersurface X ⊂ Pn−1 of degree n given by the equation W = 0, there exists an equivalence between the bounded derived category D(cohX) of coherent sheaves on X and the homotopy category HMF(W ) of graded matrix factorizations. We first give a description of this result, and present some methods for doing calculations with it. In the la...