No Arabic abstract
The goal of the article is to better understand cosupport in triangulated categories since it is still quite mysterious. We study boundedness of local cohomology and local homology functors using Koszul objects, give some characterizations of cosupport and get some results that, in special cases, recover and generalize the known results about the usual cosupport. Also we include some computations of cosupport, settle the comparison of support and cosupport of cohomologically finite objects. Finally, we assign to any object of the category a subset of $mathrm{Spec}R$, called the big cosupport.
We show that a well behaved Noetherian, finite dimensional, stable, monoidal model category is equivalent to a model built from categories of modules over completed rings in an adelic fashion. For abelian groups this is based on the Hasse square, for chromatic homotopy theory this is based on the chromatic fracture square, and for rational torus-equivariant homotopy theory this is the model of Greenlees-Shipley arXiv:1101.2511.
Given a suitable stable monoidal model category $mathscr{C}$ and a specialization closed subset $V$ of its Balmer spectrum one can produce a Tate square for decomposing objects into the part supported over $V$ and the part supported over $V^c$ spliced with the Tate object. Using this one can show that $mathscr{C}$ is Quillen equivalent to a model built from the data of local torsion objects, and the splicing data lies in a rather rich category. As an application, we promote the torsion model for the homotopy category of rational circle-equivariant spectra from [18] to a Quillen equivalence. In addition, a close analysis of the one step case highlights important features needed for general torsion models which we will return to in future work.
We study the problem of when triangulated categories admit unique infinity-categorical enhancements. Our results use Luries theory of prestable infinity-categories to give conceptual proofs of, and in many cases strengthen, previous work on the subject by Lunts--Orlov and Canonaco--Stellari. We also give a wide range of examples involving quasi-coherent sheaves, categories of almost modules, and local cohomology to illustrate the theory of prestable infinity-categories. Finally, we propose a theory of stable $n$-categories which would interpolate between triangulated categories and stable infinity-categories.
Given a bounded-above cochain complex of modules over a ring, it is standard to replace it by a projective resolution, and it is classical that doing so can be very useful. Recently, a modified version of this was introduced in triangulated categories other than the derived category of a ring. A triangulated category is emph{approximable} if this modified procedure is possible. Not surprisingly this has proved a powerful tool. For example: the fact that the derived category of a quasi compact, separated scheme is approximable has led to major improvements on old theorems due to Bondal, Van den Bergh and Rouquier. In this article we prove that, under weak hypotheses, the recollement of two approximable triangulated categories is approximable. In particular, this shows many of the triangulated categories that arise in noncommutative algebraic geometry are approximable.
Let $mathcal{C}$ be a triangulated category. We first introduce the notion of balanced pairs in $mathcal{C}$, and then establish the bijective correspondence between balanced pairs and proper classes $xi$ with enough $xi$-projectives and enough $xi$-injectives. Assume that $xi:=xi_{mathcal{X}}=xi^{mathcal{Y}}$ is the proper class induced by a balanced pair $(mathcal{X},mathcal{Y})$. We prove that $(mathcal{C}, mathbb{E}_xi, mathfrak{s}_xi)$ is an extriangulated category. Moreover, it is proved that $(mathcal{C}, mathbb{E}_xi, mathfrak{s}_xi)$ is a triangulated category if and only if $mathcal{X}=mathcal{Y}=0$; and that $(mathcal{C}, mathbb{E}_xi, mathfrak{s}_xi)$ is an exact category if and only if $mathcal{X}=mathcal{Y}=mathcal{C}$. As an application, we produce a large variety of examples of extriangulated categories which are neither exact nor triangulated.