Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOn the obscure axiom for one-sided exact categories
92
0
0.0
(
0
)
Added by
Adam-Christiaan van Roosmalen
Publication date
2020
fields
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
One-sided exact categories are obtained via a weakening of a Quillen exact category. Such one-sided exact categories are homologically similar to Quillen exact categories: a one-sided exact category $mathcal{E}$ can be (essentially uniquely) embedded into its exact hull ${mathcal{E}}^{textrm{ex}}$; this embedding induces a derived equivalence $textbf{D}^b(mathcal{E}) to textbf{D}^b({mathcal{E}}^{textrm{ex}})$. Whereas it is well known that Quillens obscure axioms are redundant for exact categories, some one-sided exact categories are known to not satisfy the corresponding obscure axiom. In fact, we show that the failure of the obscure axiom is controlled by the embedding of $mathcal{E}$ into its exact hull ${mathcal{E}}^{textrm{ex}}.$ In this paper, we introduce thr
rate research
Read More
In this paper, we introduce quotients of exact categories by percolating subcategories. This approach extends earlier localization theories by Cardenas and Schlichting for exact categories, allowing new examples. Let $mathcal{A}$ be a percolating subcategory of an exact category $mathcal{E}$, the quotient $mathcal{E} {/mkern-6mu/} mathcal{A}$ is constructed in two steps. In the first step, we associate a set $S_mathcal{A} subseteq operatorname{Mor}(mathcal{E})$ to $mathcal{A}$ and consider the localization $mathcal{E}[S^{-1}_mathcal{A}]$. In general, $mathcal{E}[S_mathcal{A}^{-1}]$ need not be an exact category, but will be a one-sided exact category. In the second step, we take the exact hull $mathcal{E} {/mkern-6mu/} mathcal{A}$ of $mathcal{E}[S_mathcal{E}^{-1}]$. The composition $mathcal{E} rightarrow mathcal{E}[S_mathcal{A}^{-1}] rightarrow mathcal{E} {/mkern-6mu/} mathcal{A}$ satisfies the 2-universal property of a quotient in the 2-category of exact categories. We formulate our results in slightly more generality, allowing to start from a one-sided exact category. Additionally, we consider a type of percolating subcategories which guarantee that the morphisms of the set $S_mathcal{A}$ are admissible. In upcoming work, we show that these localizations induce Verdier localizations on the level of the bounded derived category.
We consider the quotient of an exact or one-sided exact category $mathcal{E}$ by a so-called percolating subcategory $mathcal{A}$. For exact categories, such a quotient is constructed in two steps. Firstly, one localizes $mathcal{E}$ at a suitable class $S_mathcal{A} subseteq operatorname{Mor}(mathcal{E})$ of morphisms. The localization $mathcal{E}[S_mathcal{A}^{-1}]$ need not be an exact category, but will be a one-sided exact category. Secondly, one constructs the exact hull $mathcal{E}{/mkern-6mu/} mathcal{A}$ of $mathcal{E}[S_mathcal{A}^{-1}]$ and shows that this satisfies the 2-universal property of a quotient amongst exact categories. In this paper, we show that this quotient $mathcal{E} to mathcal{E} {/mkern-6mu/} mathcal{A}$ induces a Verdier localization $mathbf{D}^b(mathcal{E}) to mathbf{D}^b(mathcal{E} {/mkern-6mu/} mathcal{A})$ of bounded derived categories. Specifically, (i) we study the derived category of a one-sided exact category, (ii) we show that the localization $mathcal{E} to mathcal{E}[S_mathcal{A}^{-1}]$ induces a Verdier quotient $mathbf{D}^b(mathcal{E}) to mathbf{D}^b(mathcal{E}[S^{-1}_mathcal{A}])$, and (iii) we show that the natural embedding of a one-sided exact category $mathcal{F}$ into its exact hull $overline{mathcal{F}}$ lifts to a derived equivalence $mathbf{D}^b(mathcal{F}) to mathbf{D}^b(overline{mathcal{F}})$. We furthermore show that the Verdier localization is compatible with several enhancements of the bounded derived category, so that the above Verdier localization can be used in the study of localizing invariants, such as non-connective $K$-theory.
Given two small dg categories $C,D$, defined over a field, we introduce their (non-symmetric) twisted tensor product $Coverset{sim}{otimes} D$. We show that $-overset{sim}{otimes} D$ is left adjoint to the functor $Coh(D,-)$, where $Coh(D,E)$ is the dg category of dg functors $Dto E$ and their coherent natural transformations. This adjunction holds in the category of small dg categories (not in the homotopy category of dg categories $mathrm{Hot}$). We show that for $C,D$ cofibrant, the adjunction descends to the corresponding adjunction in the homotopy category. Then comparison with a result of To{e}n shows that, for $C,D$ cofibtant, $Coverset{sim}{otimes} D$ is isomorphic to $Cotimes D$, as an object of the homotopy category $mathrm{Hot}$.
In this paper, which is subsequent to our previous paper [PS] (but can be read independently from it), we continue our study of the closed model structure on the category $mathrm{Cat}_{mathrm{dgwu}}(Bbbk)$ of small weakly unital dg categories (in the sense of Kontsevich-Soibelman [KS]) over a field $Bbbk$. In [PS], we constructed a closed model structure on the category of weakly unital dg categories, imposing a technical condition on the weakly unital dg categories, saying that $mathrm{id}_xcdot mathrm{id}_x=mathrm{id}_x$ for any object $x$. Although this condition led us to a great simplification, it was redundant and had to be dropped. Here we get rid of this condition, and provide a closed model structure in full generality. The new closed model category is as well cofibrantly generated, and it is proven to be Quillen equivalent to the closed model category $mathrm{Cat}_mathrm{dg}(Bbbk)$ of (strictly unital) dg categories over $Bbbk$, given by Tabuada [Tab1]. Dropping the condition $mathrm{id}_x^2=mathrm{id}_x$ makes the construction of the closed model structure more distant from loc.cit., and requires new constructions. One of them is a pre-triangulated hull of a wu dg category, which in turn is shown to be a wu dg category as well. One example of a weakly unital dg category which naturally appears is the bar-cobar resolution of a dg category. We supply this paper with a refinement of the classical bar-cobar resolution of a unital dg category which is strictly unital (appendix B). A similar construction can be applied to constructing a cofibrant resolution in $mathrm{Cat}_mathrm{dgwu}(Bbbk)$.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
