Questions tagged [triangulated-categories]
A triangulated category is an additive category equipped with the additional structure of an autoequivalence (called the translation functor) and a class of of triangles satisfying certain axioms.
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Stable module category of non-Frobenius algebras
It is often said that the stable module category $A-\underline{\operatorname{mod}}$ for an associative algebra $A$ is triangulated if $A$ is Frobenius (i.e. over $A$ we have projective = injective). ...
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Does a fully faithful and essentially surjective exact functor between triangulated categories have a quasi-inverse the 2-cat of triangulated cats?
$\def\D{\mathcal{D}}
\def\I{\mathcal{I}}
\def\A{\mathcal{A}}$Triangulated categories are the objects of a 2-category $\mathsf{Triang}$: the 1-morphisms are the exact functors $(F,\xi)$ of triangulated ...
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The Balmer spectrum and the thick tensor ideals of the derived category of a Hopf algebra
Given a Hopf algebra $H$ over a field $\mathbb{k}$, the category of finite-dimensional left-$H$-modules naturally becomes a rigid monoidal category with exact monoidal product. Thus clearly the ...
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What is the most general notion of exactness for functors between triangulated categories?
For triangulated categories $T,T'$ I would like to define "weakly exact" functors as those that respect cones, that is, $F(Cone f)\cong Cone(F(f))$ for any $T$-morphism $f$, and I do not ...
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A particular morphism being zero in the singularity category
Let $R$ be a commutative Noetherian ring and $D^b(R)$ be the bounded derived category of finitely generated $R$-modules. Let $D_{sg}(R)$ be the singularity category, which is the Verdier localization $...
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Thick subcategory containment in bounded derived category vs. singularity category
Let $R$ be a commutative Noetherian ring, and $D^b(\operatorname{mod } R)$ the bounded derived category of the abelian category of finitely generated $R$-modules. Let me abbreviate this as $D^b(R)$. ...
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Grothendieck group and an almost localization
Let $T$ be a small triangulated category and let $S\subset T$ be a full triangulated subcategory. We denote this embedding by $I: S\rightarrow T$.
Let $F: T\rightarrow S$ be a triangulated functor ...
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Example of triangulated category with vanishing $K_0$
Let $R$ be a ring, let $\operatorname{Perf}(R)$ the category of perfect modules over $R$. Suppose we have $E$ an perfect $R$-module (concentrated in degree $0$) such that its class $[E]\in K_0(R)$ is ...
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Is there a better name for the "Mayer-Vietoris Octahedral axiom" and has it been studied?
$\newcommand{\K}{\mathcal{K}}$Say $\K$ is a triangulated category with suspension $\Sigma:\K\simeq \K$. In Iversen's book "Cohomology of Sheaves", he doesn't exactly examine triangulated ...
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Image, upto direct summands, of derived push-forward of resolution of singularities
Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all ...
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Exact sequences in Positselski's coderived category induce distinguished triangles
I am learning about Positselski's co- and contraderived categories. We know that short exact sequences do not generally induce distinguished triangles in the homotopy category but they do in the usual ...
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Comparing stabilization of stable category modulo injectives and a Verdier localization
Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as ...
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When would a left admissible triangulated subcategory be admissible
I'm walking through the proof of [1, Thm 16 at pp. 515] and am stuck at the first sentence after equation (12), where the author states that the decomposition (12) is semiorthogonal when $a\geq 0$. ...
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From exact triangles in the stable category of maximal Cohen--Macaulay modules to short exact sequences
Let $R$ be a local Gorenstein ring. Let $\underline{\text{CM}}(R)$ be the stable category of maximal Cohen--Macaulay modules, it is known to carry a triangulated structure. My question is: If $M\to N\...
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Why is this map a split monomorphism?
I have a question regarding a lemma in the proof of Hopkins-Neeman Correspondence.
It is the beginning part of Lemma 1.2 in the The Chromatic Tower for D(R)
Let $Y$ be an object of the derived ...
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