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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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Consider the bounded derived category $D^b(\operatorname{mod } R)$ of finitely generated modules over a commutative Noetherian ring $R$ and the homology functor $H_*: D^b(\operatorname{mod } R) \to D^...
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I am considering the following situation: suppose we have $(\mathcal{T}, \otimes, 1)$ an essentially small, rigid tensor triangular category with tensor prouct not necessarily symmetric and with all ...
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Let $\mathcal{A}$ an abelian category and $\mathcal{B}$ a Serre subcategory. It is known we can form the Serre quotient $Q\colon \mathcal{A}\rightarrow \mathcal{A}/\mathcal{B}$. We call $\mathcal{B}$ ...
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For a smooth Fano variety $Z$, let $X$ be the total space of its canonical bundle. Let $\operatorname{Coh}_Z(X)$ be the category of coherent sheaves that support on $Z$ set-theorically. How to show ...
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In Schwede 2008 a topological triangulated category is defined as a triangulated category equivalent to a full triangulated subcategory of the homotopy category of a stable model category. In Schwede ...
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A finite tensor category is an abelian $k$-linear (with $k$ a field) category $\mathcal{A}$ with $k$-bilinear tensor product $\otimes$, such that morphism spaces are finite-dimensional, all objects ...
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$\newcommand\Sp{\mathrm{Sp}}\DeclareMathOperator\ho{ho}$Let $\Sp_\omega$ be the stable $\infty$-category of finite spectra, and let $\ho(\Sp_\omega)$ be its homotopy category (a triangulated category)....
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In the derived category $D(A)$ of an abelian category $A$, any morphism from $a$ to $b[n]$ for $a,b\in A$ and $n>0$ may be factored as a composition of morphisms of the form $a_i\to a_{i+1}[1]$ (...
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Let $X$ be a (say, Noetherian) scheme and $U$ is its open subscheme. Is it true that the derived category of perfect complexes on $U$ is equivalent to the idempotent completion of the Verdier quotient ...
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I'm a graduate student with a background in the standard graduate-level courses in algebra, geometry, and topology (following the typical U.S. curriculum). I'd like to learn Tensor Triangulated ...
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Let $R-\operatorname{mod}$ be the category of (left) $R$-modules. Given an possibly unbounded exact sequence $C_*$ of $R$-modules, when is $C_*$ a long exact sequence associated to a short exact ...
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Consider the resolved $\mathbb{C}^3/\mathbb{Z}_3$-orbifold $X$ with exceptional divisor $E \cong \mathbb{P}^2$; it is well known that $X$ may be viewed as the total space of the line bundle ...
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The context of my question is the theory of stratification of a tensor triangulated category via Balmer-Favi support recently developed by Barthel, Heard and Sanders in their paper "...
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Let $A$ be an abelian category. Let $B$ be a weak Serre subcategory of $A$. For a symbol $*\in\{b,-\}$, let $D^*_B(A)$ be the full subcategory of the derived caetgtory $D^*(A)$ of objects with ...
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Let $k$ be a field. Suppose $A$ and $B$ are two DG-algebras over $k$. We say $A$ and $B$ are quasi-isomorphic as DG-algebras if there is a zigzag of quasi-isomorphisms of DG algebras connecting them: $...
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Let $\mathcal{C}$ be a triangulated category, and let $\mathcal{B} \subset \mathcal{C}$ be an admissible subcategory. The admissibility condition gives rise to two semi-orthogonal decompositions $\...
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The standard definition (e.g. in Gortz-Wedhorn II Definition 21.133, the Stacks Project http://stacks.math.columbia.edu/tag/08FK etc.) of a strictly perfect complex on a ringed space/site is (...
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I am new to algebraic geometry. My question is the following : We know the notion of Albanese variety and chow group for in algebraic geometry. P. Balmer defined the tensor triangular chow group for ...
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Let $\mathcal{T}$ be a triangulated category, and let $\mathcal{S} \subseteq \mathcal{T}$ be an admissible subcategory. Recall, this means that the inclusion $\mathcal{S} \hookrightarrow \mathcal{T}$ ...
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Let $\mathcal{T}$ be a triangulated category having all infinite coproducts(such triangulated category is sometimes said to be cocomplete or satisfying the TR5 axiom). We call an object $G$ tilting if ...
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Let $\mathcal{T}' \subseteq \mathcal{T}$ be a full triangulated subcategory. Recall, $\mathcal{T}'$ is called $\textit{right admissible}$ if the inclusion $\mathcal{T}' \hookrightarrow \mathcal{T}$ ...
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I asked this question on stack exchange (https://math.stackexchange.com/questions/4983535/equivalence-of-triangulated-categories-defined-by-tilting-objects), but got no responses, so I ask it here. ...
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Why are Balmer spectra important? Can someone give examples of reconstruction a category by its spectrum (in some sense)? It would also be interesting to see applications of Balmer spectra to the ...
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Let $R$ be a commutative Noetherian ring. Let $M$ be an $R$-module. It is well-known that $M$ is finitely generated if and only if the functor $M\otimes_R (-)$ preserves arbitrary products (for ...
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Let $R$ be a commutative Noetherian ring. Let $\operatorname{D}_{sg}(R)$ be the singularity category of $R$, i.e., the Verdier localization of $D_b(\text{mod } R)$ by the thick subcategory of perfect ...
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$\def\D{\mathcal{D}} \def\ind{\operatorname{Ind}} \def\K{\mathcal{K}} \def\A{\mathcal{A}}$Inside [GW, Remark F.168, p. 794], we find: [Let $\K$ be a category and let $\K_S$ be its localization with ...
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I am wondering whether, if a triangulated category $\mathcal{D}$ has a full strong exceptional collection (infinite), it is triangle-equivalent to the bounded derived category of finitely generated ...
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Let $\mathcal{A}$ be an additive category and $B \to C$ a nonzero map. Are there say "standard" techniques & criteria one should keep in mind when working with additive categories to ...
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I'm trying to prove that a certain class of morphisms in a triangulated category form a multiplicative system. More precisely: Let $\mathcal{T}$ be a triangulated category and $\mathcal{D} \subset \...
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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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$\def\D{\mathcal{D}} \def\I{\mathcal{I}} \def\A{\mathcal{A}}$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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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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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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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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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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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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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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$\newcommand{\K}{\mathcal{K}}$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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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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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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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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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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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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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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Let $\textsf{A}, \textsf{B}$ be abelian categories. Let $F: \operatorname{Ch}(\textsf{A}) \to \operatorname{Ch}(\textsf{B})$ be an additive functor of chain complexes. If $F$ preserves chain ...
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Let $\mathcal T$ be a triangulated category (or homotopy category of a stable $\infty$-category). Recall that the Freyd envelope of $\mathcal T$ is an abelian category $\mathcal A$ which is ...
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From now on, we will consider only rigid-compactly generated tensor-triangulated categories. Let $(\mathcal{T}, \otimes, 1)$ be one of these categories, it is known that the thick tensor ideals of ...
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Let $R$ be a commutative Noetherian ring, consider the bounded derived category of finitely generated $R$-modules $D^b(R)$ and consider the singularity category $D_{sg}(R):=D^b(R)/D^{perf}(R)$. Let $r\...
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$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\SW{SW}$Let $R$ be a commutative Noetherian ring. For $R$-modules $M,N$, let $\mathcal I_R(M,N)$ be the collection of all $f\in \text{Hom}_R(M,N)$ ...
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Let $\mathcal{Cat}$ denotes the $2$-category of small categories and $\mathscr{S}=\mathrm{Sch}/S$ be some category of schemes over a given scheme $S$, consider a $2$-functor $\mathscr{M}:\mathscr{S}^{...
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