# Tagged Questions

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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### Derived categories of (coherent) sheaves of modules: exceptional images, gluing, and proper descent?

I am interested in the properties of (the derived categories) of various categories of (coherent) sheaves of modules (over varieties). I would like to understand in what extent these properties are ...
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### Cohomological functor from triangulated category

Say we have a cohomological functor F from a triangulated category $C$ to the category $Ab$ of abelian groups, e.g. $F=Hom(x,-)$, where x is an object in $C$. By definition, such a functor transform ...
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### Why not define triangulated categories using a mapping cone functor?

Recall that the usual definition of a triangulated category is an additive category equipped with a self equivalence called $[1]$ in which certain diagrams, of the form $X \to Y \to Z \to X[1]$ are ...
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### Proof that the homotopy category of a stable $\infty$-category is triangulated

I've been looking at various general strategies for proving that some category is triangulated, and Lurie manages to prove that a huge class of interesting examples of categories that we know about ...
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### Does a triangulated category that possesses a subcategory $B$ of generators with no extensions of non-zero degree between them have to be isomorphic to $K^b(B)$?

Suppose that a triangulated category $C$ contains a full additive subcategory $B$ of (strong) generators (i.e. there does not exist a proper strict triangulated subcategory $C'\subset C$ that contains ...
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### Neeman's Lemma 1.4.4 (in his book on triangulated categories)

When proving Lemma 1.4.4 in his book on triangulated categories, Neeman asserts that a certain mapping cone splits into a direct sum of three candidate triangles. I'm unable to see why this is so. ...
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### On 'graded polarizable' triangulated categories; are there any mixed Galois module analogues known? Also on mixed realizations

There is a construction by Beilinson (in section 3 of "Notes on absolute Hodge cohomology") of the derived category of graded polarizable mixed Hodge complexes; he also proved that this category is ...
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### Any examples known of $K^b(B)$ localized by a set of morphisms (i.e. of complexes of length 1)?

I would like to understand the following setting: for an additive $B$ localize $K^b(B)$ by a set of $B$-morphisms (i.e. by a thick triangulated subcategory generated by some set of two-term complexes)....
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### Intuition about the triangulation of a homotopy category K(A)

Let $\cal{A}$ be an additive category. Given a morphism of (cochain) complexes $f:X\rightarrow Y$ we can form the mapping cone $C_f$, which is the complex $X[1]\oplus Y$ with differential given by \...
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### Glueing triangulated categories

Hello! Given a triangulated category, one can look for semiorthogonal decompositions into (simpler?) triangulated subcategories. I'd like to know if there's a way to attack the opposite problem, i....
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### Compact generation for modular representations

Are the derived categories of modular representations of algebraic groups compactly generated? (e.g. consider SL_2 in characteristic 2). Note modular reps of finite groups are compactly generated (by ...
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### Categories which are not compactly generated

Do you know natural examples of triangulated categories (or [presentable] stable $\infty$-categories) which are not compactly generated? (ideally they'd be defined algebraically, but curious to hear ...
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### Splitting in triangulated categories

Using the axioms for a triangulated category, is it possible to prove the following: $A\stackrel{0}{\to}B\to A\oplus B\to$ is a distinguished triangle. From the first axiom, the map ...
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### Classifying triangulated structures on a graded category

I know of several results to the effect that two triangulated categories are equivalent categories (usually one coming from algebra and one coming from topology). However, it's never been clear to me ...
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### distinguished triangles and cohomology

Start with A an abelian category and form the derived category D(A). Take a triangle (not necessarily distinguished) and take it's cohomology. We obtain a long sequence (not necessarily exact). If the ...
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### Sources for exact triangles in triangulated categories.

The other day I came across the statement that in the triangulated category $\mathfrak{KK}$ (of C*-algebras with KK-groups as morphism sets) "there are many other sources of exact triangles besides ...