**6**

votes

**3**answers

1k views

### A toy example of a tensor triangulated category?

I've been reading Paul Balmer's paper about constructing a "spectrum of prime ideals" on (essentially small) tensor triangulated category in order to then classify thick subcategories. This is all ...

**7**

votes

**2**answers

445 views

### Is the tensorproduct of a triangulated category with a ring again triangulated?

$\underline{Background}$ :
Suppose $\tau$ is a preadditive category and $R$ a ring. Then one may form a new preadditive category $\tau \otimes R$ in the following way:
$\tau \otimes R$ has the same ...

**3**

votes

**1**answer

530 views

### Do exact faithful functors between triangulated categories detect semi-simplicity?

Recall that in a triangulated category, all monomorphisms split (have a retraction). Let $F:C\to D$ be an exact functor between triangulated categories. It is an easy exercise to see that if $F$ is ...

**0**

votes

**3**answers

685 views

### The (upper hat of) an octahedral diagram in (la)tex

I would like to draw an octahedral diagram in my paper; I would prefer to present it as the 'upper hat' + the 'lower hat' (as it is common in the texts on triangulated categories). Could anyone tell ...

**5**

votes

**3**answers

513 views

### Can one extend a morphism of commutative triangles to a morphism of octahedral diagrams?

Consider two (distinct) octahedral diagrams i.e. diagrams mentioned in the octahedron axioms of triangulated categories (with four 'commutative triangular faces' and four 'distinguished triangular ...

**4**

votes

**1**answer

465 views

### Morphisms between pure complexes of sheaves

I would like to understand the theory of pure complexes of (etale?) sheaves (of geometric origin?). In particular, I would like to understand which conditions are realy necessary in (part 1 of) ...

**9**

votes

**1**answer

754 views

### How to write down the determinant of a quasi-isomorphism?

This question about the determinant of a perfect complex reminded me of an old question that I had.
The construction of the determinant (as in MR1914072 or MR0437541) is a difficult piece of ...

**13**

votes

**1**answer

563 views

### When is every “solid” perfect complex faithful?

Let $R$ be a noetherian commutative ring. Consider $D^{perf}(R)=K^b(R-proj)$ the category of bounded complexes of finitely generated projective $R$-modules, with maps of complexes up to homotopy.
...

**5**

votes

**1**answer

541 views

### Is K(R-Mod) compactly generated when R is an artin algebra?

I wonder if the triangulated category K(R-Mod) is compactly generated when R is an artin algebra? R-Mod denotes all left R-modules. I understand this would be true if R has finite representation type ...

**8**

votes

**1**answer

828 views

### Verdier duality via Brown representability?

Hello,
I wonder if the techniques introduced in Neemans paper:
"The Grothendieck duality theorem via Bousfield's techniques and Brown representability "
can be used to establish Verdier duality. More ...

**11**

votes

**2**answers

1k views

### Why do people “forget” Verdier abelianization functor?(Looking for application)

I am now learning localization theory for triangulated catgeory(actually, more general (co)suspend category) in a lecture course. I found Verdier abelianization which is equivalent to universal ...

**10**

votes

**2**answers

2k views

### What is the relationship between t-structure and Torsion pair?

I am away from Torsion theory in abelian category for some while. So it might be a stupid question.
The definition of a torsion pair in the category of modules is as follows:
Definition:
A pair ...

**12**

votes

**5**answers

1k views

### Tate Cohomology via Stable Categories

Situation
Let $G$ be a finite group and provide $G\text{-mod} := {\mathbb Z}G\text{-mod}$ with the Frobenius structure of ${\mathbb Z}$-split short exact sequences. Denote by ...

**1**

vote

**2**answers

286 views

### Relative Frobenius Structure on the Category of G-modules

Let $G$ be a group $H\leq G$ a subgroup of finite index. Further, let ${\mathcal E}^G_H$ denote the class of those short exact sequences of $G$-modules (over some fixed base ring) which split when ...

**0**

votes

**1**answer

200 views

### Motivation for Cosuspended Category Axioms

Today I was wondering about the axioms given by Bernhard Keller for Cosuspended Categories.
The axioms of a triangle feel very much like exactness, but not quite. The last axiom about the large ...

**10**

votes

**2**answers

1k views

### Is there triangulated category version of Barr-Beck's theorem?

It is well known that Beck's theorem for Comonad is equivalent to Grothendieck flat descent theory on scheme.
There are several version of derived noncommutative geometry. I wonder whether someone ...

**2**

votes

**1**answer

448 views

### Derived Functors in arbitrary triangulated categories

Let ${\mathcal D}$ be a triangulated category, ${\mathcal C}$ a triangulated subcategory and $Q: {\mathcal D}\to {\mathcal D}/{\mathcal C}$ the corresponding Verdier-localization. Now suppose we have ...

**1**

vote

**1**answer

322 views

### Any relationship of frobenius homomorphism and frobenius category?

I did not understand number theory or characteristic p-algebraic geometry at all. I just know a little about frobenius homomorphism between two schemes. On the other hand, when I learned something on ...

**4**

votes

**3**answers

317 views

### exactness in triangulated categories is reflected by hom-functor

let $T$ be a triangulated category and $A \to B \to C \to A[1]$ a triangle in $T$ such that for every $A_0 \in T$ the induced long sequence
$... \to \hom(A_0,A) \to \hom(A_0,B) \to \hom(A_0,C) \to ...

**7**

votes

**3**answers

756 views

### 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 ...

**8**

votes

**4**answers

1k views

### 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 ...

**2**

votes

**2**answers

1k views

### 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 ...

**11**

votes

**3**answers

891 views

### 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 ...

**14**

votes

**3**answers

2k views

### 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 ...

**6**

votes

**1**answer

792 views

### 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 ...

**6**

votes

**4**answers

2k views

### Localization(s) of Categories

I've been trying to read a paper by Krause and came across a strange (to me, of course) notion of localization. After looking around for a long time, and then finding this on his site, I see that ...

**12**

votes

**2**answers

562 views

### Is there a constructive description of type in the p-local stable homotopy category?

The title pretty much sums it up - but let me give a little bit of background first.
In the p-local stable homotopy category (basically one localizes away the torsion spectra which are not p-torsion) ...