**2**

votes

**1**answer

227 views

### How is this action of monoidal derived category induced?

I am reading a paper concerning the action of monoidal category to another category.
Let $k$ be a commutative ring, $R$ is a k-algebra. $A=R-mod$, $B=R^{e}-mod=R\bigotimes _{k}R^{o}-mod$.
Consider ...

**19**

votes

**6**answers

4k views

### Heuristic behind the Fourier-Mukai transform

What is the heuristic idea behind the Fourier-Mukai transform? What is the connection to the classical Fourier transform?
Moreover, could someone recommend a concise introduction to the subject?

**3**

votes

**0**answers

498 views

### Sebastiani-Thom isomorphism for D-modules

Considering $f:X\to \mathbb{C}$, $g:X\to \mathbb{C}$ and $f\oplus g:(x,y)\mapsto f(x)+g(y)$.
The Sebastiani-Thom isomorphism is an isomorphism $\Phi_{f\oplus g}(M\boxtimes N) = \Phi_{f}(M) \otimes ...

**12**

votes

**0**answers

505 views

### References for a certain generalization of Hochschild cohomology?

Let $C$ be an algebra. Let $E = C^{\otimes 2n}$ be the tensor product (over the ground field) of $2n$ copies of $C$. [EDIT: Or better, $E = C\otimes C^{op}\otimes C\otimes C^{op}\cdots\otimes C ...

**5**

votes

**2**answers

623 views

### What is the geometric meaning of reconstruction of quantum group via Ringel Hall algebra

If I remembered correctly. There are some work done by C.M.Ringel,he defined so called Ringel-Hall algebra on abelian category and then show that Ringel-hall algebra is isomorphic to positive part of ...

**16**

votes

**5**answers

2k views

### determinant of a perfect complex

Say $K_\bullet$ is a bounded complex of vector bundles. I seem to want the determinant of $K_\bullet$ to be the alternating tensor product of the terms of the complex:
$\det(K) = \bigotimes_n ...

**4**

votes

**3**answers

390 views

### Derived category with total cohomology finite dimensional: is there a better name for it?

One of the annoying things about derived categories is that they come with a host of different finiteness conditions, which are all subtlely different, and for each situation you want a particular ...

**23**

votes

**9**answers

2k views

### What is a deformation of a category?

I have several naive and possibly stupid questions about deformations of categories. I hope that someone can at least point me to some appropriate references.
What is a deformation of a (linear, dg, ...

**7**

votes

**3**answers

722 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 ...

**5**

votes

**2**answers

810 views

### Singular K3 — mathematical meaning?

There's a very interesting text by Cumrun Vafa called Geometric Physics.
Here I'm particularly interested in Chapter 4, where we take a Calabi-Yau manifold presented as a degenerating fibration:
...

**0**

votes

**1**answer

373 views

### Understanding a lemma in “Loop Spaces and Langlands Parameters” article

First, some background. I was trying to read the article Loop Spaces and Langlands Parameters but I get immediately stuck at Theorem 2.1 in the introduction.
This was actually forward-referring to ...

**11**

votes

**1**answer

923 views

### When do six operations work?

This question comes (heavily edited) from my notes, thus slightly unusual structure.
We know that algebraic maps have very strict structure, and in many settings the operations ...

**14**

votes

**3**answers

1k 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 ...

**5**

votes

**1**answer

387 views

### Equivariant Derived Categories via their properties.

There are some ways to define equivariant derived categories of all sorts. But all the ways i know of involve giving a concrete construction. Is the other way around possible? Is there some universal ...

**8**

votes

**2**answers

1k views

### Derived functors vs universal delta functors

I would like to understand the relationship between the derived category definition of a right derived functor Rf (which involves an initial natural transformation n: Qf → (Rf)Q, where Q is the ...

**16**

votes

**4**answers

3k views

### Is the Fukaya category “defined”?

Sometimes people say that the Fukaya category is "not yet defined" in general.
What is meant by such a statement? (If it simplifies things, let's just stick with Fukaya categories of compact ...

**3**

votes

**2**answers

790 views

### Equivalence of derived categories which is not Fourier-Mukai

D. Orlov proved that any equivalence of bounded derived categories F:Db(X) -> Db(Y) is a Fourier-Mukai transform, when X and Y are smooth projective varieties. Is there any example of such ...

**37**

votes

**5**answers

4k views

### Intuition about the cotangent complex?

Does anyone have an answer to the question "What does the cotangent complex measure?"
Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as ...

**26**

votes

**6**answers

2k views

### How to think about model categories?

I've read about model categories from an Appendix to one of Lurie's papers.
What are the examples of model categories? What should be my intuition about them?
E.g. I understand the typical examples ...

**40**

votes

**7**answers

3k views

### What does a projective resolution mean geometrically?

For R a commutative ring and M an R-module, we can always find a projective resolution of M which replaces M by a sequence of projective R-modules. But as R is commutative, we can consider the affine ...

**11**

votes

**3**answers

2k views

### Examples for Decomposition Theorem

There's an important piece of geometric knowledge usually quoted as Beilinson-Bernstein-Deligne.
Here's a refresher: by $IC$ one means the intersection complex, which is just $\mathbb Q$ for a smooth ...

**11**

votes

**3**answers

723 views

### Freyd-Mitchell for triangulated categories?

Is there a nice analog of the Freyd-Mitchell theorem for triangulated categories (potentially with some requirements)? Freyd-Mitchell is the theorem which says that any small abelian category is a ...

**5**

votes

**2**answers

760 views

### Higher vanishing cycles

The generalisation of the vanishing cycle formalism in SGA 7 is apparently since the 1970's an issue, Morava mentioned a connection with Bousfield localization. I find the Morava's remarks ...

**28**

votes

**7**answers

2k views

### Simplicial objects

How should one think about simplicial objects in a category versus actual objects in that category? For example, both for intuition and for practical purposes, what's the difference between a ...

**3**

votes

**4**answers

377 views

### E_\infty spectrum corresponding to Z_p

First of the questions about derived algebraic geometry from a noobie.
The way I understand it, every discrete ring R corresponds to some ring spectrum whose ...

**17**

votes

**5**answers

1k views

### Derived categories and homotopy categories

There are two constructions that look quite similar to me: the derived category of an abelian category, and the homotopy category of a model category. Is there any explicit relationship between these ...