For questions about the derived categories of various abelian categories and questions regarding the derived category construction itself.

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3
votes
2answers
773 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 ...
35
votes
5answers
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
6answers
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 ...
39
votes
7answers
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 ...
10
votes
3answers
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
3answers
707 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
2answers
749 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 ...
27
votes
7answers
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
4answers
369 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
5answers
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 ...