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0
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1answer
344 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
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1answer
841 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
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3answers
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
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1answer
360 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
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2answers
816 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 ...
14
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4answers
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
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2answers
717 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 ...
31
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5answers
3k 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 ...
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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
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7answers
2k 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 ...
9
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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
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3answers
664 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
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2answers
687 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 ...
24
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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
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4answers
361 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 ...
16
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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 ...