Timo Schürg
  • Member for 12 years, 3 months
How to think about model categories?
62 votes

Here's a bit of the historical reason why model categories came up. If you have a functor on an abelian category which doesn't quite behave as you would like it to, e.g. is not left exact, there is a ...

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Intuition about the cotangent complex?
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28 votes

One thing the cotangent complex measures is what kind of deformations a scheme has. The precise statements are in Remark 5.30 and Theorem 5.31 in Illusie's article in "FGA explained". Here's the short ...

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Homotopy pullbacks and homotopy pushouts
21 votes

Here's a simple geometric example for a homotopy pushout. This is stolen from the Dwyer-Spalinski paper on model categories. We first look at the following diagram: pt <-- S^1 --> pt. The pushout ...

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Algebraic stacks from scratch
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17 votes

Another good place to look are the notes of Master's course on stacks by Betrand Toen. I think they pretty much do exactly what you are looking for. Here's the quick summary: You will want to read ...

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Derived Algebraic Geometry and Chow Rings/Chow Motives
17 votes

I can't give you a complete answer apart from saying that this is definitely a hard problem! If you have a derived scheme $X$, you can always truncate to get an every-day scheme $t_0 (X)$. On the ...

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What is DAG and what has it to do with the ideas of Voevodsky?
17 votes

I can't tell you much about the relation to Voevodsky's work, but I can give you a quick summary of Derived Algebraic Geometry. In DAG you enlarge the category of geometric objects that you can ...

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What are Picard categories, where can I learn more about them, and why should I care to?
12 votes

I ran across Picard categories in a totally different area of mathematics, but maybe it helps. In short, a Picard category is a group object in the category of groupoids. Picard categories come up ...

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Simplicial objects
12 votes

In case you are an algebraic geometer and are you used to thinking about a commutative ring in terms of its spectrum it might be helpful to imagine the spectrum of a simplicial commutative ring A as ...

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Is there any Grothendieck Riemman Roch theorem for general stack ?
Accepted answer
12 votes

If you work with the naive Chow-groups and allow non-representable morphisms the GRR-Theorem does not hold! In the paper by Toen quoted above and in some of the papers by Joshua there are explicit ...

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Why are flat morphisms "flat?"
11 votes

One of the consequences of flatness of morphisms between projective schemes is that the dimension of the fibers stays constant. Maybe this is the reason for the term. I'm not sure whether this makes ...

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When does the cotangent complex vanish?
9 votes

Edit: Here is a possible characterization. As mentioned in the comments above, the vanishing of the 1-truncated cotangent complex $\tau_{\leq 1}L_{B/A}$ of a map of rings $f \colon A \to B$ is ...

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Functorial point of view for formal schemes
9 votes

You can try having a look at this paper: http://arxiv.org/abs/math.AT/0011121 It's the most functorial-minded paper on algebraic geometry I'ver ever seen. It's written by an algebraic topologist. He ...

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Sources for Bibtex entries
8 votes

Zotero is a nice plug-in for firefox that produces a database of your favorite publications. Every time you are on a website like arxiv, math-sci net or the homepage of some journal, it offers you to ...

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What representative examples of modules should I keep in mind?
8 votes

Another thing you can do is look at the frontispiece of Miles Reid's commutative algebra book (click the frontispiece link in the contents; Google books won't link directly to the right page). It's ...

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different "derived structure" in derived algebraic geometry
Accepted answer
5 votes

Here are the two motivations I know of: Number 1 comes from algebraic topology. The definitive reference is http://www.math.harvard.edu/~lurie/papers/survey.pdf , which explains it much better than I ...

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A sheaf-theoretic version of the proj construction?
3 votes

In one special case of the Proj construction the functor is easy to write down, namely in the case of $\mathbb{P}^n$. The functor you get in $Sh(Cring^{op})$ is: $R$ gets mapped to split inclusions ...

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Kuranishi structures vs polyfolds
3 votes

You might also enjoy this video of a talk by Katrin Wehrheim which compares the approaches a bit. It's actually also quite entertaining. http://www.msri.org/communications/vmath/VMathVideos/VideoInfo/...

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A few questions about Kontsevich formality
2 votes

Its only a reference, and I don't feel competent to give a summary. I think the answer to question 1 can be found in http://www.math.jussieu.fr/~keller/publ/emalca.pdf . At least it mentions the ...

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Do quotients of representable sheaves represent quotients?
2 votes

This is just a little side remark that in general it does make a difference whether you take quotients as representable functors or as schemes. There is a counterexample in section 4 of http://www....

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