## Schemes as a model category

I'm just learning some basics of model categories, so please forgive me if my question turns out to be trivial. I hope it does at least make sense.

A natural temptation is to relate this machinery to birational geometry; in particular one would like to find a model category structure having the birational morphisms as weak equivalences. More precisely it would be nice to have such a model structure on the category $Sch_k$ of schemes of finite type over a field $k$.

A natural problem arises: a model category is required by definition to have all small limits and colimits, and $Sch_k$ does not satisfy this. For limits the situation is not that bad. I believe the original work of Quillen required only the existence of finite limits and colimits. Since $Sch_k$ has finite products and fiber products, it has all finite limits.

On the other hand finite colimits need not exist. A simple way to see this is to realize that categorical quotients by equivalence relations do not always exist in $Sch_k$, and these are just some coequalizers. So my questions are:

1. Is there a canonical way to enlarge a category to add finite limits?

2. If this is the case, what do we obtain when applying this to $Sch_k$? The resulting category would have to contain algebraic spaces, as these arise as quotients of schemes by étale equivalence relations. How much bigger would it be?

3. Assuming one has a decent notion of birational morphism for these objects: is there a model structure on the enlarged category such that birational morphisms are the weak equivalences?

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For 1, googling for "finite limit completion" shows such a thing exists, but I am not lucky enough to get details... – Mariano Suárez-Alvarez Jun 29 2010 at 17:30
I have the vague sense that someone has thought about what happens if you formally invert the class of birational maps in the category schemes, and decided that what you get isn't very interesting. I actually thought that had been asked here before, but I can't find it. – Charles Rezk Jun 29 2010 at 18:28
What is meant by a birational map between arbitrary schemes of finite type over a field? – Tom Goodwillie Jun 29 2010 at 21:45
We might invert dense open immersions between k-schemes of finite type. A convenient way to study this localization consists to look at the model category of simplicial presheaves on $Sch/k$, and then to look at its left Bousfield localization by dense open immersions. What we get is very interesting, and leads to beautiful results and problems, related to $\pi_0$ in $A^1$-homotopy theory of schemes). This is studied by Fabien Morel and Aravind Asok in their papers arXiv:0810.0324 and arXiv:1001.4574 (even though they don't formulate things this way explicitly). – Denis-Charles Cisinski Jun 29 2010 at 22:04
@Tom: a map which is an isomorphism on dense open subschemes. Me too, I'd rather work with varieties, but then you don't have even colimits. – Andrea Ferretti Jun 29 2010 at 23:20

## 1 Answer

A canonical way would be to embed $Sch_k$ into the category of (pre)sheaves.

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To expand on this: You can embed Sch into the category of sheaves on Sch and then into the category of simplicial sheaves on Sch. One can then give the category of simplicial sheaves the structure of a closed simplicial model category. See for example section 2 of this: math.berkeley.edu/~teleman/math/simpson.pdf – Kevin Lin Jun 29 2010 at 17:36
norondion -- could you elaborate on that please? Kevin -- to define sheaves (as opposed to presheaves) one needs a (Grothendieck) topology. What is the topology on $Sch_k$ that gives birational isomorphisms as weak equivalences after applying the construction in the paper you cite? – algori Jun 29 2010 at 17:51
Some people use "space" synonymously with "etale sheaf." Then the "algebraic" in "algebraic space" becomes a finiteness condition. – Ben Wieland Jun 29 2010 at 18:12
I think this might be too big. I don't know how to make sense of the word birational in this context. I guess that the model category suggested by Kevin has little to do with birational geometry. – Andrea Ferretti Jun 29 2010 at 23:19
algori: I didn't say that this can give a model structure wherein birational morphisms are weak equivalences. I agree with Andrea that this has probably little to do with birational geometry. I was thinking that this might be an answer to Andrea's first two questions, but probably not the third. – Kevin Lin Jun 30 2010 at 0:53
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