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André and Quillen both gave constructions of the relative cotangent complex for commutative rings, so pretty immediately that gives us that we understand the cotangent complex for affine schemes. Illusie generalized the cotangent complex construction from "rings over A" for a ring A to "rings over $\mathcal{O}_X$" for a base ring object of an arbitrary Grothendieck topos. At least for ordinary schemes, it doesn't seem too hard to believe that we could glue together relative cotangent complexes along affine opens, but for things like algebraic spaces and/or formal schemes it seems conceivable to me that it might be substantially harder to glue the local modules together while preserving their simplicial structure.

What difficulties with globalizing the local definition of the cotangent complex lead to the topos-theoretic approach used by Illusie? (This is not a history question. I'm just wondering what the motivation is for the greater generality, since I'm currently reading André's book, which only covers the "classical" case of a commutative $A$-algebra for set-theoretic commutative ring $A$.)

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In particular, André and Quillen take a free or projective resolution $B$ in the category of $A$-algebras (it's slightly more subtle than an ordinary projective resolution, but you get the picture), but I have heard many times that in Grothendieck toposes, you very often may fail to have any sort of nontrivial projective resolution (with respect to the global sections functor (which is the unenriched $Hom(\mathcal{O}_X,-)$)). – Harry Gindi Nov 13 '10 at 16:46
The whole point of the cotangent complex is its applications to deformation theory of objects and morphisms, and these are often not affine (and in the case of group schemes can involve rather intricate diagrams). Gluing is also a problem in the derived category, as you must know, so it's a highly nontrivial matter to adapt the "affine" theory to the global case. Those problems have to be overcome already for schemes, regardless of any bells & whistles like algebraic spaces. Perhaps it can be done more easily nowadays with Lurie's stuff, but that may be just moving the hard work around. – BCnrd Nov 13 '10 at 17:03
Actually, I didn't realize until you just said it right now that gluing is a problem in the derived category (although it obviously is, now that I think about it). To glue in the derived category correctly, you need to look at hocolims instead of ordinary colims. Thanks for the answer! – Harry Gindi Nov 13 '10 at 17:29
Oh, by the way, BCnrd, I think that Voevodsky-Morel deals with the issue of gluing together simplicial sheaves, but that is still pretty recent compared to Illusie's work. – Harry Gindi Nov 13 '10 at 18:34
up vote 7 down vote accepted

As BCnrd points out, gluing cotangent complexes is a nontrivial thing. You might still ask whether it is really necessary for Illusie to work in the generality of a ringed topos. Would using a ringed space suffice? For standard deformation problems (deformation of a morphism or deformation of a scheme) working on the underlying ringed space would be enough. For more "interesting" deformation problems, like deformation of a morphism $X \rightarrow Y$, where $X$, $Y$, and the morphism are all allowed to vary, one needs something more sophisticated. Illusie constructs a ringed topos that encodes all of these data and then applies the machinery for ringed topoi that he has already developed.

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If you're going to bother with doing something for a ringed space, there's really no reason not to generalize to the case of a ringed topos, since the problems are actually probably clearer in that context, right? It's one of those cases where the greater generality comes basically for free. – Harry Gindi Nov 13 '10 at 17:47
Nothing interesting comes for free; there has to be a real idea (and one has to find suitable definitions, etc.) and extra generality may also entail much extra work (e.g., compare Lichtenbaum-Schlessinger with Illusie). One also has to find the right topos to assure that the extra generality gives something interesting in cases one cares about. It isn't obvious (to me...) that to do deformation theory of group schemes, one can recast it as deformation theory in some ringed topos. – BCnrd Nov 13 '10 at 17:56
Dear Harry: It is quite easy to check that a square-zero deformation of a scheme as a ringed space is a scheme whenever the corresponding square-zero ideal sheaf is quasi-coherent, but to prove the analogous result for an algebraic space (using ringed topoi) requires somewhat more care. (Try for yourself.) This is just a minor example, but it illuminates that merely passing to a more general setup doesn't obviate the need to make sure that the conclusions are actually relevant to the motivating situations one cares about. – BCnrd Nov 13 '10 at 18:03
Interesting. I'll give it a shot. – Harry Gindi Nov 13 '10 at 18:27

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