Here is what I know about the history of the cotangent complex:

Quillen did it over a point (i.e. for morphism of rings), Illusie did it in a topos (i.e. for sheaves of rings in a topos). And proved general theorems on deformation/obstructions. People have been using that to study various deformation problems of schemes. E.g. constructing obstruction theory. There are also generalization to 1-morphisms to stacks or to the logarithmic world. It seems to be a crucial object in the derived algebraic geometry setup.

My question is, has the cotangent complex used in other context with the full power of topoi? I understand that it is related to the solution of Serre's conjecture and the definition Andre-Quillen cohomology. I also understand to "glue" the affine construction is hard and you really need the power of topoi. I just want to know if this has been used in general on a morphism of rings in a topos. Or more generally, for a morphism of ringed topoi.

(I hope this question is not too vague, a more "concrete" example might be: does infinitesimal thickening of the first order of the fppf topos of a scheme $X$ (by a module in it) over that of a scheme $Y$ contains some interesting information?)

Just to be clear, by topos I always mean a Grothendieck topos, i.e. a category equivalent to the category of sheaves of sets over a site.

  • $\begingroup$ M. André, Homologie des algèbres commutatives (SPringer) $\endgroup$ May 25, 2012 at 12:59
  • $\begingroup$ @sergio: might I be so lazy to ask you to explain the idea you're thinking about when pointing out that paper (so that I don't have to look it up myself)? $\endgroup$ May 25, 2012 at 13:14
  • $\begingroup$ @temp: Which of Serre's conjectures are you referring to? $\endgroup$ May 25, 2012 at 16:05
  • $\begingroup$ In the book "Homologie des algèbres commutatives", M. Andre uses and develops the concept of cotangent complex (does emerges from the study of differential in modules) in the context of commutative and homological algebra. Although this study is strictly inherent to algebraic geometry. $\endgroup$ May 25, 2012 at 16:48
  • $\begingroup$ this also might be relevant for mentioning here: arxiv.org/abs/1004.0096 $\endgroup$ May 25, 2012 at 20:25

1 Answer 1


One does not actually need the "power of topoi" to make Illusie's definition of the cotangent complex work. One does need sheaves, though. Illusie's constructions would have worked equally well had he used ringed spaces instead of ringed topoi. Some constructions related to diagrams of schemes might be easier to phrase in terms of topoi, and of course they become essential if one wants to work in, say, the étale topology.

Illusie makes use of the generality of his construction in a number of places. By treating ringed topoi that are not the étale or Zariski topoi of schemes he is able to treat the deformation theory of diagrams, which he applies to deformations of group schemes, commutative group schemes, and modules.

I do not know if it would be useful to study deformations of the structure sheaf on the fppf topos of a scheme $S$. Such a deformation would correspond to a consistent collection of deformations of every scheme over $S$.

  • $\begingroup$ Thanks. Actually this question was partly raised by reading your paper on THE DEFORMATION THEORY OF SHEAVES OF COMMUTATIVE RINGS. Somehow I feel like the power of the cotangent complex hasn't been fully unleashed due to various reasons (or I feel that way due to my limited knowledge...). (Also, Illusie said using the cotangent complex part of EGA IV could be rewritten, and presumably shorter, do you know which part exactly does he mean?) $\endgroup$
    – temp
    May 26, 2012 at 1:19
  • 1
    $\begingroup$ Where did Illusie say this? As far as I know, EGA IV does not contain much about the cotangent complex. $\endgroup$ May 28, 2012 at 6:45

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