I would like to know if one can compute all the cohomology sheaves of the cotangent complex of a subvariety of the affine space once a resolution of its ideal sheaf is given?


In my precise situation, I work with $X \subset \mathbb{A}^n$ Gorenstein of codimension 3 and I have a resolution:


$$0 \rightarrow \mathcal{O}\_{\mathbb{A}^n} \rightarrow E^\* \rightarrow E \rightarrow I\_X \rightarrow 0$$


Is it possible to compute the cotangent complex of the closed immersion $ X \hookrightarrow \mathbb{A}^n$ from this resolution?


Many thanks!


EDIT : I really want the cotangent complex of the closed immersion $X \hookrightarrow \mathbb{A}^n$ (and not of the map $X \rightarrow spec(k)$).



Computing cotangent complex
Your X is not a local complete intersection in A^n. Then the cotangent complex is going to have cohomology in infinitely many degrees. (Follows from a conjecture by Quillen proved by Avramov IIRC.) Your best bet would be to use Quillen's spectral sequence relating Tor\_\*(O/I, O/I) to the cohomology of the cotangent complex. You should read the original, but if you are in a hurry, take a look at [Example Tag 08RG](http://stacks.math.columbia.edu/tag/08RG)
of the Stacks project. If you only want to compute the first 3 terms, then I suggest looking at the method of Lichtenbaum-Schlessinger, see [Section Tag 09AM](http://stacks.math.columbia.edu/tag/09AM).


