Cotangent Complex in Analytic Category

Question

I am looking for a reference which develops the theory of the cotangent complex for complex analytic spaces. I need this to justify some computations I did assuming some formal properties which hold in the algebraic category. Here is a list of properties I want (some might be deducible from others, I haven't checked this carefully):

(1) To each analytic space $X$, assign a well-defined object $\mathbb{L}^\bullet_X$ in the derived category of complexes of $\mathcal O_X$-modules with coherent cohomology sheaves, supported in degrees $\le 0$.

(2) For smooth $X$, we must have a (quasi-)isomorphism $\mathbb L^\bullet_X = \Omega^1_X[0]$, where the right hand side is the holomorphic cotangent sheaf in degree $0$. For each morphism $f:X\to Y$, we have a morphism in the derived category $f^\dagger:Lf^*\mathbb L^\bullet_Y\to\mathbb L^\bullet_X$ (which is the obvious one if $X,Y$ are smooth, i.e., the adjoint of $df$ between the cotangent spaces) and whose cone defines the relative cotangent complex $\mathbb L^\bullet_{X/Y}$ (or $\mathbb L^\bullet_f$), which is an object living on $X$. The assignment $f\mapsto f^\dagger$ respects composition of maps.

(3) For a closed immersion $X\subset Y$ with $Y$ smooth and $\mathscr I$ the ideal sheaf of $X$ in $Y$, we have $h^0(\mathbb L^\bullet_{X/Y})=0$ and $h^{-1}(\mathbb L_{X/Y}^\bullet) = \mathscr I/\mathscr I^2$. Further, the truncation $\tau_{\ge-1}\mathbb L^\bullet_X$ can be identified with $d:\mathscr I/\mathscr I^2\to\Omega^1_Y|_X$ (living in degrees $-1$ and $0$) in a way that is compatible with the exact triangle $Lf^*\mathbb L^\bullet_Y\to\mathbb L^\bullet_X\to\mathbb L^\bullet_{X/Y}\to Lf^*\mathbb L^\bullet_Y[1]$ coming from part (2).

(4) More generally, for any sequence of maps $X\to Y\to Z$, there's an exact sequence $Lf^*\mathbb L^\bullet_{Y/Z}\to\mathbb L^\bullet_{X/Z}\to\mathbb L^\bullet_{X/Y}\to Lf^*\mathbb L^\bullet_{Y/Z}[1]$ where $f:X\to Y$ is the first map in the above.

(5) Crucially, for a morphism $X\to Y$ of algebraic schemes, $\mathbb L^\bullet_{X/Y}$ should be functorially (quasi-)isomorphic to the analytification of the cotangent complex defined in the Stacks Project (tag 08T1).

I'm actually happy if I just have a reference which treats the corresponding statements for just $\tau_{\ge-1}\mathbb L^\bullet_X$ (called the "Naive cotangent complex" in the Stacks Project). Also, if you can't find any reference which treats this, I'd also find useful a sketch of how to go about proving this in the analytic category.

Apologies in advance if this is too basic a question, I'm rather new to these concepts.

Answer 1

I will attack the problem three ways, in increasing level of elaboration.

1. Here is a recent paper that proves all this. Compared to older sources, this paper uses more machinery. It uses stabilization / Goodwillie calculus, which is more directly about deformations and has to be proved to be about tangents, rather than vice versa. It uses derived algebraic geometry, which is hardly more work because, say, Illusie, was secretly building DAG. But it puts that in the foreground, especially the language of ∞-categories. Whereas, you might prefer black box statements about ordinary categories that bury the ∞-categories in proofs.

2. Lots of sources (eg, the Stacks project and Illusie) define the cotangent complex for ringed spaces (and ringed topoi). So that gives you a definition. They probably prove (4) in that generality. Illusie II 6.6.2.3 explicitly mentions the analytic case, but maybe he just says that it’s easy and doesn’t prove anything. Someone must have elaborated on this, but I don’t know where.

2a. As I understand it, Illusie takes the cotangent complex for rings and (homotopy) sheafifies it. This is difficult, but once he has created a coherent global object, we can study it locally and reduce to statements about rings. I believe that I have a slick proof of the comparison (5). Consider the map from the analytic space to the algebraic space. This has a relative cotangent complex. Since the map of rings is formally étale this cotangent complex is locally zero (ie, acyclic) to the sheaf is globally zero. Now apply (4) and the acyclic cone implies that the comparison map (5) is a quasi-isomorphism. Moreover, (5) implies (2)+(3) because every analytic morphism is locally isomorphic to an algebraic morphism.

2b. To build the analytic theory on its own and calculate the cotangent complex (2) and (3) without recourse to comparison (5), we would like to reduce to the case of rings. Rings of analytic rings are very similar to algebraic rings and (2) and (3) follow mutatis mutandis. We start with rings and sheafify, but we have to worry that sheafification is destructive. What we really want for a nice theory is that the presheaf that we are sheafifying was already a (homotopy) sheaf (ie, satisfied Mayer-Vietoris); we want sheafification to extend it to a global object, but not to change it on nice open sets (ie, Stein or affine ones). This is not true in general, as in the example of big topoi. But I believe that it is true for the Zariski and analytic topologies because the restriction map of the structure sheaf is formally smooth, so the restriction of the cotangent complex is just given by tensoring. Thus we can reduce to the cotangent complex of rings. (Actually, maybe you don’t need the general statement, but could do (2) and (3) by hand because (2) is a sheaf in a single degree, which is a homotopy sheaf and (3) is more subtle, but you only care about a single cohomology sheaf, which is a sheaf; you just need to check that the lower cohomology doesn’t mess it up.)

Luc Illusie. Complexe cotangent et déformations. I, 1971, LNM. 239; II, 1972, LNM 283.