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5 votes
0 answers
133 views

What classifies deformations of group schemes (or Hopf algebras)?

The cotangent complex of a scheme classifies its deformations. That is, if $X$ is a scheme over a field $k$ (with conditions?) and $\mathbf{T}^*_X\in D^b(\text{QCoh}(X))$ its cotangent complex, the ...
3 votes
0 answers
164 views

Linear deformations of a morphism between stacks

Given smooth algebraic stacks $\mathcal{X}$, $\mathcal{Y}$ what are the linear deformations $\operatorname{Def}^1(f: \mathcal{X} \to \mathcal{Y})$ of a morphism $f:\mathcal{X} \to \mathcal{Y}$? In ...
3 votes
0 answers
214 views

formal smoothness and cotangent complex

If $k$ is a field and $A$ is a formally smooth $k$-algebra, then we know that $\Omega^{1}_{A/k}$ is projective. What about its cotangent complex $L_{A/k}$? When is it quasi-isomorphic to $\Omega^{1}_{...
64 votes
5 answers
9k views

Intuition about the cotangent complex?

Does anyone have an answer to the question "What does the cotangent complex measure?" Algebraic intuitions (like "homology measures how far a sequence is from being exact") are as welcome as ...
17 votes
1 answer
848 views

Are deformations of a scheme some kind of a "derived gerbe" under the cotangent complex?

(This is probably a very naive question. My understanding of the cotangent complex is quite vague.) Let me first recall the picture for deformations of a smooth morphism: If $f:X_0\to S_0$ is a ...
15 votes
1 answer
770 views

Cotangent Complex in Analytic Category

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 ...
8 votes
1 answer
2k views

Elementary (English) reference for the cotangent complex?

I'm trying to understand cotangent complexes and their role in deformation theory, and later the statement that they're somehow natural in a derived scheme/stack. I understand that the standard ...