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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 …

Let $L$ be the complex $[ f^\ast \Omega_X \rightarrow \Omega_C(D) ]$, concentrated in degrees $[-1,0]$ on $C$. One way to understand the obstructions is to understand the deformations for affine curv …

The answer is yes. Let $[E^{-1} \rightarrow E^0]$ be a perfect obstruction theory on $M$. After localizing in $M$ we can assume that the map $E^0 \rightarrow \Omega_M$ is induced as $\mathcal{O}_M \ …

Perhaps one way of reading this question is "Why is it important to think about complexes when doing deformation theory?" One must certainly accept that deformation theory is cohomological in nature: …

The cotangent bundle $\Omega_X$ of a smooth scheme $X$ shows up in two places in my understanding of algebraic geometry. The first is deformation theory, where maps out of $\Omega_X$ control the infi …

I'd like to know about the history of obstruction theories in algebraic geometry, as well as the relationship with concepts of the same name in topology. I would also like to know where obstruction t …

A representation of G on a vector space V is a descent datum for V, viewed as a vector bundle over a point, to BG. That is, linear representations of G are "the same" as vector bundles on BG. So the …

First a correction: the cotangent complex of a local complete intersection embedding is concentrated in degree -1, not in degree 1. In general, the cotangent complex of an algebraic space can be supp …