Introduction to Derived Algebraic Geometry

This master thesis is an attempt to give a gentle introduction to some modern topics of algebraic geometry and algebraic topology. In a certain sense, derived algebraic geometry is an application of tools borrowed from algebraic topology by algebraic geometry (while viceversa, spectral algebraic geometry is often understood as an application of algebraic geometry to algebraic topology). Derived Algebraic Geometry naturally arose as a generalization of scheme theory by replacing ordinary rings with more sophisticated objects (like simplicial rings, cdga’s or E1-rings). Derived schemes were constructed to give a right setting for dealing with some problems such as intersection theory of singular varieties, deformation problems or obstruction theories. Even though it is a relatively recent subject, plenty of literature already exists: it was simultaneously developed by Lurie in the language of 1-categories (see Lurie’s PhD thesis) and by Töen and Vezzosi in the language of model categories (in Homotopical Algebraic Geometry II); then it was carried on by the same authors cited and by many others, such as Gaitsgory, Rozenblyum, Ben-Zvi, Nadler, et al. Along this thesis, we cover the two parallel and equivalent approaches to derived algebraic geometry: the one that uses model categories and the one that uses 1-categories. Only in the last chapter we show an application of the machinery developed using the language of 1-category alone: in the proof of the Multiplicative HKR Theorem, this modern language is particularly ductile and expressive, and in some sense present some advantages with respect to the model categorical one. In the first chapter we give some concrete motivations to the study of this recent subject: problems arising from intersection theory to deformation moduli problems were solved by this new framework. We start from Serre’s intersection formula (that in the classical world has no geometric interpretation), then we take a glimpse of deformation theory and obstruction theories seeing that this new framework gives us conceptual advantages. In the second chapter we review the basic affine theory for derived algebraic geometry: as for the theory of ordinary schemes, one has to develop the affine theory first and then one can glue things together. We will see the basic notions for these new objects and their morphisms. The analogy with ordinary scheme theory carries on in many occasions, so very often the definitions are the ones expected, but there are some things to point out when making comparisons. In the third chapter we present the non-affine theory and a couple of concrete computations, just to have a taste of the theory exposed. In the derived settings things get a little bit more complicated and so the functorial point of view is fundamental. We generalize the various notions given in the affine case and we define the cotangent complex for derived stacks (one of the main characters of derived algebraic geometry). In the fourth and last chapter, we talk about some more involved applications. The main result, as we previously said, is the Multiplicative HKR Theorem (that has been proved by J. António, F. Petit and M. Porta in an article not yet submitted) which relates the Hochschild cohomology to the algebraic De Rham cohomology. We show first the plain HKR Theorem looking only to de Rham algebra functor and to the Hochschild homology functor, then we take into account the structure of mixed algebra and the S1-action. They key point is that the space of Hopf structures on the de Rham algebra is contractible so the mixed and the S1 structures coincide. This last result is heavily based on the ductility of the 1-categorical language and is a good example where one can see the usefulness of this modern language