DAG I: the cotangent complex and derived de Rham cohomology

In this series of lectures, I will give an introduction to derived algebraic geometry aimed at algebraic geometers. The first lecture will introduce simplicial commutative rings and use them to define the cotangent complex and derived de Rham cohomology with several examples. The second lecture will introduce derived stacks and the moduli stack of objects in a derived category. Then, I will give the geometricity theorem of Toën--Vaquié and describe the cotangent complex to the moduli stack. In the third lecture, we will use the machinery developed in the first two lectures to study three examples: cohomology as maps to a geometric derived stack, the (derived) Picard stack, and the stack of Fourier--Mukai equivalences.

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