Notes on derived algebraic geometry

These are notes on derived algebraic geometry in the context of animated rings. More precisely, we recall the proof of Toën-Vaquié that the derived stack of perfect complexes is locally geometric in the language of $\mathrm{\infty }$-categories. Along the way, we recall the necessary notions in derived commutative algebra and derived algebraic geometry. We also analyze the deformation theory and quasi-coherent modules over derived stacks.