Derived Algebraic Geometry

Derived algebraic geometry is a very natural generalization of ordinary algebraic geometry. It takes its origin in two phenomena:

• • The fiber product of ordinary schemes often needs to be understood in the derived sense (or otherwise, base change would fail),

  • Systematically thinking about deformation theory requires derived objects.

Any classical scheme can be viewed as a derived scheme; conversely, any derived scheme has a classical scheme `underlying' it. However, the (derived) categories of quasi-coherent sheaves on the derived scheme and on the classical scheme lying under it would be very different.

Many of the naturally defined moduli problems are represented by derived stacks; this includes many stacks that appear on the `spectral' side of the Langlands correspondence. As long as we care about quasi-coherent sheaves on them (and we do), we need to admit them into our world, together with their derived structure.

Similar to the case of infinity-categories and DG categories, it is hard to find perfect references. But the only thing you really need is to believe that the derived algebraic geometry (DAG) exists, that is:

• • To a commutative DG algebra A living in non-positive cohomological degrees you can assign an affine derived scheme Spec(A) (its spectrum)

  • You can glue arbitrary DG schemes from affine DG schemes

  • DG schemes form an infinity-category

The most important thing to know is that the fiber products of affine DG schemes is formed by taking the derived tensor product of the corresponding commutative DG algebras.

The more sophisticated objects such as derived Artin stacks, etc. are constructed starting from affine DG schemes just as in the usual algebraic geometry.

• 1. That said, a good survey is this paper by Toen.

  2. Another possible set of references (close to the spirit of this workshop) are Gaitsgory's notes on stacks and quasi-coherent sheaves.

  3. A more direct approach is to take a paper using the derived algebraic geometry (such as one of the papers by Nadler and Ben-Zvi) and work through it.