Derived algebraic geometry

Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide a local chart, are replaced by ring spectra in algebraic topology, whose higher homotopy accounts for the non-discreteness (e.g, Tor) of the structure sheaf. Grothendieck's scheme theory allows the structure sheaf to carry nilpotent elements. Derived algebraic geometry can be thought of as an extension of this, and provides a natural setting for cotangent complexes and such in deformation theory (cf. F. Francis).

Introduction

Basic objects of study in the field are derived schemes and derived stacks; they generalize, for instance, differential graded schemes. The oft-cited example is Serre's intersection formula.^[1] In the usual formulation, the formula involves Tor functor and thus, unless higher Tor vanish, the scheme-theoretic intersection (i.e., fiber product of immersions) does not yield the correct intersection number. In the derived context, one takes the derived tensor product $A \otimes^L B$, whose higher homotopy is higher Tor, whose Spec is not a scheme but a derived scheme. Hence, the "derived" fiber product yields the correct intersection number. (Currently this is just a theory; the derived intersection theory has yet to be developed.)

The term "derived" comes from derived category. It is classic that many operations in algebraic geometry make sense only in the derived category of say quasi-coherent sheaves, rather than the category of such. In the much same way, one usually talks about the ∞-category of derived schemes, etc, as opposed to ordinary category.

See also

• Noncommutative algebraic geometry
• Simplicial commutative ring
• Algebra over an operad
• En-ring
• Higher Topos Theory
• ∞-topos

Notes

1. http://mathoverflow.net/questions/12236/serre-intersection-formula-and-derived-algebraic-geometry

References

• Ben-Zvi, D., Francis, J., and D. Nadler. Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry.
• Francis, John; Derived Algebraic Geometry Over ${\mathcal {E}}_{n}$-Rings
• Toën, Bertrand (2014-01-06). "Derived Algebraic Geometry". arXiv:1401.1044.
• Toën, Bertrand; Vezzosi, Gabriele (2004). "From HAG to DAG: derived moduli stacks". In Greenlees, J. P. C. Axiomatic, enriched and motivic homotopy theory. Proceedings of the NATO Advanced Study Institute, Cambridge, UK, September 9–20, 2002. NATO Science Series II: Mathematics, Physics and Chemistry. 131. Dordrecht: Kluwer Academic Publishers. pp. 173–216. ISBN 1-4020-1833-9. Zbl 1076.14002.
• Vezzosi, Gabriele (2011). "What is ...a derived stack?" (PDF). Notices Am. Math. Soc. 58 (7): 955–958. Zbl 1228.14004.

External links

• Jacob Lurie's Home Page
• DAG reading group at Harvard
• http://ncatlab.org/nlab/show/derived+algebraic+geometry
• Michigan Derived Algebraic Geometry RTG Learning Workshop, 2012
• http://mathoverflow.net/questions/217792/derived-algebraic-geometry-how-to-reach-research-level-math/219361#219361
• Gabriele Vezzosi, An overview of derived algebraic geometry, October 2013