# Derived algebraic geometry

Derived algebraic geometry (also called spectral algebraic geometry[1]) is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, 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 idea, and provides natural settings for intersection theory (or motivic homotopy theory[2]) of singular algebraic varieties and cotangent complexes in deformation theory (cf. J. Francis), among the other applications. No applications for problems in classical algebraic geometry are known (in contrast to scheme theory).

## 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.[3] 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 ${\displaystyle A\otimes ^{L}B}$, 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 hypothetical; 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.

According to Justin Curry:[4]

The need for a derived perspective can be stated with one picture. In Figure 25 two maps are drawn to the two-sphere S².

One is defined on the wedge sum S²∨S¹ and maps the S¹ to a point. The other is defined on the closed disk ${\displaystyle \mathbb {D} ^{2}}$ and maps the boundary circle to a point. If one is only allowed to look at the homology of the fiber for both of these maps, they will not be able to tell them apart. The derived category is the universal solution to this problem, as well as many others.

1. ^ Some authors (e.g., Lurie) use the term "derived algebraic geometry" for the approach based on simplicial commutative rings and the term "spectral algebraic geometry" for the approach based on ${\displaystyle {\textbf {E}}_{\infty }}$-ring spectra. Over a field of characteristic zero, the distinction is usually insignificant.