Derived algebraic geometry

Derived algebraic geometry is a branch of mathematics that generalizes algebraic geometry to a situation where commutative rings, which provide local charts, are replaced by either differential graded algebras (over ${\displaystyle \mathbb {Q} }$), simplicial commutative rings or ${\displaystyle E_{\infty }}$-ring spectra from algebraic topology, whose higher homotopy groups account 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) of singular algebraic varieties and cotangent complexes in deformation theory (cf. J. Francis), among the other applications.

Basic objects of study in the field are derived schemes and derived stacks. The oft-cited motivation is Serre's intersection formula. In the usual formulation, the formula involves the 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. See Theorem 3.22 in Khan, where derived intersection theory has been developed.

The term "derived" is used in the same way as derived functor or derived category, in the sense that the category of commutative rings is being replaced with a ∞-category of "derived rings." In classical algebraic geometry, the derived category of quasi-coherent sheaves is viewed as a triangulated category, but it has natural enhancement to a stable ∞-category, which can be thought of as the ∞-categorical analogue of an abelian category.

Derived algebraic geometry is fundamentally the study of geometric objects using homological algebra and homotopy. Since objects in this field should encode the homological and homotopy information, there are various notions of what derived spaces encapsulate. The basic objects of study in derived algebraic geometry are derived schemes, and more generally, derived stacks. Heuristically, derived schemes should be functors from some category of derived rings to the category of sets

which can be generalized further to have targets of higher groupoids (which are expected to be modelled by homotopy types). These derived stacks are suitable functors of the form

Many authors model such functors as functors with values in simplicial sets, since they model homotopy types and are well-studied. Differing definitions on these derived spaces depend on a choice of what the derived rings are, and what the homotopy types should look like. Some examples of derived rings include commutative differential graded algebras, simplicial rings, and ${\displaystyle E_{\infty }}$-rings.

Over characteristic 0 many of the derived geometries agree since the derived rings are the same. ${\displaystyle E_{\infty }}$ algebras are just commutative differential graded algebras over characteristic zero. We can then define derived schemes similarly to schemes in algebraic geometry. Similar to algebraic geometry, we could also view these objects as a pair ${\displaystyle (X,{\mathcal {O}}_{X}^{\bullet })}$ which is a topological space ${\displaystyle X}$ with a sheaf of commutative differential graded algebras. Sometimes authors take the convention that these are negatively graded, so ${\displaystyle {\mathcal {O}}_{X}^{n}=0}$ for ${\displaystyle n>0}$. The sheaf condition could also be weakened so that for a cover ${\displaystyle U_{i}}$ of ${\displaystyle X}$, the sheaves ${\displaystyle {\mathcal {O}}_{U_{i}}^{\bullet }}$ would glue on overlaps ${\displaystyle U_{ij}}$ only by quasi-isomorphism.

Unfortunately, over characteristic p, differential graded algebras work poorly for homotopy theory, due to the fact ${\displaystyle d[x^{p}]=pd[x^{p-1}]}$[1]. This can be overcome by using simplicial algebras.