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, simplicial commutative rings or -ring spectra from algebraic topology, whose higher homotopy groups account for the non-discreteness 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 of singular algebraic varieties and cotangent complexes in deformation theory, among the other applications.

Introduction

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 does not yield the correct intersection number. In the derived context, one takes the derived tensor product, 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.
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.

Definitions

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. 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 -rings.

Derived geometry over characteristic 0

Over characteristic 0 many of the derived geometries agree since the derived rings are the same. 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 which is a topological space with a sheaf of commutative differential graded algebras. Sometimes authors take the convention that these are negatively graded, so for. The sheaf condition could also be weakened so that for a cover of, the sheaves would glue on overlaps only by quasi-isomorphism.
Unfortunately, over characteristic p, differential graded algebras work poorly for homotopy theory, due to the fact . This can be overcome by using simplicial algebras.

Derived geometry over arbitrary characteristic

Derived rings over arbitrary characteristic are taken as simplicial commutative rings because of the nice categorical properties these have. In particular, the category of simplicial rings is simplicially enriched, meaning the hom-sets are themselves simplicial sets. Also, there is a canonical model structure on simplicial commutative rings coming from simplicial sets. In fact, it is a theorem of Quillen's that the model structure on simplicial sets can be transferred over to simplicial commutative rings.

Higher Stacks

It is conjectured there is a final theory of higher stacks which model homotopy types. Grothendieck conjectured these would be modelled by globular groupoids, or a weak form of their definition. Simpson gives a useful definition in the spirit of Grothendieck's ideas. Recall that an algebraic stack is called representable is the fiber product of any two schemes is isomorphic to a scheme. If we take the ansatz a 0-stack is just an algebraic space, a 1-stack is just a stack, we can recursively define an n-stack as an object such that the fiber product along any two schemes is an -stack. If we go back the definition of an algberaic stack, this new definition agrees.

Spectral schemes

Another theory of derived algebraic geometry is encapsulated by the theory of spectral schemes. Their definition requires a fair amount of technology in order to precisely state. But, in short, spectral schemes are given by a spectrally ringed -topos together with a sheaf of -rings on it subject to some locality conditions similar to the definition of affine schemes. In particular
  1. must be equivalent to the -topos of some topological space
  2. There must exist a cover of such that the induced topos is equivalent to a spectrally ringed topos for some -ring
Moreover, the spectral scheme is called non-connective if for.

Examples

Recall that the topos of a point is equivalent to the category of sets. Then, in the -topos setting, we instead consider -sheaves of -groupoids, denoted, giving an analogue of the point topos in the -topos setting. Then, the structure of a spectrally ringed space can be given by attaching an -ring. Notice this implies spectrally ringed spaces generalize -rings since every -ring can be associated with a spectrally ringed site.
This spectrally ringed topos can be a spectral scheme if the spectrum of this ring gives an equivalent -topos, so its underlying space is a point. For example, this can be given by the ring spectrum, called the Eilenberg-Maclane spectrum, constructed from the Eilenberg-Maclane spaces.

Applications

En and E∞ -rings