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 formula_1), simplicial commutative rings or formula_2-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.
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 (i.e., fiber product of immersions) "does not" yield the correct intersection number. In the derived context, one takes the derived tensor product formula_3, 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" 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 (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 formula_6-rings.
Derived geometry over characteristic 0.
Over characteristic 0 many of the derived geometries agree since the derived rings are the same. formula_6 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 formula_8 which is a topological space formula_9 with a sheaf of commutative differential graded algebras. Sometimes authors take the convention that these are negatively graded, so formula_10 for formula_11. The sheaf condition could also be weakened so that for a cover formula_12 of formula_9, the sheaves formula_14 would glue on overlaps formula_15 only by quasi-isomorphism.
Unfortunately, over characteristic p, differential graded algebras work poorly for homotopy theory, due to the fact formula_16. 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 (here a 1-stack) is called representable if the fiber product of any two schemes is isomorphic to a scheme. If we take the ansatz that a 0-stack is just an algebraic space and 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 (n-1)-stack. If we go back to the definition of an algebraic 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 formula_17 are given by a spectrally ringed formula_18-topos formula_19 together with a sheaf of formula_20-rings formula_21 on it subject to some locality conditions similar to the definition of affine schemes. In particular
Moreover, the spectral scheme formula_9 is called non-connective if formula_31 for formula_32.
Examples.
Recall that the topos of a point formula_33 is equivalent to the category of sets. Then, in the formula_18-topos setting, we instead consider formula_18-sheaves of formula_18-groupoids (which are formula_18-categories with a single object), denoted formula_38, giving an analogue of the point topos in the formula_18-topos setting. Then, the structure of a spectrally ringed space can be given by attaching an formula_20-ring formula_41. Notice this implies spectrally ringed spaces generalize formula_20-rings since every formula_20-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 formula_18-topos, so its underlying space is a point. For example, this can be given by the ring spectrum formula_45, called the Eilenberg-Maclane spectrum, constructed from the Eilenberg-Maclane spaces formula_46.