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
-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[1]) of singular algebraic varieties and cotangent complexes in deformation theory (cf.
The oft-cited motivation is Serre's intersection formula.
[2] 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.
Hence, the "derived" fiber product yields the correct intersection number.
(Currently this is hypothetical; the derived intersection theory has yet to be developed.)
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.
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
Similar to algebraic geometry, we could also view these objects as a pair
with a sheaf of commutative differential graded algebras.
Sometimes authors take the convention that these are negatively graded, so
Unfortunately, over characteristic p, differential graded algebras work poorly for homotopy theory, due to the fact
Derived rings over arbitrary characteristic are taken as simplicial commutative rings because of the nice categorical properties these have.
[3] In fact, it is a theorem of Quillen's that the model structure on simplicial sets can be transferred over to simplicial commutative rings.
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[4] 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.
[5] 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.
Their definition requires a fair amount of technology in order to precisely state.
on it subject to some locality conditions similar to the definition of affine schemes.
Then, the structure of a spectrally ringed space can be given by attaching an
Notice this implies that spectrally ringed spaces generalize