derived smooth geometry
symmetric monoidal (∞,1)-category of spectra
What has been called derived algebraic geometry by (Kapranov), (Toën-Vezzosi), (Lurie) is higher geometry locally modeled on formal duals of commutative simplicial rings, dg-algebras and generally E-∞ rings, hence the derived geometry refinement of traditional algebraic geometry.
The subject of derived algebraic geometry stem from (related) attempts to
(cf. Tabuada, Kontsevich, Enumeration 1994) to generalize the algebraic geometry allowing the notion of derived moduli spaces (moduli stacks if you wish), which extend or replace the usual moduli spaces. The classical moduli spaces are in some cases their truncations. Locally these moduli spaces should correspond to the deformation theory problem which geometrically realizes the cotangent complex for the deformation problem; the derived moduli spaces often exhibit smoothness which is lost at the truncated level (hidden smoothness principle). This should make sense to the virtual fundamental class and virtual dimension to moduli spaces related to GW and Donaldson-Thomas invariants.
to correct the intersection theory to a derived version, which would relax special position/transversality requirements and make the (co)homological formulas for intersection multiplicities (cf. Serre intersection formula, Kontsevich, Enumeration 1994 1.4.2) more natural.
(for noncommutative flavour) to study the part of the geometry of schemes captured cohomologically (by the derived category of coherent sheaves), which presents the kind of data very close to the interests of the classical Italian school; more generally a “derived” variety is defined as an -category with certain good properties (analogues of proper, smooth etc.), Fukaya category of a symplectic manifold gives an example.
The adjective “derived” means pretty much the same as the ”-” in ∞-category, so this is higher algebraic geometry in the sense being locally represented by higher algebras. The word stems from the use of “derived” as in derived functor. This came from the study of derived moduli problem. Namely to parametrize the moduli, one first looks at some space of “cochains” which are candidates for structures to parametrize. then one cuts those which indeed satisfy the axioms (“equations”) for the structures. Satisfying these equations is a limit-type construction, hence left exact and one is lead to right derived functors to improve; exactness on the right; this leads to use cochain complexes. The obtained moduli is too big as there are many isomorphic structures, so one needs to quotient by the automorphisms; this is a colimit type construction hence right exact. The improved quotient is the left derived functor, what is obtained by passing to stacks.
Cohomological information of a scheme leads to replacing variety by a stable -category of objects which generalize quasicoherent sheaves. The traditional variant are the enhanced triangulated categories or -categories. This is usually called derived noncommutative algebraic geometry.
The derived moduli spaces are realized by as derived stacks locally representable by commutative dgas.
On the other hand there is a closely related effort to include sheaves of commutative dg-algebras as structure sheaves (dg-schemes of Kapranov, Ciocan-Fontaine, and Kontsevich) and more generally to allow higher categorical structured spaces of algebraic type, generalizing algebraic stacks, schemes and algebraic spaces. This is a higher categorical version of algebraic geometry: its vertical categorification is also called derived algebraic geometry. Notice that in that sense, there is no loss of information in a passage from a scheme to its natural extension to a derived scheme.
This second school has been, after the original ideas of Deligne, Drinfel’d and Kontsevich advanced by Carlos Simpson (who introduced also basic prerequisited like algebraic and geometric n-stacks), and later Bertrand Toen and coworkers. One of the main motivations for both variants of derived algebraic geometry is to develop a satisfactory deformation theory and on its basis the theory of moduli stacks in algebraic geometry beyond the few examples which work in classical language of algebraic spaces and algebraic 1-stacks.
Sometimes, but not always getting rid of limitations coming from 1-categorical truncations removes non-smoothness, but the expectations in that directions (hidden smoothness principle) failed in generality expected at the beginning. The construction of the derived moduli spaces relies, similarly to the classical moduli theory in algebraic geometry, on the infinitesimal case – the deformation theory (cf. cotangent complex).
Where ordinary algebraic geometry uses algebra to describe geometry, derived algebraic geometry uses higher algebra. Where ordinary algebraic geometry uses schemes modeled on commutative rings, derived algebraic geometry uses structured (∞,1)-toposes modeled on E-∞ rings and noncommutative derived algebraic geometry models on A-infinity rings, but no topos theory is developed there.
For the derived noncommutative geometry see there.
In Lurie, Structured spaces a definition of derived algebraic scheme? and derived Deligne-Mumford stack is given in the wider context of generalized schemes realized as locally affine structured (∞,1)-toposes.
See these links for more details.
This definition is based on the observation that it is a deficiency of the ordinary definition of scheme to demand that underlying a scheme is a topological space and that a better definition is obtained by demanding it to have an underlying locale. But a locale is a 0-topos. This motivates then the definition of a generalized scheme as a (locally affine, structured) (∞,1)-topos.
Gromov-Witten invariants and homological mirror symmetry is one of the main motivations and statements of the derived noncommutative algebraic geometry via -categories, see there for references. It also lead to the study of derived moduli spaces (Kontsevich, Enumeration 1994) developed later by Kapranov, Toën, Lurie and others.
More recent big success of derived algebraic geometry locally modeled on -rings was elliptic cohomology and the construction and study of the tmf spectrum as a certain derived moduli “of derived elliptic curves”. This construction of moduli space is based on earlier Lurie result (not available in full) in which Lurie has proved an analogue of the Artin’s representability theorem from the algebraic geometry of Grothendieck school. For more on that see
A prediction of derived moduli spaces is spelled out (in a bit different language) in
An early variant, the dg-schemes, were used to construct some derived moduli spaces for the first time in the works of Kapranov and Ciocan-Fontanine:
A survey of derived category apsect of the algebraic geometry and related physics (mirror symmetry, Landau-Ginzburg models) is
A major case when derived geometry in the first sense gives full information is given by a reconstruction theorem of Bondal-Orlov:
A. I. Bondal, D. O. Orlov, Reconstruction of a variety from the derived category and groups of autoequivalences, Compos. Math. 125 (2001), 327–344 doi:10.1023/A:1002470302976
A. I. Bondal, D. O. Orlov, Derived categories of coherent sheaves, Proc. Internat. Congress of Mathematicians (Beijing, 2002)
The higher stacks and algebraic stacks were pioneered by ideas of Simpson’s school. Here is one of the first successes, used later by Toen et al.:
Then the major systematic work is
A set of lecture notes on the model structure on simplicial presheaves with an eye towrads algebraic sites and derived algebraic geometry is
A general theory of derived geometry is developed in
and specialized to geometry over -rings in
which merges the structural theory developed in
with the algebraic theory developed in
A discussion of derived algebraic geometry over E-infinity rings is in
A framework of derived noncommutative algebraic geometry in the language of -categories can be found in
This formal aspect is supposedly related to the infinitesimal picture of the moduli stacks considered by Toen et al. and it generalizes more classical approaches to the deformation theory like Illusie’s cotangent complex (cf. also perfect obstruction theory of Fantechi-Behrend). See also motivic aspects in the entry noncommutative motive and
Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435
Anatoly Preygel, Thom-Sebastiani and duality for matrix factorizations, arxiv/1101.5834