nLab
derived algebraic geometry

Idea

The term derived algebraic geometry is used in two closely related but logically different notions:

1. as 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 and

2. as the generalizations of the algebraic geometry allowing the notion of derived moduli spaces (moduli stacks if you wish), which extend or replace the usual moduli spaces.

derived categories of sheaves on a space

In the early works of the Russian school (Kapranov, Bondal, Orlov, Kontsevich) it meant, replacing a variety by the derived category of coherent sheaves (or quasicoherent sheaves on that variety, or dg-category (or A-infinity category) enhancements thereof. There are also noncommutative deformations of such derived categories and analogues like the categories corresponding to the so-called Landau-Ginzburg models. Therefore noncommutative derived geometry? (and even noncommutative motives).

Notice that the derived category of coherent sheaves on a variety does not remember all the structure of the original variety hence derived geometry loses often some information (sometimes not); thus derived algebraic geometry is sometimes easier than nonderived.

derived noncommutative geometry: derived structure sheaves

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 nonsmoothness, 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).

“derived” in the second sense versus ”$\mathrm{\infty }$-”

The adjective “derived” means pretty much the same as the ”$\mathrm{\infty }$-” in ∞-category. It stems from the use of “derived” as in derived category and derived functor. But incidentally, derived algebraic geometry is honestly higher categorical, whereas derived categories and derived functors are really more like 1-categorical shadows of higher categorical structures, as described in detail at homotopy category.

relation to higher algebra

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 .

Definitions

derived noncommutative geometry: stable $(\mathrm{\infty },1)$-categories

details to be inserted here and harmonized with derived noncommutative geometry:

Basic idea is to identify triangulated dg-categories, categories and other models for stable (∞,1)-categories with generalized “derived” spaces and to describe morphism between them in terms of geometric morphisms between these categories. It might be noteworthy that a (accessible) stable $(\mathrm{\infty },1)$-category is much like a (Grothendieck) (∞,1)-topos. See the definition below.

structured $(\mathrm{\infty },1)$-toposes

In

• Jacob 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.

Remark (derived scheme are $(\mathrm{\infty },1)$-toposes)

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.

Examples

Homological mirror symmetry

Homological mirror symmetry is one of the main motivations and statements of the derived algebraic geometry of the first kind.

• Maxim Kontsevich, Homological algebra of mirror symmetry, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Zürich, 1994), 120–139, Birkhäuser, Basel, 1995.

• Maxim Kontsevich, Yan Soibelman, Homological mirror symmetry and torus fibrations, in Symplectic geometry and mirror symmetry (Seoul, 2000), 203–263, World Sci. Publ., River Edge, NJ, 2001.

elliptic cohomology

More recent big success of derived algebraic geometry of the second kind 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

• Jacob Lurie, A Survey of Elliptic Cohomology

References

A prediction of derived moduli spaces is spelled out (in a bit different language) in

• M. Kontsevich, Enumeration of rational curves via torus actions. The moduli space of curves (Texel Island, 1994), 335–368, Progr. Math. 129, Birkhäuser 1995. MR1363062 (97d:14077), hep-th/9405035.

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:

• M. Kapranov, Injective resolutions of BG and derived moduli spaces of local systems, J. Pure Appl. Algebra 155 (2001), no. 2-3, 167–179; math/alg-geom/9710027, MR1801413 (2002b:18017)

• I. Ciocan-Fontanine, M. Kapranov, Derived Hilbert scheme math.AG/0005155, Derived Quot scheme, math.AG/9905174

A survey of derived category apsect of the algebraic geometry and related physics (mirror symmetry, Landau-Ginzburg models) is

• A. N. Kapustin, D. O. Orlov, Lectures on mirror symmetry, derived categories, and D-branes (Russian. Russian summary) Uspekhi Mat. Nauk 59 (2004), no. 5(359), 101–134; translation in Russian Math. Surveys 59 (2004), no. 5, 907–940

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.:

• André Hirschowitz, Carlos Simpson, Descente pour les n-champs (Descent for n-stacks), math/9807049

Then the major systematic work is

• Bertrand Toën, Gabriele Vezzosi, From HAG to DAG: derived moduli stacks, in Axiomatic, enriched and motivic homotopy theory, 173–216, NATO Sci. Ser. II Math. Phys. Chem., 131, Kluwer Acad. Publ., Dordrecht, 2004, math.AG/0210407.

A set of lecture notes on the model structure on simplicial presheaves with an eye towrads algebraic sites and derived algebraic geometry is

• Bertrand Toen, Simplicial presheaves and derived algebraic geometry , lecture at Simplicial methods in higher categories (pdf)

The theory of derived algebraic geometry in the second sense is given yet another general framework in

• Jacob Lurie, Structured Spaces

which merges the structural theory developed in

• Jacob Lurie, Higher Topos Theory

with the theory developed in

• Jacob Lurie, Noncommutative algebra, Commutative algebra (see higher algebra).

A discussion of derived algebraic geometry over E-infinity rings is in

• Paul Goerss, Topological Algebraic Geometry - A Workshop

A part of (derived) algebraic geometry in the framework of $A_{\mathrm{\infty }}$-category can be found in

• L.Katzarkov, M.Kontsevich, T.Pantev, Hodge theoretic aspects of mirror symmetry arxiv:0806.0107

and a bit earlier treatise on formal (infinitesimal in the sense of formal schemes) aspect as used in the deformation theory is in

• M. Kontsevich, Y. Soibelman, Notes on A-infinity algebras, A-infinity categories and non-commutative geometry. I, math.AG/0606241

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 smooth obstruction theory of Fantechi-Behrend). See also motivic aspects in

• Maxim Kontsevich, Yan Soibelman, Stability structures, motivic Donaldson-Thomas invariants and cluster transformations, arXiv:0811.2435

The relations to tropical and symplectic geometry are in recent Kontsevich’s talk at 2009 Arbeitstagung:

• M. Kontsevich, Mathematische Arbeitstagung 2009, Symplectic geometry of homological algebra, preprint MPIM2009-40a, pdf