higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
derived smooth geometry
symmetric monoidal (∞,1)-category of spectra
Derived algebraic geometry is the specialization of higher geometry and homotopical algebraic geometry to the (infinity,1)-category of simplicial commutative rings (or sometimes, coconnective commutative dg-algebras). Hence it is a generalization of ordinary algebraic geometry where instead of commutative rings, derived schemes are locally modelled on simplicial commutative rings.
Derived algebraic geometry is the correct setting for certain problems arising in algebraic geometry that involve intersection theory and deformation theory (see below).
Sometimes the term derived algebraic geometry is also used for the related subject of spectral algebraic geometry, where commutative ring spectra are used instead of simplicial commutative rings. Sometimes it may also refer to the subject of derived noncommutative algebraic geometry.
There are several motivations for the study of derived algebraic geometry.
The hidden smoothness principle of Maxim Kontsevich, which conjectures that in classical algebraic geometry, the non-smoothness? of certain moduli spaces is a consequence of the fact that they are in fact truncations of derived moduli stacks (which are smooth).
Intersection theory: a geometric interpretation of the Serre intersection formula for non-flat intersections.
Deformation theory: a geometric interpretation of the cotangent complex. (In derived algebraic geometry, the cotangent complex $\mathbf{L}_X$ of $X$ is its cotangent space.
For more detail on the final two points, see (Vezzosi, 2011).
The original approach to derived algebraic geometry was via dg-schemes, introduced by Maxim Kontsevich. Using dg-schemes, Ionut Ciocan-Fontanine and Mikhail Kapranov constructed the first derived moduli spaces (derived Hilbert scheme and derived Quot scheme).
Bertrand Toen and Gabriele Vezzosi developed homotopical algebraic geometry, which is algebraic geometry in any HAG context, i.e. over any symmetric monoidal model category satisfying certain assumptions. As special cases they recover the algebraic geometry of Grothendieck and the higher stacks of Carlos Simpson, and also develop new theories of derived algebraic geometry, complicial algebraic geometry, and brave new algebraic geometry?.
In his thesis Jacob Lurie also developed fundamentals of derived algebraic geometry, using the language of structured (infinity,1)-toposes where Toen-Vezzosi used model toposes. He also developed a version of derived algebraic geometry which is locally modelled on E-∞ rings, called spectral algebraic geometry.
The adjective “derived” means pretty much the same as the “$\infty$-” 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.
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.
Under some conditions, derived schemes $X$ in the sense of (Lurie, Structured Spaces) are faithfully encoded by their stable (∞,1)-categories $QCoh(X)$ of quasicoherent sheaves. This is the content of Tannaka duality for geometric stacks, (Lurie, Quasi-Coherent Sheaves and Tannaka Duality Theorems). Therefore one can turn this around and declare that a suitable stable (∞,1)-category $\mathcal{A}$ which is not of the form $QCoh(X)$ for an actual derived scheme $X$ represents a generalized, “noncommutative” derived scheme. This is much like a 2-category theory or rather (∞,2)-category theory analog of how in algebraic geometry the opposite of the category of monoids (algebras) is regarded as a category of generalized spaces. This might be (and has been) called 2-algebraic geometry.
Accordingly, one can decide to regard the opposite (∞,1)-category of suitable (e.g. monoidal) stable (∞,1)-categories as being a category of “noncommutative derived schemes”. This is effectively the perspective on noncommutative algebraic geometry that Maxim Kontsevich has been promoting.
Often and traditionally, all this is expressed in terms of certain presentations for these stable (∞,1)-categories by triangulated derived categories or better, enhancements as dg-categories.
In this fashion then in derived noncommutative algebraic geometry, a space is by definition a dg-category that is smooth and proper in an appropriate sense. The relation between noncommutative algebraic geometry and derived algebraic geometry may then be summed up by the adjunction
where $Pf(X)$ denotes the dg-category of perfect complexes on the derived stack $X$, and $\mathcal{M}_\mathcal{C}$ denotes the derived moduli stack of objects in the dg-category $\mathcal{C}$. See derived moduli stack of objects in a dg-category for details.
(For references on dg-schemes, the historical precursor to derived schemes, see there.)
The main references are
Bertrand Toën, Gabriele Vezzosi, Homotopical algebraic geometry II: geometric stacks and applications, 2004, arXiv:math/0404373.
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.
The following notes deal with the theory modelled on coconnective commutative dg-algebras.
See also the surveys
Bertrand Toën, Higher and derived stacks: a global overview, lecture notes, 2006, arXiv:math/0604504.
Gabriele Vezzosi, What is a derived stack?, 2011, pdf.