derived smooth geometry
symmetric monoidal (∞,1)-category of spectra
Just as algebraic geometry is the study of spaces locally modelled on commutative rings, derived algebraic geometry is the study of spaces locally modelled on “derived commutative rings”, which means either simplicial commutative rings (or equivalently when working over a base field of characteristic zero, commutative dg-rings), or E-∞ rings. The former approach has been developed by Bertrand Toen and Gabriele Vezzosi, and the latter by Jacob Lurie. Both are known to be equivalent, up to some minor differences.
In Grothendieck’s functor of points approach to the theory of schemes, a scheme over a field is viewed as a kind of functor from the category of commutative -algebras to the category of sets. Motivated by moduli problems, people have enlarged the target category to the category of groupoids, arriving at the notion of a stack. Carlos Simpson extended this further by replacing by the category of simplicial sets, arriving at the notion of higher stacks. Derived algebraic geometry may be viewed as the next step in this natural progression, replacing the source category by , the category of simplicial commutative k-algebras, or by the category of E-∞ rings. Hence a derived stack over is a kind of functor
The term derived algebraic geometry sometimes also refers to the study of derived categories of coherent sheaves on varieties (Alexei Bondal, Dmitri Orlov, …). This is related to noncommutative algebraic geometry and to derived noncommutative algebraic geometry, where one replaces a scheme by its triangulated category of perfect complexes (or derived category of quasi-coherent sheaves) or a (∞,1)-categorical enhancement thereof.
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?).
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?.
Jacob Lurie developed a version of derived algebraic geometry which is locally modelled on E-∞ rings. The derived algebraic geometries of Toen-Vezzosi and of Lurie turn out to be equivalent, up to minor differences. Their approaches differ in that the former is based on homotopy theory, in the sense that it uses the language of model toposes, while the latter is based on higher category theory and uses the language of structured (∞,1)-toposes.
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.
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.
In derived noncommutative algebraic geometry, a space is by definition a triangulated category which is smooth and proper in an appropriate sense. Given a scheme of finite type over a field, it has an associated noncommutative space which is its triangulated category of perfect complexes. admits a generator , and is equivalent to the category of perfect? dg-modules over the dg-algebra . Hence one may associate to the spectrum of this dg-algebra, which is an affine? derived scheme (when working over characteristic zero). In this way the spaces of derived noncommutative geometry that come from commutative schemes can indeed be viewed as derived schemes.
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
The two main references are
In the first volume, they develop the theory of stacks over simplicially enriched sites? and model sites?. In the second, they develop linear and commutative algebra over symmetric monoidal model categories, and then apply the theory of the first volume to develop the theory of geometric stacks over a symmetric monoidal model category.
In the following lecture notes, they study in detail various derived moduli spaces.
The following is a short exposition on some of the motivation behind derived algebraic geometry.
A general theory of derived geometry is developed in
and specialized to geometry over -rings – E-∞ geoometry? – 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