higher geometry / derived geometry
geometric little (∞,1)-toposes
geometric big (∞,1)-toposes
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, nonpositively? graded? commutative dg-rings. The fundamentals of the subject have been developed by Bertrand Toen and Gabriele Vezzosi, and by Jacob Lurie.
In Grothendieck’s functor of points approach to the theory of schemes, a scheme $X$ over a field $k$ is viewed as a functor from the category $CAlg_k$ of commutative $k$-algebras to the category $Set$ of sets, satisfying a descent condition. Motivated by moduli problems, people have enlarged the target category to the category $Grpd$ of groupoids, arriving at the notion of a stack. Carlos Simpson extended this further by replacing $Grpd$ by the category $SSet$ of simplicial sets, arriving at the notion of higher stacks. Derived algebraic geometry may be viewed as a further step in this progression, replacing the source category $CAlg_k$ by $SAlg_k$, the category of simplicial commutative k-algebras. This is motivated by classical concerns involving 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 E-infinity rings are used instead of simplicial rings. Derived algebraic geometry may also refer to the study of derived categories of coherent sheaves on varieties as studied in noncommutative algebraic geometry, where one replaces a scheme $X$ by its triangulated category of perfect complexes.
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).
Universal elliptic cohomology (topological modular forms). See below.
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.
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.
More recent big success of derived algebraic geometry locally modeled on $E_\infty$-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
For references on dg-schemes, the historical precursor to derived stacks, see there.
A general compact survey is
The two main references are
Bertrand Toën, Gabriele Vezzosi, Homotopical algebraic geometry I: topos theory, 2002, arXiv:math/0207028.
Bertrand Toën, Gabriele Vezzosi, Homotopical algebraic geometry II: geometric stacks and applications, 2004, arXiv:math/0404373.
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.
The following is an exposition of the theory of derived schemes modelled on nonpositively graded dg-algebras.
A general theory of derived geometry is developed in
and specialized to geometry over $E_\infty$-rings – E-∞ geometry – 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