This entry is about the notion of module or quasi-coherent complexes in the setting of higher geometry and more specifically, E-infinity geometry and derived algebraic geometry.
In this setting, modules are already derived, in the sense that modules over an ordinary scheme or stack, viewed as a discrete derived scheme or derived stack, are complexes of quasi-coherent sheaves.
Let $Sch^{aff}$ and $Stk$ denote the (infinity,1)-categories of affine derived schemes and derived stacks, respectively. Consider the (infinity,1)-prestack of stable (infinity,1)-categories
which associates to a commutative ring spectrum $A$ the stable (infinity,1)-category $Mod(A)$.
By taking the right Kan extension of this prestack along the (opposite of the) Yoneda embedding
one gets an (infinity,1)-prestack
In other words, for a derived stack $X$, $Mod(X)$ is given by the limit
In the case of ordinary affine schemes, modules in this sense, i.e. modules over Eilenberg-Mac Lane spectra, correspond by the stable Dold-Kan correspondence to chain complexes. The corresponding notion of module over an ordinary scheme or stack is then a quasi-coherent complex. That is, for a commutative ring $A$,
(the derived category of chain complexes of $A$-modules), and for a classical scheme $X$,
(the derived category of chain complexes of quasi-coherent sheaves).
The (infinity,1)-prestack $Mod$ satisfies Zariski descent and even Nisnevich descent; this is due to Jacob Lurie and Vladimir Drinfeld.
$Mod$ lifts to a prestack of symmetric monoidal (infinity,1)-categories. The dualizable objects are precisely the perfect modules. In good cases, the stable (infinity,1)-category $Mod(X)$ is compactly generated and the compact objects are precisely the perfect modules.
Dennis Gaitsgory, Notes on geometric Langlands: quasi-coherent sheaves.
Bertrand Toën, Gabriele Vezzosi, Homotopical algebraic geometry II: geometric stacks and applications, 2004, arXiv:math/0404373.
Jacob Lurie, Derived Algebraic Geometry, thesis, pdf.
David Ben-Zvi, John Francis, David Nadler, section 3.1 of Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry, J. Amer. Math. Soc. 23 (2010), no. 4, 909-966 (arXiv:0805.0157)
B. Toen, Derived Azumaya algebras and generators for twisted derived categories, arXiv:1002.2599.
Last revised on February 7, 2015 at 10:42:39. See the history of this page for a list of all contributions to it.