nLab module in higher geometry

Idea

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.

Definition

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

$$Mod : (Sch^{aff})^{op} \to Cat^{stab}_\infty$$

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

$$(Sch^{aff})^{op} \hookrightarrow Stk^{op},$$

one gets an (infinity,1)-prestack

$$Mod : Stk^{op} \to Cat^{stab}_\infty.$$

In other words, for a derived stack $X$, $Mod(X)$ is given by the limit

$$Mod(X) = lim_{Spec(A) \to X} Mod(A).$$

In ordinary algebraic geometry

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 $A$,

$$Mod(H A) = D(Mod(A))$$

(the derived category of chain complexes of $A$-modules), and for a classical scheme $X$,

$$Mod(X) = D(QCoh(A))$$

(the derived category of chain complexes of quasi-coherent sheaves).

Properties

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.

Related concepts

• perfect module
• triangulated categories of sheaves

References

• 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.