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 $X$,

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

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.