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 affSch^{aff} and StkStk 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) opCat stab Mod : (Sch^{aff})^{op} \to Cat^{stab}_\infty

which associates to a commutative ring spectrum AA the stable (infinity,1)-category Mod(A)Mod(A).

By taking the right Kan extension of this prestack along the (opposite of the) Yoneda embedding

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

one gets an (infinity,1)-prestack

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

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

Mod(X)=lim Spec(A)XMod(A). 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 AA,

Mod(HA)=D(Mod(A)) Mod(H A) = D(Mod(A))

(the derived category of chain complexes of AA-modules), and for a classical scheme XX,

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

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

Properties

The (infinity,1)-prestack ModMod satisfies Zariski descent and even Nisnevich descent; this is due to Jacob Lurie and Vladimir Drinfeld.

ModMod 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)Mod(X) is compactly generated and the compact objects are precisely the perfect modules.

References

Last revised on February 7, 2015 at 10:42:39. See the history of this page for a list of all contributions to it.