nLab
simplicial ring

Idea

One way to think of simplicial rings is this:

the category Ring${}^{\mathrm{op}}$ is naturally equipped with the structure of a pregeometry. The corresponding geometry (for structured (∞,1)-toposes) is ${\mathrm{SRing}}^{\mathrm{op}}$, the opposite of the category of simplicial rings.

Definition

A simplicial ring is a simplicial object in the category Ring of rings.

Modules over simplicial rings

There is an evident notion of (∞,1)-category of modules over a simplicial ring. The corresponding bifibration $\mathrm{SMod}\to \mathrm{SRing}$ of modules over simplicial ring has the remarkable property that it is equivalently to that tangent (∞,1)-category of the (∞,1)-category of simplicial rings.

Model category presentation

We describe the model category presentation of the (∞,1)-category of modules over simplicial rings.

Let $A$ be a simplicial commutative algebra. Write $A\mathrm{SMod}$ for the category which, with $A$ regarded as a monoid in the category $\mathrm{SAb}$ of abelian simplicial groups is just the category of $A$-modules in $\mathrm{SAb}$. This means that

• objects are abelian simplicial groups $N$ equipped with an action morphism $A\otimes N\to N$ of simplicial abelian groups;

Equip $A\mathrm{SMod}$ with the structure of a model category by setting:

• a morphism $N_1\to N_2$ of $A$-modules is a weak equivalence resp. a fibration precisely if the underlying morphism of simplicial sets is a weak equivalence, resp. fibration, in the standard model structure on simplicial sets.

Proposition This defines a model category structure which is

• cofibrantly generated;

• proper;

• cellular.

In higher geometry

Simplicial rings are one notion of vertical categorification of rings, another generalization is E-infinity-rings. As such, simplicial rings come with their notion of higher geometry, in particular their notion of derived schemes, derived stacks and quasicoherent sheaves. An application of this appears for instance in geometric infinity-function theory.

References

For instance chapter 4 of

• Bertrand Toen, Simplicial presheaves and derived algebraic geometry , lecture at Simplicial methods in higher categories (pdf)