nLab
simplicial ring

Idea

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

It may be understood conceptually as follows:

• as ordinary rings are algebras over the ordinary algebraic theory $T$ of rings, if we regard this as an (∞,1)-algebraic theory then simplicial rings model the $(\mathrm{\infty },1)$-algebras over that;

• 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.

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 is equivalently to the tangent (∞,1)-category of the (∞,1)-category of simplicial rings.

Properties

Homotopy groups

Given a simplicial ring $A_{•}$, it is standard that its connected components (the 0th “homotopy group”) ${\pi }_0(A)$ is an ordinary ring and ${\pi }_n(A)$ is a module over it, so $A_{•}$ is weakly contractible iff ${\pi }_0(A)=0$.

Also, ${\pi }_0(A)$ is just the cokernel of $d_0-d_1:A_1\to A_0$. We can take any chain of face maps $A_n\to A_0$ and compose with the projection to ${\pi }_0(A)$, and this will be independent of which chain we chose. These maps $A_n\to {\pi }_0(A)$ give a surjective map from $A_{•}$ to the constant simplicial ring ${\pi }_0(A)$.

(This is just the simplest piece of the Postnikov tower.) If ${\pi }_0(A)\ne 0$ we can then compose with a surjective map to a constant simplicial field.

All simplicial fields are constant. This is just because the composite $A_0\stackrel{s_0^n}{\to }{A}_n\stackrel{d_0^n}{\to }{A}_0$ is the identity, so $d_0^n$ is surjective, but all field homomorphisms are injective, so $d_0^n$ is an isomorphism.

Model category structure

There is a model category structure on simplicial rings that presents $\mathrm{\infty }$-rings. See model structure on simplicial T-algebras for more.

We describe here 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.

Related concepts

• simplicial algebra

• model structure on simplicial T-algebras

References

An introduction is in chapter 4 of

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

Some of the above material is taken from this MO entry.