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simplicial ring

Context

Homotopy theory

Higher algebra

Contents

Idea

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

This may be understood conceptually as follows:

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 sModsRingsMod \to sRing of modules over simplicial ring is equivalently to the tangent (∞,1)-category of the (∞,1)-category of simplicial rings.

Properties

Connected components

Given a simplicial ring A=A A = A_{\bullet}, its connected components (the 0th “homotopy group”)) is an ordinaty ring

π 0(A)Ring. \pi_0(A) \in Ring \,.

Forming connected components is a functor from simplicial rings to plain rings, which is left adjoint to the inclusion of ordinary rings as simplicially constant simplicial rings, exhibiting a reflective subcategory inclusion

Ringπ 0Ring Δ op. Ring \stackrel{\stackrel{\pi_0}{\longleftarrow}}{\hookrightarrow} Ring^{\Delta^{op}} \,.

(Toën 14, section 2.2)

So on formal duals of commutative (simplicial) rings this is a coreflection of affine schemes in affine derived schemes

CRing op(CRing Δ op) op. CRing^{op} \stackrel{\hookrightarrow}{\longleftarrow} (CRing^{\Delta^{op}})^{op} \,.

Notice that coreflective embeddings are also given for instance by the inclusion of manifolds into formal manifolds. This is one way in which formal duals of simplicial rings manifest themselves as infinitesimal neighbourhoods of formal duals of plain rings.

Higher homotopy groups

The higher homotopy groups π n(A)\pi_n(A) of a simplicial ring A A_\bullet are naturally modules over the ring π 0(A)\pi_0(A) of connected components, so A A_\bullet is weakly contractible, as a simplicial set, iff π 0(A)=0\pi_0(A)=0.

(Again this is a manifestation of the simplicial ring being just an infinitesimal thickening of its connected components.)

Notice that π 0(A)\pi_0(A) is equivalently the cokernel of d 0d 1:A 1A 0d_0-d_1 \colon A_1 \longrightarrow A_0. Accordingly, any chain of face maps A nA 0A_n \longrightarrow A_0 compose with the projection to π 0(A)\pi_0(A) is independent of the choices. These maps

A nπ 0(A) A_n \longrightarrow \pi_0(A)

give a surjective map from A A_\bullet to the constant simplicial ring π 0(A)\pi_0(A).

(This is just the simplest piece of the Postnikov tower.) If π 0(A)0\pi_0(A)\neq 0 we can then compose with a surjective map to a constant simplicial field.

Model category structure

There is a model category structure on simplicial rings that presents \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 AA be a simplicial commutative algebra. Write ASModA SMod for the category which, with AA regarded as a monoid in the category SAbSAb of abelian simplicial groups is just the category of AA-modules in SAbSAb. This means that

Equip ASModA SMod with the structure of a model category by setting:

Proposition This defines a model category structure which is

Examples

Simplicial fields

All simplicial fields are simplicially constant. This is because the composite A 0s 0 nA nd 0 nA 0A_0\xrightarrow{s_0^n}A_n\xrightarrow{d_0^n}A_0 is the identity, so d 0 nd_0^n is surjective, but all field homomorphisms are injective, so d 0 nd_0^n is an isomorphism.

References

Introduction and survey includes

See model structure on simplicial algebras for references on the model structure discussed above.

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

Discussion in the context of homotopy theory, hence for simplicial ring spectra includes

Revised on August 19, 2016 11:45:13 by anonymousbutnotacoward? (50.5.224.8)