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simplicial ring

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Homotopy theory

Higher algebra

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Idea

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

It 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

Homotopy groups

Given a simplicial ring A A_\bullet, it is standard that its connected components (the 0th “homotopy group”) π 0(A)\pi_0(A) is an ordinary ring and π n(A)\pi_n(A) is a module over it, so A A_\bullet is weakly contractible iff π 0(A)=0\pi_0(A)=0.

Also, π 0(A)\pi_0(A) is just the cokernel of d 0d 1:A 1A 0d_0-d_1:A_1\to A_0. We can take any chain of face maps A nA 0A_n\to A_0 and compose with the projection to π 0(A)\pi_0(A), and this will be independent of which chain we chose. These maps A nπ 0(A)A_n\to\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.

All simplicial fields are constant. This is just 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.

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

References

An introduction is in chapter 4 of

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

Revised on December 29, 2012 17:15:27 by Urs Schreiber (89.204.155.26)