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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

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.

Revised on November 7, 2014 15:35:43 by Adeel Khan (77.182.65.125)