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core of a ring

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Definition

Definition

For RR a commutative ring, its core cRc R is the regular image of the unique ring homomorphism eR\mathbb{Z} \overset{e}{\longrightarrow} R (note that \mathbb{Z} is the initial commutative ring). That is, it is the smallest regular monomorphism into RR in the category CRing.

By the general construction of regular images (here), this can be computed as the equalizer of the two inclusions from RR into the pushout R RR\sqcup_{\mathbb{Z}} R. Since \mathbb{Z} is initial, this is just the coproduct RRR\sqcup R in CRingCRing, which is the tensor product of abelian groups RRR\otimes R equipped with its canonically induced commutative ring structure (prop.).

Thus the most explicit definition of cRc R is that it is the equalizer in

cRequRRR, c R \overset{equ}{\longrightarrow} R \stackrel{\longrightarrow}{\longrightarrow} R \otimes R \,,

where the top morphism is

RReidRR R \simeq \mathbb{Z} \otimes R \overset{e \otimes id}{\longrightarrow} R \otimes R

and the bottom one is

RRideRR. R \simeq R \otimes \mathbb{Z} \overset{id \otimes e}{\longrightarrow} R \otimes R \,.

A ring which is isomorphic to its core is called a solid ring.

(Bousfield-Kan 72, §1, def. 2.1, Bousfield 79, 6.4)

Remark

We may think of the opposite category CRing opCRing^{op} as that of affine arithmetic schemes. Here for RCRingR \in CRing we write Spec(R)Spec(R) for the same object, but regarded in CRing opCRing^{op}.

So the initial object \mathbb{Z} in CRing becomes the terminal object Spec(Z) in CRing opCRing^{op}, and so for every RR there is a unique morphism

Spec(R)Spec(Z) Spec(R) \longrightarrow Spec(Z)

in CRing opCRing^{op}, exhibiting every affine arithmetic scheme Spec(R)Spec(R) as equipped with a map to the base scheme Spec(Z).

Since the coproduct in CRing is the tensor product of rings (prop.), this is the dually the Cartesian product in CRing opCRing^{op} and hence

Spec(RR)Spec(R)×Spec(R) Spec(R \otimes R) \simeq Spec(R) \times Spec(R)

exhibits RRR \otimes R as the ring of functions on Spec(R)×Spec(R)Spec(R) \times Spec(R).

Hence the terminal morphism Spec(R)Spec()Spec(R) \to Spec(\mathbb{Z}) induced the corresponding Cech groupoid internal to CRing opCRing^{op}

Spec(R)×Spec(R)×Spec(R) Spec(R)×Spec(R) s t Spec(R). \array{ Spec(R) \times Spec(R) \times Spec(R) \\ \downarrow \\ Spec(R) \times Spec(R) \\ {}^{\mathllap{s}}\downarrow \uparrow \downarrow^{\mathrlap{t}} \\ Spec(R) } \,.

This exhibits RRR \otimes R (the ring of functions on the scheme of morphisms of the Cech groupoid) as a commutative Hopf algebroid over RR.

Moreover, the arithmetic scheme of isomorphism classes of this groupoid is the coequalizer of the source and target morphisms

Spec(R)×Spec(R)AAstSpec(R)coeqSpec(cR), Spec(R) \times Spec(R) \underoverset {\underset{s}{\longrightarrow}} {\overset{t}{\longrightarrow}} {\phantom{AA}} Spec(R) \overset{coeq}{\longrightarrow} Spec(c R) \,,

also called the coimage of Spec(R)Spec()Spec(R) \to Spec(\mathbb{Z}). Since limits in the opposite category CRing opCRing^{op} are equivaletly colimits in CRingCRing, this means that the ring of functions on the scheme of isomorphism classes of the Cech groupoid is precisely the core cRc R or RR according to def. .

This is morally the reason why for EE a homotopy commutative ring spectrum then the core cπ 0(E)c \pi_0(E) of its underlying ordinary ring in degree 0 controls what the EE-Adams spectral sequence converges to (Bousfield 79, theorems 6.5, 6.6, see here), because the EE-Adams spectral sequence computes E-nilpotent completion which is essentially the analog in higher algebra of the above story: namely the coimage ((infinity,1)-image) of Spec(E)Spec(E) \to Spec(S) (see here).

Classification

Theorem

The following is the complete list of solid rings (def. ) up to isomorphism:

  1. The localization of the ring of integers at a set JJ of prime numbers

    [J 1]; \mathbb{Z}[J^{-1}] \,;
  2. the cyclic rings

    /n \mathbb{Z}/n\mathbb{Z}

    for nn \in \mathbb{N}, n2n \geq 2;

  3. the product rings

    [J 1]×/n, \mathbb{Z}[J^{-1}] \times \mathbb{Z}/n\mathbb{Z} \,,

    for n2n \geq 2 such that each prime factor of nn is contained in the set of primes JJ;

  4. the ring cores of product rings

    c([J 1]×pK/p e(p)), c(\mathbb{Z}[J^{-1}] \times \underset{p \in K}{\prod} \mathbb{Z}/p^{e(p)}) \,,

    where KJK \subset J are infinite sets of primes and e(p)e(p) are positive natural numbers.

(Bousfield-Kan 72, prop. 3.5, Bousfield 79, p. 276)

Properties

Proposition

The core of any ring RR is solid (def. ):

ccRcR. c c R \simeq c R \,.

(Bousfield-Kan 72, prop. 2.2)

References

Last revised on July 18, 2016 at 06:27:14. See the history of this page for a list of all contributions to it.