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core of a ring

Definition

Definition

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

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

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

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

where the top morphism is

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

and the bottom one is

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

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 $CRing^{op}$ as that of affine arithmetic schemes. Here for $R \in CRing$ we write $Spec(R)$ for the same object, but regarded in $CRing^{op}$.

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

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

in $CRing^{op}$, exhibiting every affine arithmetic scheme $Spec(R)$ 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 $CRing^{op}$ and hence

$$Spec(R \otimes R) \simeq Spec(R) \times Spec(R)$$

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

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

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

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

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

$$Spec(R) \times Spec(R) \underoverset {\underset{s}{\longrightarrow}} {\overset{t}{\longrightarrow}} {\phantom{AA}} Spec(R) \overset{coeq}{\longrightarrow} Spec(c R) \,,$$

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

This is morally the reason why for $E$ a homotopy commutative ring spectrum then the core $c \pi_0(E)$ of its underlying ordinary ring in degree 0 controls what the $E$-Adams spectral sequence converges to (Bousfield 79, theorems 6.5, 6.6, see here), because the $E$-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 $Spec(E) \to$ 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 $J$ of prime numbers

   $$\mathbb{Z}[J^{-1}] \,;$$

2. the cyclic rings

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

   for $n \in \mathbb{N}$, $n \geq 2$;

3. the product rings

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

   for $n \geq 2$ such that each prime factor of $n$ is contained in the set of primes $J$;

4. the ring cores of product rings

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

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

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

Properties

Proposition

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

$$c c R \simeq c R \,.$$

(Bousfield-Kan 72, prop. 2.2)

Related concepts

• nilpotent completion of spectra

• Adams spectral sequence

References

• Aldridge Bousfield, Daniel Kan, The core of a ring, Journal of Pure and Applied Algebra, Volume 2, Issue 1, April 1972, Pages 73-81 (link)

• Aldridge Bousfield, The localization of spectra with respect to homology, Topology 18 (1979), no. 4, 257–281. (pdf)