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core of a ring

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Definition

Core of a ring

Definition

(core of a ring)
For RR a unital commutative ring, its core is the following sub-ring of the tensor product of abelian groups R RR \otimes_{{}_{\mathbb{Z}}} R:

cR{rR|1r=r1R R} c R \;\coloneqq\; \Big\{ \, r \in R \,\left\vert\, 1 \otimes r \,=\, r \otimes 1 \;\in\; R \otimes_{{}_{\mathbb{Z}}} R \right. \, \Big\}

(Bousfield & Kan 1972, Sec. 1)

Remark

(core as equalizer and as regular image)
In category-theoretic terminology, Def. describes the equalizer (Bousfield 1979, 6.4)

cRequRAAAAAAR R, c R \xrightarrow{\;equ\;} R \overset{\phantom{AAAAAA}}\rightrightarrows R \otimes_{{}_{\mathbb{Z}}} R \,,

where the top morphism is

RReidRR R \;\simeq\; \mathbb{Z} \otimes R \xrightarrow{e \otimes id} R \otimes R

and the bottom one is

RRideRR, R \;\simeq\; R \otimes \mathbb{Z} \xrightarrow{\;id \otimes e\;} R \otimes R \,,

with

(1) e R 1 1 \array{ \mathbb{Z} &\xrightarrow{ \;e\; } & R \\ 1 &\mapsto& 1 }

denoting the unique ring homomorphism form the commutative ring of integers, which is the initial object in CommutativeRings.

Observing (by this Prop.) that the tensor product of abelian groups R RR \otimes_{{}_{\mathbb{Z}}} R equipped with its canonically induced commutative ring structure is the coproduct in CommutativeRings

R RRR R \otimes_{{}_{\mathbb{Z}}} R \;\simeq\; R \sqcup R

this means equivalently that the core is the equalizer of the two coprojections into the coproduct:

cRequRAAAAAARRRR, c R \xrightarrow{\;equ\;} R \overset{\phantom{AAAAAA}}\rightrightarrows R \sqcup R \;\simeq\; R \underset{\mathbb{Z}}{\coprod} R \,,

hence – since \mathbb{Z} is the initial object in CommutativeRings – into the cofiber coproduct of eR\mathbb{Z} \xrightarrow{e} R (1) with itself.

In this form, the core is manifestly (here) the regular image of the initial morphism eR\mathbb{Z} \xrightarrow{e} R:

cRIm reg(eR)AAR, c R \;\simeq\; Im_{reg} \left( \mathbb{Z} \xrightarrow{e} R \right) \xhookrightarrow{\phantom{AA}} R \,,

hence the smallest regular monomorphism into RR in the category of CommutativeRings.

Remark

(geometric interpretation)
By duality between algebra and geometry, we may think of the opposite category CommutativeRings opCommutativeRings^{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() Spec(R) \longrightarrow Spec(\mathbb{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}) induces 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).

Solid rings

Definition

(solid rings)
A commutative ring RR which is isomorphic to its core (Def. ), RcRR \,\simeq\, c R is called a solid ring.

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

Proposition

(solidity means that multiplication is isomorphism)
A commutative ring RR is solid (Def. ) iff its multiplication morphism is an isomorphism:

Ris solidR R()()R. R\; \text{is solid} \;\;\;\;\;\;\;\;\; \Leftrightarrow \;\;\;\;\;\;\;\;\; R \otimes_{{}_{\mathbb{Z}}} R \underoverset {\simeq} {\;(-) \cdot (-)\;} {\longrightarrow} R \,.

(Bousfield & Kan 1972, 2.4 – this is called a T-ring in Bowshell & Schultz 1977, Def. 1.6)
Proof

In one direction, assume that multiplication is an isomorphism. Since both r1rr \mapsto 1 \otimes r and rr1r \mapsto r \otimes 1 are right inverses

and since right inverses of isomorphisms are unique (this Prop.), it follows that these two morphisms on the left are in fact equal, and therefore that their equalizer, hence the core of RR (by Rem. ), is RR.

In the other direction, assume that 1r=r11 \otimes r \,=\, r \otimes 1 for all rRr \in R. Then

r 1r 2 =(r 11)(1r 2) =(1r 1)(1r 2) =1(r 1r 2), \begin{aligned} r_1 \,\otimes\, r_2 & \;=\; \big( r_1 \,\otimes\, 1 \big) \cdot \big( 1 \otimes r_2 \big) \\ & \;=\; \big( 1 \,\otimes\, r_1 \big) \cdot \big( 1 \otimes r_2 \big) \\ & \;=\; 1 \,\otimes\, \big( r_1 \cdot r_2 \big) \,, \end{aligned}

and therefore ()()(-)\cdot(-) is a bijection, hence an isomorphism of rings.

Properties

Proposition

(cores are solid)
The core (Def. ) of any ring RR is solid (Def. ):

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

(Bousfield-Kan 72, prop. 2.2)

Examples

Theorem

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

  1. The localization of the ring of integers at a subset PrPr of prime numbers

    [Pr 1]; \mathbb{Z}\big[Pr^{-1}\big] \,;
  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)

In particular:

Example

(rational numbers form a solid ring)
The ring \mathbb{Q} of rational numbers is a solid ring.

Proof

Since every rational number may be written as

r=q/p=1pq r \,=\,q / p \,=\, \tfrac{1}{p} \cdot q \,\in\, \mathbb{Q}

for some p,qp, q \,\in\, \mathbb{Z}, we have

1r =1(1pq) =(1pp)(1pq) =1p(p1pq) =1p(q1) =(1pq)1 =r1 . \begin{aligned} 1 \otimes r & \;=\; 1 \,\otimes\, \big( \tfrac{1}{p} \cdot q \big) \\ & \;=\; \big( \tfrac{1}{p} \cdot p \big) \,\otimes\, \big( \tfrac{1}{p} \cdot q \big) \\ & \;=\; \tfrac{1}{p} \,\otimes\, \big( p \cdot \tfrac{1}{p} \cdot q \big) \\ & \;=\; \tfrac{1}{p} \,\otimes\, \big( q \cdot 1 \big) \\ & \;=\; \big( \tfrac{1}{p} \cdot q \big) \,\otimes\, 1 \\ & \;=\; r \,\otimes\, 1 \;\;\;\;\;\; \in \; \mathbb{Q} \otimes_{{}_{\mathbb{Z}}} \mathbb{Q} \,. \end{aligned}

Here the third and the fifth line use the equivalence relation defining the tensor product of abelian groups for the case of rings:

r 1(nr 2)=(r 1n)r 2 ,forn. r_1 \,\otimes\, \big( n \cdot r_2 \big) \,=\, \big( r_1 \,\cdot\, n \big) \,\otimes\, r_2 \;\;\; \in \; \mathbb{Q} \otimes_{{}_{\mathbb{Z}}} \mathbb{Q} \,, \;\;\;\; \text{for}\; n \,\in\, \mathbb{Z} \,.

Remark

(no other field of characteristic zero is a solid ring)
The same argument shows that no other field of characteristic zero kk is a solid ring, since, being a proper superset 𝕂\mathbb{K} \supset \mathbb{Q}, it contains elements k𝕂k \in \mathbb{K} for which there are no pairs of integers qq, pp such that pk=qp \cdot k \,=\, q. In particular, the real numbers are not a solid ring.

This is, ultimately, the reason why the derived PL de Rham-Quillen adjunction is idempotent only over k=k = \mathbb{Q} (where it models rationalization of homotopy types, see at fundamental theorem of dg-algebraic rational homotopy theory).

References

The original articles:

The concept re-appears under the name “T-rings” in

  • R. A. Bowshell and P. Schultz, Unital rings whose additive endomorphisms commute, Mathematische Annalen volume 228, pages 197–214 (1977) (doi10.1007/BF01420290)

and under the name “\mathbb{Z}-epimorphs” in:

  • Warren Dicks, W. Stephenson, Epimorphs and Dominions of Dedekind Domains, Journal of the London Mathematical Society, Volume s2-29, Issue 2, April 1984, Pages 224–228 (doi:10.1112/jlms/s2-29.2.224)

Generalization to monoids in monoidal categories:

Last revised on September 16, 2021 at 03:57:11. See the history of this page for a list of all contributions to it.