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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 15th 2016

gave core of a ring some minimum content

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJul 15th 2016

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJul 15th 2016

Ah, thanks for catching that.

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeJul 15th 2016

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJul 15th 2016

Thanks! I hadn’t realized that relation.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeJul 15th 2016

It makes me wonder, what are the regular monos in general in $CRing$?

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJul 16th 2016

I have added a remark here on the dual interpretation. (my battery is dying right now, possibly there are typos left):

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. \ref{CoreOfARing}.

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 alegbra of the above story: namely the coimage ((infinity,1)-image) of $Spec(E) \to$ Spec(S) (see here).

• CommentRowNumber8.
• CommentAuthorUrs
• CommentTimeJul 18th 2016
• (edited Jul 18th 2016)

Is there established terminology for the generalization of the concept to commutative Hopf algebroids?

I.e. for

$A \underoverset {\underset{\eta_R}{\longrightarrow}} {\overset{\eta_L}{\longrightarrow}} {\phantom{AA}} \Gamma$

the left and right unit maps of a commutative Hopf algebroid $\Gamma$ over $A$, is there established terminology for their equalizer?

(So that the core of a ring $R$ would be the special case for the commutative Hopf algebroid $R \otimes R$ over $R$…)

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJul 17th 2021

Linked the remark on the geometric interpretation to duality between algebra and geometry.

• CommentRowNumber10.
• CommentAuthorUrs
• CommentTimeJul 17th 2021
• (edited Jul 17th 2021)

I have slightly reworked the Definition-section for readability:

Gave the definition as a subset, stated up-front, then followed by a remark which expands on the category-theoretic formulation as an equalizer and regular image.

Also added the statement that a commutative ring is solid iff its multiplication is an isomorphism.

• CommentRowNumber11.
• CommentAuthorUrs
• CommentTimeJul 17th 2021

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:

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeJul 17th 2021
• (edited Jul 17th 2021)

For what it’s worth, I have made more explicit (here) why the rationals are solid, but no other char=0 field is.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeJul 17th 2021

For what it’s worth, I have spelled out a proof (here) that ring is solid iff its multiplication is an isomorphism.

• CommentRowNumber14.
• CommentAuthorJohn Baez
• CommentTimeSep 18th 2021

Added “idea” of the core of a ring.

• CommentRowNumber15.
• CommentAuthorJohn Baez
• CommentTimeSep 18th 2021

Added an “idea” of the core.

1. I have created the page idempotent monoid in a monoidal category and linked to it in the related references section of this page.

Solid rings are an example of these, as are idempotent monads.

• CommentRowNumber17.
• CommentAuthorUrs
• CommentTimeSep 23rd 2021

cross-linked the remark on $\mathbb{R}$ not being solid (here) with real homotopy theory