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gave core of a ring some minimum content
I added the word “commutative”, and added a link to the Bousfield-Kan paper.
Ah, thanks for catching that.
I added a remark about regular images.
Thanks! I hadn’t realized that relation.
It makes me wonder, what are the regular monos in general in $CRing$?
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).
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 $R$…)
Linked the remark on the geometric interpretation to duality between algebra and geometry.
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.
Added these references:
The concept re-appears under the name “T-rings” in
and under the name “$\mathbb{Z}$-epimorphs” in:
Generalization to monoids in monoidal categories:
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.
cross-linked the remark on $\mathbb{R}$ not being solid (here) with real homotopy theory
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