cross-linked the remark on not being solid (here) with real homotopy theory
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]]>Added an “idea” of the core.
]]>Added “idea” of the core of a ring.
]]>For what it’s worth, I have spelled out a proof (here) that ring is solid iff its multiplication is an isomorphism.
]]>For what it’s worth, I have made more explicit (here) why the rationals are solid, but no other char=0 field is.
]]>Added these references:
The concept re-appears under the name “T-rings” in
and under the name “-epimorphs” in:
Generalization to monoids in monoidal categories:
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.
]]>Linked the remark on the geometric interpretation to duality between algebra and geometry.
]]>Is there established terminology for the generalization of the concept to commutative Hopf algebroids?
I.e. for
the left and right unit maps of a commutative Hopf algebroid over , is there established terminology for their equalizer?
(So that the core of a ring would be the special case for the commutative Hopf algebroid over …)
]]>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 as that of affine arithmetic schemes. Here for we write for the same object, but regarded in .
So the initial object in CRing becomes the terminal object Spec(Z) in , and so for every there is a unique morphism
in , exhibiting every affine arithmetic scheme 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 and hence
exhibits as the ring of functions on .
Hence the terminal morphism induced the corresponding Cech groupoid internal to
This exhibits (the ring of functions on the scheme of morphisms of the Cech groupoid) as a commutative Hopf algebroid over .
Moreover, the arithmetic scheme of isomorphism classes of this groupoid is the coequalizer of the source and target morphisms
also called the coimage of . Since limits in the opposite category are equivaletly colimits in , this means that the ring of functions on the scheme of isomorphism classes of the Cech groupoid is precisely the core or according to def. \ref{CoreOfARing}.
This is morally the reason why for a homotopy commutative ring spectrum then the core of its underlying ordinary ring in degree 0 controls what the -Adams spectral sequence converges to (Bousfield 79, theorems 6.5, 6.6, see here), because the -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(S) (see here).
]]>It makes me wonder, what are the regular monos in general in ?
]]>Thanks! I hadn’t realized that relation.
]]>I added a remark about regular images.
]]>Ah, thanks for catching that.
]]>I added the word “commutative”, and added a link to the Bousfield-Kan paper.
]]>gave core of a ring some minimum content
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