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nLab > Latest Changes: core of a ring

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2016

    gave core of a ring some minimum content

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 15th 2016

    I added the word “commutative”, and added a link to the Bousfield-Kan paper.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJul 15th 2016

    Ah, thanks for catching that.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeJul 15th 2016

    I added a remark about regular images.

    • 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 CRingCRing?

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

    So the initial object ℤ\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(Z) Spec(R) \longrightarrow Spec(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(R⊗R)≃Spec(R)×Spec(R) Spec(R \otimes R) \simeq Spec(R) \times Spec(R)

    exhibits R⊗RR \otimes R as the ring of functions on Spec(R)×Spec(R)Spec(R) \times Spec(R).

    Hence the terminal morphism Spec(R)→Spec(ℤ)Spec(R) \to Spec(\mathbb{Z}) induced 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 R⊗RR \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)AA⟶s⟶tSpec(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(ℤ)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. \ref{CoreOfARing}.

    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 alegbra of the above story: namely the coimage ((infinity,1)-image) of Spec(E)→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

    AAA⟶η R⟶η LΓ A \underoverset {\underset{\eta_R}{\longrightarrow}} {\overset{\eta_L}{\longrightarrow}} {\phantom{AA}} \Gamma

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

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

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJul 17th 2021

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

    diff, v9, current

    • 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.

    diff, v10, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeJul 17th 2021

    Added these references:

    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 “ℤ\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:

    diff, v11, current

    • 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.

    diff, v11, current

    • 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.

    diff, v11, current

    • CommentRowNumber14.
    • CommentAuthorJohn Baez
    • CommentTimeSep 18th 2021

    Added “idea” of the core of a ring.

    diff, v14, current

    • CommentRowNumber15.
    • CommentAuthorJohn Baez
    • CommentTimeSep 18th 2021

    Added an “idea” of the core.

    diff, v14, current

    • CommentRowNumber16.
    • CommentAuthorDELETED_USER_2018
    • CommentTimeSep 18th 2021
    • (edited Apr 11th 2023)

    [deleted]

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeSep 23rd 2021

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

    diff, v17, current

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