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Finally added to fracture theorem the basic statement of the “arithmetic fracture square”, hence the following discussion.
The number theoretic statement is the following:
+– {: .num_prop #ArithmeticFractureSquare}
The integers $\mathbb{Z}$ are the fiber product of all the p-adic integers $\underset{p\;prime}{\prod} \mathbb{Z}_p$ with the rational numbers $\mathbb{Q}$ over the rationalization of the former, hence there is a pullback diagram in CRing of the form
$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{Q}\otimes_{\mathbb{Z}}\underset{p\;prime}{\prod} \mathbb{Z}_p && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \underset{p\;prime}{\prod} \mathbb{Z}_p } \,.$Equivalently this is the fiber product of the rationals with the integral adeles $\mathbb{A}_{\mathbb{Z}}$ over the ring of adeles $\mathbb{A}_{\mathbb{Q}}$
$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{A}_{\mathbb{Q}} && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \mathbb{A}_{\mathbb{Z}} } \,.$=–
In the context of a modern account of categorical homotopy theory this appears for instance as (Riehl 14, lemma 14.4.2).
+– {: .num_remark}
Under the function field analogy we may think of
$Spec(\mathbb{Z})$ as an arithmetic curve over F1;
$\mathbb{A}_{\mathbb{Z}}$ as the ring of functions on the formal disks around all the points in this curve;
$\mathbb{Q}$ as the ring of functions on the complement of a finite number of points in the curve;
$\mathbb{A}_{\mathbb{Q}}$ is the ring of functions on punctured formal disks around all points, at most finitely many of which do not extend to the unpunctured disk.
Under this analogy the arithmetic fracture square of prop. \ref{ArithmeticFractureSquare} says that the curve $Spec(\mathbb{Z})$ has a cover whose patches are the complement of the curve by some points, and the formal disks around these points.
This kind of cover plays a central role in number theory, see for instance thr following discussions:
=–
[ Fixed. ]
That fracture theorem pullback is also a pushout in the category of topological rings if we give $\mathbb{Q}$ the discrete topology.
Ah, thanks, that’s an excellent point!
I have added to the proposition here the following paragraph:
Since the ring of adeles is the rationalization of the integral adeles $\mathbb{A}_{\mathbb{Q}} = \mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{A}_{\mathbb{Z}}$, this is also a pushout diagram in CRing, and in fact in topological commutative rings (for $\mathbb{Q}$ with the discrete topology and $\mathbb{A}_{\mathbb{Z}}$ with its profinite/completion topology).
In the classical fracture square for spectra, the adic completion and the $\mathbb{Q}$-acyclicity-ization functors appear where in cohesion the flat modality and the shape modality appear, respectively.
Is there any context in which adic completion becomes adjoint to rational acylitization? (Maybe a stupid question, but anyway.)
I have expanded slightly at fracture theorem, trying to bring the section on the fracture squares in homotopy theory a bit more into shape. But it is still deserves more attention.
In remark 4, you’ve moved to a product over all primes in the diagram, but have the single prime $p$ in $G_{H \mathbb{F}_p}(X)$ as cofiber. Is there a way to allow this cofiber to cope with all primes?
Oh, does Neil Strickland’s answer for Bousfield localization help? Does $G_{H \mathbb{Q}/\mathbb{Z}}(X)$ work?
Ah, sorry, I see that I created a mess with some last minute changes. Yes, you are right. I’ll fix this when I am back online in about 90 minutes from now.
By the way, what is in that MO comment is mostly prop. 2.2 in Bauer’s notes linked to from the nLab entry but the comment about Q/Z is good. What might be a source for that?
In the single prime case, do we see $H \mathbb{F}_p$ anyway?
I can see $E = M(\mathbb{Z}/p)$ (Moore spectrum), but I don’t see this equated in the way it is for $\mathbb{Q}$, i.e., we find $M(\mathbb{Q}) = H \mathbb{Q}$.
So globally now we need $E = \vee_p M(\mathbb{Z}/p)$. Is this $M(\mathbb{Q}/\mathbb{Z})$? From here, Strickland uses ’S’ for the Moore spectrum. So I guess that tallies with his MO answer.
There seems to be a ’Serre’s theorem’ which states that:
rational homology is the same as rational stable homotopy,
and which induces an equivalence $M(\mathbb{Q}) = H \mathbb{Q}$.
Thanks for all this, David! I just came online, but am again in a rush.
The business of distinuishing Moore from EM was also an issue at localization of a space. I have tried to fix all occurences.
Then I edited fracture theorem, added the statement that we talked about and then below that fixed the missing globalization in the top right entry of the big diagram.
Again, I am doing all this in a rush and haven’t proof-read yet, so your careful eye might usefully be exercised further. I will be offline now for another hour or more, then I hope to come back with more leisure.
OK. I put ’since’ rather than ’let’ in
Since $S (\mathbb{Q}/\mathbb{Z}) = \vee_p S \mathbb{F}_p$,
because it’s a result rather than something in our power.
Thanks! We should say it’s a Bousfield equivalence, I put that in now.
Of course $\mathbb{Q}/\mathbb{Z}$ also comes in if one just takes the plain square
$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{Q}\otimes_{\mathbb{Z}}\underset{p\;prime}{\prod} \mathbb{Z}_p && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \underset{p\;prime}{\prod} \mathbb{Z}_p }$and then extends the diagonals to their long homotopy fiber sequences in chain complexes, simply, induced from the short exact sequence
$\mathbb{Z} \longrightarrow \mathbb{Q} \longrightarrow \mathbb{Q}/\mathbb{Z}$I had been wondering about what to do with that, but I suppose that Bousfield equivalence $S (\mathbb{Q}/\mathbb{Z}) \simeq_B \vee_p S \mathbb{F}_p$ is related.
I have now
added a proper Idea-section;
added the statement of the generalized fracture theorem (for any suitable pair of localizations);
rewrote the section In cohesive homotopy theory to make it more to the point;
gave the Example of tmf a little bit of accompanying text.
There were still several instances of EM-spectrum where Moore spectrum wanted at localization of a space, Bousfield localization of spectra, and fracture theorem. Hope I’ve got those changes right. Do people ever localize at EM-spectra?
So can we complete the hexagon of the fracture diagram below remark 4?
I need to go to the original articles. Van Koughnett has a brief lecture note known by Google which in prop. 4.2 has one relevant statement. Will include it a little later.
If the hope is to put arithmetic fracturing into line with differential cohesive fracturing, presumably we’re looking for a monad/comonad pair amongst the two pairs of $L_E$ and $G_E$s, and a comparison map between an acyclification and a localization.
Yes, exactly.
Or rather, it seems (looking at the variance of the morphisms) we should think of the homotopy-theoretic arithmetic fracture square (involving formal completion $X_p^\wedge$ of spectra $X$) as the image of the number-theoretic one (involving the integral adeles $\mathbb{A}_{\mathbb{Z}}$) as the image under mapping the geometric one (involving $Spf(\mathbb{A}_{\mathbb{Z}})$) into the arithmetic presheaf of spectra which is induced from $X$ by
$X : Spec(R) \mapsto X \otimes R \,.$For instance homming
$\array{ Spec(\mathbb{Q}) \\ & \searrow \\ && Spec(\mathbb{Z}) }$into this would yield
$\array{ X_{\mathbb{Q}} \\ & \nwarrow \\ && X }$(the top right piece of the fracture square, am too rushed now to draw all of it).
From this point of view it would be sufficient to realize arithmetic geometry as cohesive over something, such that the flat modality forms “arithmetic jets”. A homotopy-theoretic version would then be induced by passing to sheaves of spectra and homming.
It seems that Borger’s absolute geometry might be getting in this direction, but something is missing still.
Added a ’chromatic fracture square’ diagram. I wonder how the text should flow after this into the tmf example.
Thanks!
I have now added in this remark a third item noticing that by the pasting law there is indeed an “homotopy exact two-third hexagon”
$\array{ && X_{\mathbb{Q}} && \longleftarrow && G_{S (\mathbb{Q}/\mathbb{Z})}(X) \\ & \swarrow && \nwarrow && \swarrow \\ (\prod_p X_p^\wedge)_{\mathbb{Q}} && && X \\ & \nwarrow && \swarrow && \nwarrow \\ && \prod_p X_p^\wedge && \longleftarrow && G_{H\mathbb{Q}}(X) }$in that it follows that also the top and bottom outer sequences are homotopy exact.
(And yes, this means I am finally back online :-)
The remaining question is then to which extent the two operations in the bottom row of the above diagram are adjoint.
To which extent is passing to $p^\nu$-torsion adjoint to $p$-completion?
Should it not be true for suitably co-small objects which may be taken into the filtered limit which defines the completion?
[Edit: I see something going in this direction in Porta et al.: “On the homology of completion and torsion.” ]
Now in New York and online again, I have briefly noted the following (to be polished/expanded):
Remark 4 suggests that arithmetic fracturing naturally goes along with adjoint pairs of (derived/(∞,1)-functorial) idempotent (co-)-monads given by formal completion and by taking torsion subgroups, respectivly.
Whether such adjoint modalities exist for spectra…, but in (Porta-Shaul-Yekutieli 10) they are essentially shown to exist on chain complexes (hence, by the stable Dold-Kan correspondence, at least for $H A$-module spectra ).
(Porta-Shaul-Yekutieli 10, corollary 3.30, prop. 6.10) says that for commutative rings $A$ with suitable (namely “weakly pro-regular”) ideals $\mathfrak{a}$, then the derived $\mathfrak{a}$-torsion subgroup-functor and the derived $\mathfrak{a}$-completion functor on the derived category of $A$-modules are idempotent via their unit/counit maps, respectively. Moreover, (Porta-Shaul-Yekutieli 10, theorem 6.12) says that the sub-categories of modal types are canonically equivalent, as befits an adjoint modality
$\mathfrak{a}completion \dashv \mathfrak{a}torsion \,.$Have added here the following more formal version of the above statement (check):
+– {: .num_defn}
Let $A$ be a commutative ring, let $\mathfrak{a} \subset A$ be be an ideal generated by a single element. Write $A Mod_{\infty}^{op}$ for the opposite (∞,1)-category of the (∞,1)-category of modules over $A$.
Write
$\flat_{\mathfrak{a}}\colon A Mod_\infty^{op} \to A Mod_{\infty}^{op}$ for the derived functor of formal completion (adic completion) of modules at $\mathfrak{a}$;
with canonical natural transformation
$\epsilon_{\mathfrak{a}} \colon \flat_{\mathfrak{a}} \longrightarrow id$
$\Pi_{\mathfrak{a}} \colon A Mod_\infty^{op} \to A Mod_\infty^{op}$ for the total derived functor of the $\mathfrak{a}$-torsion subgroup-functor;
with canonical natural transformation
$\eta_{\mathfrak{a}}\colon id \longrightarrow \Pi_{\mathfrak{a}}$
Finally write
$(A Mod_\infty^{op})^{\mathfrak{a}com}, (A Mod_\infty^{op})^{\mathfrak{a}tor} \hookrightarrow A Mod_\infty$for the full (∞,1)-subcategories of objects $X$ for which, $\epsilon_{\mathfrak{a}}(X)$ or $\eta_{\mathfrak{a}}(X)$ is an equivalence in an (∞,1)-category, respectively.
=–
+– {: .num_prop}
The transformation $\epsilon_{\mathfrak{a}}$ exhibits $(A Mod_\infty^{op})^{\mathfrak{a}com}\hookrightarrow A Mod_\infty$ as a reflective (∞,1)-subcategory, hence $\flat_{\mathfrak{a}}$ as an idempotent (∞,1)-monad.
The transformation $\eta_{\mathfrak{a}}$ exhibits $(A Mod_\infty^{op})^{\mathfrak{a}tor}\hookrightarrow A Mod_\infty$ as a co-reflective $(\infty,1)$-category, hence $\Pi_{\mathfrak{a}}$ as an idempotent $(\infty,1)$-comonad.
Restricted to these sub-$(\infty,1)$-categories both $\flat_{\mathfrak{a}}$ as well as $\Pi_{\mathfrak{a}}$ become equivalences of (∞,1)-categories, hence exhibiting $(\Pi_{\mathfrak{a}} \dashv \flat_{\mathfrak{a}})$ as a higher adjoint modality.
=–
+– {: .proof}
This is a paraphrase of the results in (Porta-Shaul-Yekutieli 10):
First of all, by our simplifying assumption that $\mathfrak{a}$ is generated by a single element the running assumption of “weak proregularity” in (Porta-Shaul-Yekutieli 10, def.3.21) is trivially satisfied.
Then in view of (Porta-Shaul-Yekutieli 10, corollary 3.31) the statement of (Porta-Shaul-Yekutieli 10, theorem 6.12) is the characterization of reflectors as discussed at reflective sub-(∞,1)-category, and formally dually so for the coreflection. With the fully faithfulness that goes with this the equivalence of the two inclusions on the level of homotopy categories given by (Porta-Shaul-Yekutieli 10, theorem 6.11) implies the canonical equivalence of the two sub-(∞,1)-categories and this means that $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$ are the adjoint pair induced from the reflection/coreflection adjoint triple.
=–
Turns out the story is in full beauty in section 4.2 of DAG XII. I had missed that.
In particular remark 4.2.6 there should give that we have cohesion on $E_\infty Ring^{op}$ over $E_\infty Ring^{op}_{torsion}$, hence after passing to toposes $Sh_\infty(-)$ this should induce arithmetic cohesion on $E_\infty$-geometry. It seems. Whose differential cohomology hexagons in the left part are the traditional arithmetic fracture squares and their generalizations.
I have that statement to the entry, here:
+– {: .num_prop #CompletionTorsionAdjointModalityForModuleSpectra}
Let $A$ be an E-∞ ring and let $\mathfrak{a} \subset \pi_0 A$ be a finitely generated ideal in its underlying commutative ring.
Then there is an adjoint triple of adjoint (∞,1)-functors
$\array{ \underoverset{ A Mod_{\mathfrak{a}com}^{op}} {A Mod_{\mathfrak{a}tors}^{op}} {\simeq} &\stackrel{\overset{\Pi_{\mathfrak{a}}}{\longleftarrow}}{\stackrel{\hookrightarrow}{\underset{\flat_{\mathfrak{a}}}{\longleftarrow}}}& A Mod^{op} }$where
$A Mod$ is the stable (∞,1)-category of modules, i.e. of ∞-modules over $A$;
$A Mod_{\mathfrak{a}tor}$ and $A Mod_{\mathfrak{a} comp}$ are the full sub-(∞,1)-categories of $\mathfrak{a}$-torsion and of $\mathfrak{a}$-complete $A$-∞-modules, respectively;
$(-)^{op}$ denotes the opposite (∞,1)-category;
the equivalence of (∞,1)-categories on the left is induced by the restriction of $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$.
=–
+– {: .proof}
This is effectively the content of (Lurie “Proper morphisms”, section 4):
the existence of $\Pi_{\mathfrak{a}}$ is corollary 4.1.16 and remark 4.1.17
the existence of $\flat_{\mathfrak{a}}$ is lemma 4.2.2 there;
the equivalence of sub-$\infty$-categories is proposition 4.2.5 there.
=–
+– {: .num_cor}
The traditional arithmetic fracture square of prop. \ref{SullivanArithmeticFracture}, regarded as in remark \ref{TwoThirdHexagon}, is the left part of the
“differential cohomology diagram” induced by the adjoint modality $(\Pi_{\mathfrak{a}} \dashv \flat_{\mathfrak{a}} )$ of prop. \ref{CompletionTorsionAdjointModalityForModuleSpectra}, for the special case that $A = \mathbb{S}$ is the sphere spectrum and $\mathfrak{a} = (p)$ a prime ideal.
=–
added the remark that the above adjoint modality is monoidal and therefore lifts to $E_\infty$-algebras to yield
$(\Pi_{\mathfral{a}} \dashv \flat_{\mathfrak{a}}) \colon E_\infty Alg^{op} \to E_\infty Alg^{op}$.
I got a hint of something fracture-ish in the ’broad’ part of this MO answer, but don’t know.
Thanks for the pointer. I agree, that does look suggestive.
Myself, though, I have absolutely no time to look into this right now, alas.
There’s an article out ’Local duality in algebra and topology’ arXiv:1511.03526 presenting an abstract framework which generalizes a bunch of things including Greenlees-May duality.
local duality admits a uniform and conceptual explanation when working in a homotopically enriched context, at the same time simplifying the arguments and uncovering new instances of the theory.
Since Greenlees-May duality occurs on the fracture theorem page, I mention it here. Maybe there should be an entry for ’local duality’?
in the comment section to this MO reply some “FrankScience” points out that a condition is missing on the page here (missing also in Bauer’s notes, which I probably followed) and Neil Strickland then gives a counter-example showing that condition is in fact necessary.
Somebody should add this here.
added pointer to
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