A basic fact in number theory is that the natural numbers may be decomposed into the rational numbers and the p-adic integers for all prime numbers . Dually in arithmetic geometry this says that Spec(Z) has a cover by all its formal disks and the complements of finitely many points, a fact that is crucial in the geometric interpretation of the function field analogy and which motivates for instance the geometric Langlands correspondence. (See below.)
Lifting this statement to stable homotopy theory and “higher arithmetic geometry” the arithmetic fracture theorem says that stable homotopy types (and suitably tame plain homotopy types) canonically decompose into their rationalization and their p-completion for all primes , hence into their images in rational homotopy theory and p-adic homotopy theory. Since these images are typically simpler than the original homotopy type itself, this decomposition is a fundamental computational tool in stable homotopy theory, often known under the slogan of “working one prime at a time”. (See below.)
One finds that this arithmetic fracturing in stable homotopy theory is really a statement about the Bousfield localization of spectra with respect to the Moore spectrum for and that of . Viewed this way there is a a more general fracture theorem which says that for any suitable pair of spectra/homology theories the Bousfield localization at their coproduct decomposes into the separate Bousfield localizations. This generalized fracture theorem appears for instance in chromatic homotopy theory for localization at Morava K-theory and Morava E-theory. (See below.)
In cohesive homotopy theory every stable homotopy type canonically sits in a fracture square formed from the localizations exhibited by the shape modality and the flat modality. For differential cohesion over infinitesimal cohesion this is a higher geometric analog of the classical artihmetic fracture. (See below.)
Since the ring of adeles is the rationalization of the integral adeles , this is also (by the discussion here) a pushout diagram in CRing, and in fact in topological commutative rings (for with the discrete topology and with its profinite/completion topology).
Under the function field analogy we may think of
as the ring of functions on the complement of a finite number of points in the curve;
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.
This kind of cover plays a central role in number theory, see for instance the following discussions:
Then is the homotopy fiber product
but more generally there are fracture squares for the coproduct homology theory whenever -localization is -acyclic:
For a prime number write
(Sullivan arithmetic square)
For every spectrum the canonical square
(“one prime at a time”)
Both the rationalization and the p-completion are typically much easier to analyze than itself and so the fracture theorem gives a way to decompose the remaining hard part of study of homotopy types into that of -local/-complete spaces. This procedure is known in homotopy theory as working “one prime at a time”.
Moreover there is a Bousfield equivalence
and therefore also an equivalence
By the discussion at Bousfield localization of spectra and at localization of a space, the rationalization and the p-completion maps on spectra are homotopy cofibers of -acyclifications , for and the Moore spectra of and of the cyclic group/finite field , respectively (e.g. Lurie 10, lecture 20).
Including this into the statement of prop. 2 says that for spectra satisfying sufficient conditions as above, then the canonical diagram
is homotopy exact, in that
Together this is like two-thirds of a differential cohomology hexagon – to the extent that -adic completion is adjoint to -torsion approximation. This is indeed the case, as the next proposition asserts
Notice that in view of remark 1 then is like the restriction of from Spec(Z) to all formal disks around the points , and hence is like the restriction to the “complement of all formal disks”. Finally may be understood as the restriction to the Ran space of (Gaitsgory 11), roughly the colimit of the restriction of to the complement of finitely many points, as this set of points ranges through all points.
In view of remark 4 we may regard the following fact as an refinement of the traditional arithmetic fracture theorem.
This is effectively the content of (Lurie “Proper morphisms”, section 4):
the existence of is corollary 4.1.16 and remark 4.1.17
the existence of is lemma 4.2.2 there;
the equivalence of sub--categories is proposition 4.2.5 there.
The traditional arithmetic fracture square of prop. 2, regarded as in remark 4, is the left part of the “differential cohomology diagram” induced by the adjoint modality of prop. 8, for the special case that is the sphere spectrum and a prime ideal
|flat modality||formal completion at|
|shape modality||torsion approximation|
|dR-shape modality||localization away|
|dR-flat modality||adic residual|
The special case of prop. 8 where is the Eilenberg-MacLane spectrum of a plain commutative ring, and hence – by the stable Dold-Kan correspondence – the case where -∞-modules are equivalently chain complexes, has a longer tradition in the existing literature. This we highlight separately below.
By the discussion at completion of modules in the section Monoidalness, the adjoint modality of prop. 8 is a monoidal (∞,1)-functor without, possibly, respect the tensor unit in . This means that passes to “commutative ∞-monoids-without unit” in , hence to (formal duals of) nonunital E-∞ algebras. By this proposition (Lurie “Algebra”, prop. 18.104.22.168) nonunital E-∞ rings are equivalent to augmented E-∞ rings over the sphere spectrum, hence this is E-∞ arithmetic geometry under .
Notice that in addition here should preserve finite products (because by the discussion at completion of a module – monoidalness the underlying preserves all small (∞,1)-colimits and because by this proposition finite coproducts in are computed in the underlying .
(It may be entertaining to note that on the level of ∞-groups of units then E-∞ arithmetic geometry under translates to abelian ∞-groups of twists over the sphere spectrum – which has been argued to be the homotopy-theoretic incarnation of superalgebra, see at superalgebra – abstract idea for more on this.)
For more discussion of this see also differential cohesion and idelic structure.
We discuss here arithmetic fracturing on chain complexes of modules. Under the stable Dold-Kan correspondence this is a special case of prop. 8, but this special case has a longer tradition in the literature – often called Greenlees-May duality due to (Greenlees-May 92) – and we point to these original proofs.
with canonical natural transformation
with canonical natural transformation
The transformation exhibits as a co-reflective -category, hence as an idempotent -comonad.
First of all, by our simplifying assumption that is generated by a single regular element, the running assumption of “weak proregularity” in (Porta-Shaul-Yekutieli 10, def.3.21) is 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-category#CharacterizationOfReflectors) 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 (Hovey-PalieriS-trickland 97, 3.3.5, Dwyer-Greenlees 99, theorem 2.1 Porta-Shaul-Yekutieli 10, theorem 6.11) implies the canonical equivalence of the two sub-(∞,1)-categories and this means that and are the adjoint pair induced from the reflection/coreflection adjoint triple.
Therefore arithmetic fracture squares in the homotopy theory of chain complexes are induced by this as in corollary 1 above.
In this form the statement holds much more generally:
e.g. (Bauer 11, prop. 2.2)
From another perspective:
Then there is a fracture diagram of operations
With (Bousfield 79, prop.2.5)
This is effectively the content of (Lurie “Completions”, section 4):
Here is such that for any other stable cohesive homotopy type then functions are equivalent to functions , where is a generalized form of rationalization in the sense discussed at differential cohomology hexagon. In particular if is a Hopkins-Singer-type differential cohomology refinement of a plain spectrum , then is its ordinary rationalization given by the Chern character and is the corresponding map on Chern curvature forms.
Moreover, if the ambient cohesion is differential cohesion over a base of infinitesimal cohesion, then the flat modality takes any space to the union of all its formal disks. (See at differential cohesion and idelic structure.) Accordingly the collection of functions in this case behave like the product of all formal power series of -valued functions around all global points of , analogous to remark 1.
An example of this are synthetic differential ∞-groupoids regarded as cohesive over their formal moduli problems, as its its complex analytic incarnation by synthetic differential complex analytic ∞-groupoids. In this context if is a complex curve then is precisely the analog of the integral adeles as it is predicted by the function field analogy.
The construction of the tmf-spectrum – the spectrum of global sections of the derived Deligne-Mumford stack of derived elliptic curves – as described in (Behrens 13) proceeds by first applying the arithmetic fracture square of prop. \ref to decompose the moduli stack of elliptic curves into rational and -adic curves, and then in a second step in applying in turn the general fracture square of prop. 6 for Morava K-theory to the remaining -adic pieces.
See at tmf – Decomposition via arithmetic fracture squares for more on this.
Dennis Sullivan, Geometric topology: localization, periodicity and Galois symmetry, volume 8 of K- Monographs in Mathematics. Springer, Dordrecht, 2005. The 1970 MIT notes, Edited and with a preface by Andrew Ranicki (pdf)
Emily Riehl, Categorical homotopy theory, new mathematical monographs 24, Cambridge University Press 2014 (published version)
Discussion of rational functions as functions on the Ran space is in