nLab fracture theorem

Idea

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 $p$. 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 $p$, 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 $\mathbb{Q}$ and that of $\mathbb{Q}/\mathbb{Z}$. Viewed this way there is a more general fracture theorem which says that for any suitable pair $E,F$ 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 arithmetic fracture. (See below.)

Statement

In number theory and arithmetic geometry

The statement in number theory/arithmetic geometry is the following:

Proposition

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

$$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{Q}\otimes_{\mathbb{Z}}\left(\underset{p\;prime}{\prod} \mathbb{Z}_p \right) && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \underset{p\;prime}{\prod} \mathbb{Z}_p } \,.$$

Equivalently this is the fiber product of the rationals with the integral adeles $\mathbb{A}_{\mathbb{Z}}$ over the ring of adeles $\mathbb{A}_{\mathbb{Q}}$

$$\array{ && \mathbb{Q} \\ & \swarrow && \nwarrow \\ \mathbb{A}_{\mathbb{Q}} && && \mathbb{Z} \\ & \nwarrow && \swarrow \\ && \mathbb{A}_{\mathbb{Z}} } \,,$$

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 (by the discussion here) 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).

An original discussion is (Sullivan 05, prop. 1.18). Review includes (Riehl 14, lemma 14.4.2).

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. 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 the following discussions:

• Moduli stack of bundles over curves;

• geometric Langlands correspondence;

• Weil conjecture on Tamagawa numbers.

In homotopy theory

In homotopy theory the corresponding statement is that homotopy types may be decomposed into that of rational homotopy types and p-complete homotopy types of p-local homotopy types.

Proposition

Let $p$ be a prime number. Let $X$ be a homotopy type/∞-groupoid satisfying at least one of the following sufficient conditions

• $X$ is a connected, nilpotent space with finitely generated homotopy groups;

• $X$ is p-local homotopy type;

Then $X$ is the homotopy fiber product

$$X \simeq X_{\mathbb{Q}} \underset{(X_p^\wedge)_{\mathbb{Q}}}{\times} X_p^\wedge$$

of its rationalization $X_{\mathbb{Q}}$ with its p-completion $X_p^\wedge$ over the rationalization $(X_p^\wedge)_{\mathbb{Q}}$of the $p$-completions.

This originates around (Bousfield-Kan 72, VI.8.1). A detailed more modern account is in (May-Ponto, theorem 13.1.4). A quick survey is in (Riehl 14, theorem 14.4.14).

In stable homotopy theory

Similar statements hold in stable homotopy theory for spectra. There is a stable version of

• The arithmetic fracture square

but more generally there are fracture squares for the coproduct homology theory $E \vee F$ whenever $F$-localization is $E$-acyclic:

• General fracture squares

warning: a condition missing in the following, see the comment section of this MO reply. Somebody should add the relevant clause here…

The arithmetic fracture square for spectra

For $p$ a prime number write

• $L_p$ for Bousfield localization of spectra at the Moore spectrum $S \mathbb{F}_p$, hence for p-completion $(-)_p^\wedge$;

• $L_{\mathbb{Q}}$ for the Bousfield localization of spectra at the Moore spectrum/Eilenberg-MacLane spectrum $S \mathbb{Q} \simeq H \mathbb{Q}$, hence for rationalization.

Proposition

(Sullivan arithmetic square)

For every spectrum $X$ the canonical square

$$\array{ && L_{\mathbb{Q}}X \\ & \swarrow && \nwarrow \\ L_{\mathbb{Q}} \left( \prod_p L_p X \right) && && X \\ & \nwarrow && \swarrow \\ && \prod_p L_p X }$$

is a homotopy pushout (hence also a homotopy pullback).

Original statements of this include (Bousfield 79, Sullivan 05, prop. 3.20). Review includes (van Koughnett 13, prop. 4.5, Bauer 11, lemma 2.1).

Remark

(“one prime at a time”)

The impact of prop. is that it decomposes the study of (stable) homotopy theory into that of

1. rational homotopy theory and

2. p-adic homotopy theory for each prime $p$.

Both the rationalization $X_{\mathbb{Q}}$ and the p-completion $X_{p}^\wedge$ are typically much easier to analyze than $X$ itself and so the fracture theorem gives a way to decompose the remaining hard part of study of homotopy types into that of $p$-local/$p$-complete spaces. This procedure is known in homotopy theory as working “one prime at a time”.

More generally:

Proposition

The product of all p-completions is equivalently the Bousfield localization of spectra at the wedge sum $\vee_p S \mathbb{F}_p$ of all Moore spectra

$$\prod_p L_p X \simeq L_{\vee_p S \mathbb{F}_p} X \,.$$

Moreover there is a Bousfield equivalence

$$S (\mathbb{Q}/\mathbb{Z}) \simeq_{Bousf} \vee_p S \mathbb{F}_p \,,$$

and therefore also an equivalence

$$\prod_p L_p X \simeq L_{S (\mathbb{Q}/\mathbb{Z})} X \,.$$

The first statement originates around (Bousfield 79, prop. 2.6), review includes (van Koughnett 13, prop. 4.4, Bauer 11, below prop. 2.2); the second is highlighted in (Strickland 12, MO comment).

Remark

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 $E$-acyclifications $G_E(X) \to X$, for $E = S \mathbb{Q} \simeq H \mathbb{Q}$ and $E = S \mathbb{F}_p$ the Moore spectra of $\mathbb{Q}$ and of the cyclic group/finite field $\mathbb{F}_p = \mathbb{Z}/(p)$, respectively (e.g. Lurie 10, lecture 20).

Including this into the statement of prop. says that for spectra $X$ satisfying sufficient conditions as above, then the canonical diagram

$$\array{ && X_{\mathbb{Q}} && \longleftarrow && G_{S (\mathbb{Q}/\mathbb{Z})}(X) \\ & \swarrow && \nwarrow && \swarrow \\ (\prod_p X_p^\wedge)_{\mathbb{Q}} && && X \\ & \nwarrow && \swarrow && \nwarrow \\ && \prod_p X_p^\wedge && \longleftarrow && G_{H\mathbb{Q}}(X) }$$

is homotopy exact, in that

1. the square is a homotopy pullback and hence also a homotopy pushout (this is prop. );

2. the diagonals are homotopy cofiber sequences and hence also homotopy fiber sequences (by this proposition at Bousfield localization of spectra);

3. the top and bottom outer composite sequences are homotopy fiber sequences (and hence homotopy cofiber sequences) (by applying the pasting law to the previous two items).

Together this is like two-thirds of a differential cohomology hexagon – to the extent that $p$-adic completion is adjoint to $p$-torsion approximation. This is indeed the case, as the next proposition asserts

Notice that in view of remark then $X_p^\wedge$ is like the restriction of $X$ from Spec(Z) to all formal disks around the points $(p)$, and hence $G_{S\mathbb{F}_p}$ is like the restriction to the “complement of all formal disks”. Finally $X_{\mathbb{Q}}$ may be understood as the restriction to the Ran space of $Spec(\mathbb{Z})$ (Gaitsgory 11), roughly the colimit of the restriction of $X$ to the complement of finitely many points, as this set of points ranges through all points.

In view of remark we may regard the following fact as an refinement of the traditional arithmetic fracture theorem.

Proposition

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

$$\array{ \underoverset{ A Mod_{\mathfrak{a}comp}^{op}} {A Mod_{\mathfrak{a}tors}^{op}} {\simeq} &\stackrel{\overset{\Pi_{\mathfrak{a}}}{\longleftarrow}}{\stackrel{\hookrightarrow}{\underset{\flat_{\mathfrak{a}}}{\longleftarrow}}}& A Mod^{op} }$$

where

• $A Mod$ is the stable (∞,1)-category of modules, i.e. of ∞-modules over $A$;

• $A Mod_{\mathfrak{a}tors}$ 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.

Corollary

The traditional arithmetic fracture square of prop. , regarded as in remark , is the left part of the “differential cohomology diagram” induced by the adjoint modality $(\Pi_{\mathfrak{a}} \dashv \flat_{\mathfrak{a}} )$ of prop. , for the special case that $X = \mathbb{S}$ is the sphere spectrum and $\mathfrak{a} = (p)$ a prime ideal

$$\array{ && \Pi_{\mathfrak{a}dR} X && \stackrel{\mathbf{d}}{\longrightarrow} && \flat_{\mathfrak{a}dR} X \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat X && \Downarrow && X && \Downarrow && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && \flat_{\mathfrak{a}} X && \longrightarrow && \Pi_{\mathfrak{a}} X } \,,$$

cohesion in E-∞ arithmetic geometry:

cohesion modality	symbol	interpretation
flat modality	$\flat$	formal completion at
shape modality	$ʃ$	torsion approximation
dR-shape modality	$ʃ_{dR}$	localization away
dR-flat modality	$\flat_{dR}$	adic residual

the differential cohomology hexagon/arithmetic fracture squares:

$$\array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ \Pi_{\mathfrak{a}dR} \flat_{\mathfrak{a}} X && && X && && \Pi_{\mathfrak{a}} \flat_{\mathfrak{a}dR} X \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,$$

Remark

The special case of prop. where $A$ is the Eilenberg-MacLane spectrum of a plain commutative ring, and hence – by the stable Dold-Kan correspondence – the case where $A$-∞-modules are equivalently chain complexes, has a longer tradition in the existing literature. This we highlight separately below.

Remark

By the discussion at completion of modules in the section Monoidalness, the adjoint modality of prop. is a monoidal (∞,1)-functor without, possibly, respect the tensor unit in $A Mod$. This means that $(\Pi_{\mathfrak{a}}\dashv \flat_{\mathfrak{a}})$ passes to “commutative ∞-monoids-without unit” in $A Mod$, hence to (formal duals of) nonunital E-∞ algebras. By this proposition (Lurie “Algebra”, prop. 5.2.3.15) nonunital E-∞ rings are equivalent to augmented E-∞ rings over the sphere spectrum, hence this is E-∞ arithmetic geometry under $Spec(\mathbb{S})$.

Notice that in addition $\Pi_{\mathfrak{a}}$ here should preserve finite products (because by the discussion at completion of a module – monoidalness the underlying $\Pi_{\mathfrak{a}} \colon A Mod \to A Mod$ preserves all small (∞,1)-colimits and because by this proposition finite coproducts in $CRng(A Mod)$ are computed in the underlying $A Mod$.

Therefore we may think of $\Pi_{\mathfrak{a}}$ as a shape modality and of $\flat_{\mathfrak{a}}$ as a sharp modality on affine E-∞-arithmetic geometry under $Spec(\mathbb{S})$ – namely on formal duals of nonunital E-∞ rings .

(It may be entertaining to note that on the level of ∞-groups of units then E-∞ arithmetic geometry under $Spec(\mathbb{S})$ 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.

The arithmetic fracture square for chain complexes

We discuss here arithmetic fracturing on chain complexes of modules. Under the stable Dold-Kan correspondence this is a special case of prop. , but this special case has a longer tradition in the literature – going back to Grothendieck local duality and sometimes called Greenlees-May duality due to (Greenlees-May 92) – and we point to these original proofs.

Definition

Let $A$ be a commutative ring, let $\mathfrak{a} \subset A$ be be an ideal generated by a single regular element (i.e. not a zero divisor). 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 approximation-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.

Proposition

1. 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.

2. 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.

3. 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 (Dwyer-Greenlees 99) and (Porta-Shaul-Yekutieli 10) from the language of derived categories to (∞,1)-category theory.

First of all, by our simplifying assumption that $\mathfrak{a}$ 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 $\Pi_{\mathfrak{a}}$ and $\flat_{\mathfrak{a}}$ are the adjoint pair induced from the reflection/coreflection adjoint triple.

This adjoint triple is stated more explicitly in (Dwyer-Greenlees 99, section 4), see also (Porta-Shaul-Yekutieli 10, end of remark 6.14).

Therefore arithmetic fracture squares in the homotopy theory of chain complexes are induced by this as in corollary above.

General fracture squares of spectra

By prop. the arithmetic fracture square of prop. is equivalently of the form

$$\array{ && L_{H\mathbb{Q}}X \\ & \swarrow && \nwarrow \\ L_{H\mathbb{Q}} L_{S \mathbb{Q}/\mathbb{Z}} X && && X \\ & \nwarrow && \swarrow \\ && L_{S \mathbb{Q}/\mathbb{Z}} X } \,.$$

In this form the statement holds much more generally:

Proposition

Let $E, F, X$ be spectra such that the $F$-localization of $X$ is $E$-acyclic, i.e. $E_\bullet(L_F X) \simeq 0$, then the canonical square diagram

$$\array{ && L_F X \\ & \swarrow && \nwarrow \\ L_F L_E X && && L_{E\vee F} X \\ & \nwarrow && \swarrow \\ && L_E X }$$

is a homotopy pullback (and hence by stability also a homotopy pushout).

e.g. (Bauer 11, prop. 2.2)

Remark

The general version of the fracture statement in prop. is used frequently in chromatic homotopy theory for decomposition in Morava K-theory and Morava E-theory-localizations. For example there is a chromatic fracture square:

$$\array{ && L_{E(n-1)} X \\ & \swarrow && \nwarrow \\ L_{E(n-1)} L_{K(n)} X && && L_{E(n)} X \\ & \nwarrow && \swarrow \\ && L_{K(n)} X }$$

In particular it is used for instance in the construction of tmf, see example below.

From another perspective:

Claim

Suppose that $L \colon Spectra \to Spectra$ is a smashing localization given by smash product with some spectrum $T$. Write $F$ for the homotopy fiber

$$F \longrightarrow \mathbb{S} \longrightarrow T \,.$$

Then there is a fracture diagram of operations

$$\array{ T \wedge (-) && \longleftarrow && [T,-] \\ & \nwarrow && \swarrow \\ && \mathbb{S} \\ & \swarrow & & \nwarrow \\ [F,-] && \longleftarrow && F \wedge (-) }$$

where $[F,-]$ and $T \wedge (-) \colon Spectra \to Spectra$ are idempotent (∞,1)-monads and $[T,-]$, $F \wedge (-)$ are idempotent $\infty$-comonads, the diagonals are homotopy fiber sequences.

(Charles Rezk, MO comment,August 2014)

Example

For $T = S \mathbb{Z}[p^{-1}]$ the Moore spectrum of the integers localized away from $p$, then

$$F = \Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}) \to \mathbb{S} \to S \mathbb{Z}[p^{-1}]$$

and hence

• $\Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}) \wedge (-)$ is $p$-torsion approximation;

• $[\Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}),-]$ is $p$-completion;

• $S \mathbb{Z}[p^{-1}] \wedge (-)$ is $p$-rationalization;

• $[T,-]$ is forming $p$-adic residual.

$$\array{ && localization\;away\;from\;\mathfrak{a} && \stackrel{}{\longrightarrow} && \mathfrak{a}\;adic\;residual \\ & \nearrow & & \searrow & & \nearrow && \searrow \\ && && X && && \\ & \searrow & & \nearrow & & \searrow && \nearrow \\ && formal\;completion\;at\;\mathfrak{a}\; && \longrightarrow && \mathfrak{a}\;torsion\;approximation } \,,$$

With (Bousfield 79, prop.2.5)

For $E_\infty$-modules

Proposition

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

$$\array{ \underoverset{ A Mod_{\mathfrak{a}comp}^{op}} {A Mod_{\mathfrak{a}tors}^{op}} {\simeq} &\stackrel{\overset{\Pi_{\mathfrak{a}}}{\longleftarrow}}{\stackrel{\hookrightarrow}{\underset{\flat_{\mathfrak{a}}}{\longleftarrow}}}& A Mod^{op} }$$

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 $\flat_{\mathfrak{a}}$.

This is effectively the content of (Lurie “Completions”, section 4):

In cohesive (stable) homotopy theory

In cohesive homotopy theory every stable homotopy type $X$ sits in a fracture square of the form

$$\array{ && \Pi_{dR} X && \longrightarrow && \flat_{dR} X \\ & \nearrow & & \searrow && \nearrow \\ \Pi_{dR} \flat X && && X \\ & \searrow & & \nearrow && \searrow \\ && \flat X && \longrightarrow && \Pi X }$$

where $\flat$ is the flat modality and $\Pi_{dR}$ the homotopy fiber of the unit $X\to \Pi X$ of the shape modality. This is the left part of the differential cohomology hexagon for $X$, see there for details.

Here $\Pi_{dR} X$ is such that for any other stable cohesive homotopy type $\hat E$ then functions $\Pi_{dR} X \to \hat E$ are equivalent to functions $X \to \flat_{dR} \hat E$, where $\hat E \to \flat_{dR} \hat E$ is a generalized form of rationalization in the sense discussed at differential cohomology hexagon. In particular if $\hat E$ is a Hopkins-Singer-type differential cohomology refinement of a plain spectrum $E$, then $E\to \flat_{dR} E$ is its ordinary rationalization given by the Chern character and $\hat E \to \flat_{dR} \hat E$ 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 $\flat$ takes any space $X$ to the union of all its formal disks. (See at differential cohesion and idelic structure.) Accordingly the collection of functions $\flat X \to \hat E$ in this case behave like the product of all formal power series of $\hat E$-valued functions around all global points of $X$, analogous to remark .

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 $X = \Sigma$ is a complex curve then $\flat \Sigma$ is precisely the analog of the integral adeles as it is predicted by the function field analogy.

Examples

Example

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. , prop. to decompose the moduli stack of elliptic curves into rational and $p$-adic curves, and then in a second step in applying in turn the general fracture square of prop. for Morava K-theory to the remaining $p$-adic pieces.

See at tmf – Decomposition via arithmetic fracture squares for more on this.

Related concepts

• p-adic homotopy theory, p-completion

• rational homotopy theory, rationalization

References

• Aldridge Bousfield, Daniel Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol 304, Springer 1972

• Aldridge Bousfield, The localization of spectra with respect to homology , Topology vol 18 (1979) (pdf)

• 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)

• Tilman Bauer, Bousfield localization and the Hasse square (2011) (pdf, pdf), chapter 6 in: Christopher Douglas, John Francis, André Henriques, Michael Hill (eds.), Topological Modular Forms, Mathematical Surveys and Monographs Volume 201, AMS 2014 (ISBN:978-1-4704-1884-7)

• Paul VanKoughnett, Spectra and localization, 2013 (pdf)

• Emily Riehl, Categorical homotopy theory, new mathematical monographs 24, Cambridge University Press 2014 (published version)

• Peter May, Kate Ponto, chapters 7 and 8 of More concise algebraic topology: Localization, completion, and model categories (pdf)

• Michael Shulman, The Propositional Fracture Theorem, (blog post)

Related MO-discussion:

• Fracture squares of Bousfield Localizations of Spectra

Discussion of rational functions as functions on the Ran space is in

• Dennis Gaitsgory, Contractibility of the space of rational maps (arXiv:1108.1741)

Discussion of $\mathfrak{a}$-adic completion and $\mathfrak{a}$-torsion approximation as derived idempotent (co-)monads on a derived category of chain complexes of modules – Greenlees-May duality – is in

• Marco Porta, Liran Shaul, Amnon Yekutieli, On the Homology of Completion and Torsion (arXiv:1010.4386)

building on

• John Greenlees, Peter May, Derived functors of I-adic completion and local homology, J. Algebra 149 (1992), 438–453 (pdf)

• Leovigildo Alonso, Ana Jeremías, Joseph Lipman, Local Homology and Cohomology on Schemes (arXiv:alg-geom/9503025)

• Mark Hovey, John Palmieri, Neil Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 128 (1997), no. 610, x+114.

• William Dwyer, John Greenlees, Complete modules and torsion modules, Amer. J. Math. 124, No. 1, (1999) (pdf)

Discussion of this in stable homotopy theory and the full generality of higher algebra is in

• Jacob Lurie, section 4 of Proper Morphisms, Completions, and the Grothendieck Existence Theorem

And in the context of commutative DG-rings in

• Liran Shaul, Completion and torsion over commutative DG rings, arXiv:1605.07447

This and further generalization is in

• Tobias Barthel, Drew Heard, Gabriel Valenzuela, Local duality in algebra and topology (arXiv:1511.03526)

Discussion in homotopy type theory:

• Luis Scoccola, Nilpotent Types and Fracture Squares in Homotopy Type Theory (arXiv:1903.03245)

Exposition from the point of view of modal homotopy theory:

• Urs Schreiber, Fractures, Ideles and the Differential Hexagon, talk at Workshop on differential cohomologies, CUNY Graduate Center, August 12-14 2014 (video recording)

• Urs Schreiber, Differential cohesion and Arithmetic geometry, talk at Karslruher Weihnachts-Workshop 2015

Formalization in cohesive modal homotopy type theory homotopy type theory:

• David Jaz Myers, Modal Fracture of Higher Groups, talk at CMU-HoTT Seminar, 2021 (pdf, pdf)

See also

• John Greenlees, Tate cohomology in axiomatic stable homotopy theory (pdf)