fracture theorem in nLab

Context

Homotopy theory

Arithmetic geometry

• Idea
• Statement
• In number theory and arithmetic geometry
• In homotopy theory
• In stable homotopy theory
• The arithmetic fracture square for spectra
• The arithmetic fracture square for chain complexes
• General fracture squares of spectra
• Claim
• For E ∞E_\infty-modules
• In cohesive (stable) homotopy theory
• Examples
• Related concepts
• References

Proposition

The integers ℤ\mathbb{Z} are the fiber product of all the p-adic integers ∏pprimeℤ p\underset{p\;prime}{\prod} \mathbb{Z}_p with the rational numbers ℚ\mathbb{Q} over the rationalization of the former, hence there is a pullback diagram in CRing of the form

ℚ ↙ ↖ ℚ⊗ ℤ(∏pprimeℤ p) ℤ ↖ ↙ ∏pprimeℤ p. \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 𝔸 ℤ\mathbb{A}_{\mathbb{Z}} over the ring of adeles 𝔸 ℚ\mathbb{A}_{\mathbb{Q}}

ℚ ↙ ↖ 𝔸 ℚ ℤ ↖ ↙ 𝔸 ℤ, \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 𝔸 ℚ=ℚ⊗ ℤ𝔸 ℤ\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 ℚ\mathbb{Q} with the discrete topology and 𝔸 ℤ\mathbb{A}_{\mathbb{Z}} with its profinite/completion topology).

Proposition

(Sullivan arithmetic square)

For every spectrum XX the canonical square

L ℚX ↙ ↖ L ℚ(∏ pL pX) X ↖ ↙ ∏ pL pX \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).

Proposition

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

∏ pL pX≃L ∨ pS𝔽 pX. \prod_p L_p X \simeq L_{\vee_p S \mathbb{F}_p} X \,.

Moreover there is a Bousfield equivalence

S(ℚ/ℤ)≃ Bousf∨ pS𝔽 p, S (\mathbb{Q}/\mathbb{Z}) \simeq_{Bousf} \vee_p S \mathbb{F}_p \,,

and therefore also an equivalence

∏ pL pX≃L S(ℚ/ℤ)X. \prod_p L_p X \simeq L_{S (\mathbb{Q}/\mathbb{Z})} X \,.

Proof

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

• the existence of Π 𝔞\Pi_{\mathfrak{a}} is corollary 4.1.16 and remark 4.1.17

• the existence of ♭ 𝔞\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 (Π 𝔞⊣♭ 𝔞)(\Pi_{\mathfrak{a}} \dashv \flat_{\mathfrak{a}} ) of prop. , for the special case that X=𝕊X = \mathbb{S} is the sphere spectrum and 𝔞=(p)\mathfrak{a} = (p) a prime ideal

Π 𝔞dRX ⟶d ♭ 𝔞dRX ↗ ↘ ↗ ↘ Π 𝔞dR♭X ⇓ X ⇓ Π 𝔞♭ 𝔞dRX ↘ ↗ ↘ ↗ ♭ 𝔞X ⟶ Π 𝔞X, \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 } \,,

Definition

Let AA be a commutative ring, let 𝔞⊂A\mathfrak{a} \subset A be be an ideal generated by a single regular element (i.e. not a zero divisor). Write AMod ∞ opA Mod_{\infty}^{op} for the opposite (∞,1)-category of the (∞,1)-category of modules over AA.

Write

• ♭ 𝔞:AMod ∞ op→AMod ∞ op\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 𝔞\mathfrak{a};

  with canonical natural transformation

  ϵ 𝔞:♭ 𝔞⟶id\epsilon_{\mathfrak{a}} \colon \flat_{\mathfrak{a}} \longrightarrow id

• Π 𝔞:AMod ∞ op→AMod ∞ op\Pi_{\mathfrak{a}} \colon A Mod_\infty^{op} \to A Mod_\infty^{op} for the total derived functor of the 𝔞\mathfrak{a}-torsion approximation-functor;

  with canonical natural transformation

  η 𝔞:id⟶Π 𝔞\eta_{\mathfrak{a}}\colon id \longrightarrow \Pi_{\mathfrak{a}}

Finally write

(AMod ∞ op) 𝔞com,(AMod ∞ op) 𝔞tor↪AMod ∞ (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 XX for which, ϵ 𝔞(X)\epsilon_{\mathfrak{a}}(X) or η 𝔞(X)\eta_{\mathfrak{a}}(X) is an equivalence in an (∞,1)-category, respectively.

Proposition

1. The transformation ϵ 𝔞\epsilon_{\mathfrak{a}} exhibits (AMod ∞ op) 𝔞com↪AMod ∞(A Mod_\infty^{op})^{\mathfrak{a}com}\hookrightarrow A Mod_\infty as a reflective (∞,1)-subcategory, hence ♭ 𝔞\flat_{\mathfrak{a}} as an idempotent (∞,1)-monad.

2. The transformation η 𝔞\eta_{\mathfrak{a}} exhibits (AMod ∞ op) 𝔞tor↪AMod ∞(A Mod_\infty^{op})^{\mathfrak{a}tor}\hookrightarrow A Mod_\infty as a co-reflective (∞,1)(\infty,1)-category, hence Π 𝔞\Pi_{\mathfrak{a}} as an idempotent (∞,1)(\infty,1)-comonad.

3. Restricted to these sub-(∞,1)(\infty,1)-categories both ♭ 𝔞\flat_{\mathfrak{a}} as well as Π 𝔞\Pi_{\mathfrak{a}} become equivalences of (∞,1)-categories, hence exhibiting (Π 𝔞⊣♭ 𝔞)(\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 𝔞\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 Π 𝔞\Pi_{\mathfrak{a}} and ♭ 𝔞\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).

Proposition

Let E,F,XE, F, X be spectra such that the FF-localization of XX is EE-acyclic, i.e. E •(L FX)≃0E_\bullet(L_F X) \simeq 0, then the canonical square diagram

L FX ↙ ↖ L FL EX L E∨FX ↖ ↙ L EX \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).

Claim

Suppose that L:Spectra→SpectraL \colon Spectra \to Spectra is a smashing localization given by smash product with some spectrum TT. Write FF for the homotopy fiber

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

Then there is a fracture diagram of operations

T∧(−) ⟵ [T,−] ↖ ↙ 𝕊 ↙ ↖ [F,−] ⟵ F∧(−) \array{ T \wedge (-) && \longleftarrow && [T,-] \\ & \nwarrow && \swarrow \\ && \mathbb{S} \\ & \swarrow & & \nwarrow \\ [F,-] && \longleftarrow && F \wedge (-) }

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

Example

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

F=Σ −1S(ℤ[p −1]/ℤ)→𝕊→Sℤ[p −1] F = \Sigma^{-1} S (\mathbb{Z}[p^{-1}]/\mathbb{Z}) \to \mathbb{S} \to S \mathbb{Z}[p^{-1}]

and hence

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

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

• Sℤ[p −1]∧(−)S \mathbb{Z}[p^{-1}] \wedge (-) is pp-rationalization;

• [T,−][T,-] is forming pp-adic residual.

localizationawayfrom𝔞 ⟶ 𝔞adicresidual ↗ ↘ ↗ ↘ X ↘ ↗ ↘ ↗ formalcompletionat𝔞 ⟶ 𝔞torsionapproximation, \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 } \,,