nLab
torsion approximation

Contents

Idea

In stable homotopy theory the rational Bousfield localization of spectra, hence \mathbb{Q}-localization L SL_{S\mathbb{Q}}, is accompanied dually by “\mathbb{Q}-acyclification” G SG_{S\mathbb{Q}}, forming a natural homotopy fiber sequence

G SXXL SX G_{S \mathbb{Q}} X \longrightarrow X \longrightarrow L_{S\mathbb{Q}} X

(by this proposition).

This G SG_{S\mathbb{Q}} is the operation of universal torsion approximation. In the special case of chain complexes it corresponds to the derived functor that forms torsion subgroups (see the discussion at fracture theorem – Arithmetic fracturing of chain complexes).

Similarly, if one already looks at p-local spectra then torsion approximation is the homotopy fiber of S[p 1]S\mathbb{Z}[p^{-1}]-localization.

Definition

Let AA be an E-∞ ring and 𝔞π 0A\mathfrak{a} \subset \pi_0 A a finitely generated ideal of its underlying commutative ring.

Definition

An AA-∞-module NN is an 𝔞\mathfrak{a}-torsion module if for all elements nπ kNn \in \pi_k N and all elements a𝔞a \in \mathfrak{a} there is kk \in \mathbb{N} such that a kn=0a^k n = 0.

(Lurie “Completions”, def. 4.1.3).

Proposition

The full sub-(∞,1)-category

AMod 𝔞torAMod A Mod_{\mathfrak{a}tor} \hookrightarrow A Mod

is co-reflective and the co-reflector ʃ 𝔞ʃ_{\mathfrak{a}} – the torsion approximation – is smashing.

(Lurie “Completions”, prop. 4.1.12).

Proposition

For NAMod 0N \in A Mod_{\leq 0} then torsion approximation, prop. , intuced a monomorphism on π 0\pi_0

π 0ʃ 𝔞Nπ 0N \pi_0 ʃ_{\mathfrak{a}} N \hookrightarrow \pi_0 N

including the 𝔞\mathfrak{a}-nilpotent elements of π 0N\pi_0 N.

(Lurie “Completions”, prop. 4.1.18).

Properties

Relation to localization

Proposition

There is a natural homotopy fiber sequence

ʃ 𝔞idʃ 𝔞dR ʃ_{\mathfrak{a}} \longrightarrow id \longrightarrow ʃ_{\mathfrak{a}dR}

relating 𝔞\mathfrak{a}-torsion approximation on the left with 𝔞\mathfrak{a}-localization on the right.

(Lurie “Completions”, remark 4.1.20)

As a modality in arithmetic cohesion

Under suitable conditions, torsion approximation forms an adjoint modality with adic completion.

cohesion in E-∞ arithmetic geometry:

cohesion modalitysymbolinterpretation
flat modality\flatformal completion at
shape modalityʃʃtorsion approximation
dR-shape modalityʃ dRʃ_{dR}localization away
dR-flat modality dR\flat_{dR}adic residual

the differential cohomology hexagon/arithmetic fracture squares:

localizationawayfrom𝔞 𝔞adicresidual Π 𝔞dR 𝔞X X Π 𝔞 𝔞dRX formalcompletionat𝔞 𝔞torsionapproximation, \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 } \,,

References

Discussion for chain complexes is in

Discussion in the generality of E-∞ rings and ∞-modules is in

Last revised on November 11, 2017 at 13:46:51. See the history of this page for a list of all contributions to it.