nLab torsion approximation

Idea

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

$$G_{S \mathbb{Q}} X \longrightarrow X \longrightarrow L_{S\mathbb{Q}} X$$

(by this proposition).

This $G_{S\mathbb{Q}}$ 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 $S\mathbb{Z}[p^{-1}]$-localization.

Definition

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

Definition

An $A$-∞-module $N$ is an $\mathfrak{a}$-torsion module if for all elements $n \in \pi_k N$ and all elements $a \in \mathfrak{a}$ there is $k \in \mathbb{N}$ such that $a^k n = 0$.

(Lurie “Completions”, def. 4.1.3).

Proposition

The full sub-(∞,1)-category

$$A Mod_{\mathfrak{a}tor} \hookrightarrow A Mod$$

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

(Lurie “Completions”, prop. 4.1.12).

Proposition

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

$$\pi_0 ʃ_{\mathfrak{a}} N \hookrightarrow \pi_0 N$$

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

(Lurie “Completions”, prop. 4.1.18).

Properties

Relation to localization

Proposition

There is a natural homotopy fiber sequence

$$ʃ_{\mathfrak{a}} \longrightarrow id \longrightarrow ʃ_{\mathfrak{a}dR}$$

relating $\mathfrak{a}$-torsion approximation on the left with $\mathfrak{a}$-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 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:

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

Related concepts

• Pontryagin duality for torsion abelian groups

References

Discussion for chain complexes is in

• William Dwyer, John Greenlees, section 4.1 of Complete modules and torsion modules, Amer. J. Math, 1999 (pdf)

• Carles Casacuberta et al., Models for torsion homotopy types (pdf)

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

• Jacob Lurie, section 4.1 of Proper Morphisms, Completions, and the Grothendieck Existence Theorem