nLab torsion module



Linear algebra

homotopy theory, (∞,1)-category theory, homotopy type theory

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see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




For modules over rings

Given a ring RR, an element mm in an RR-module MM is torsion element if there is a nonzero element rr in RR such that rm=0r m=0. In constructive mathematics, given a ring RR with a tight apartness relation #\#, an element mm in an RR-module MM is a torsion element if there is a element rr in RR such that r#0r \# 0 and rm=0r m=0.

A torsion module is a module whose elements are all torsion. A torsion-free module is a module whose elements are not torsion, other than 00.

More generally, given an ideal 𝔞R\mathfrak{a} \subset R then an 𝔞\mathfrak{a}-torsion module is one all whose elements are annihilated by some power of elements in 𝔞\mathfrak{a}.

For \infty-modules over E E_\infty-rings

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


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).


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).


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).


Last revised on May 19, 2022 at 06:09:45. See the history of this page for a list of all contributions to it.