nLab torsion module

Definition

For modules over rings

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

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 $0$.

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

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

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).

Related concdepts

• torsion subgroup

• torsion-free module

References

• Daniel Quillen, Module theory over nonunital rings, August 1996 (pdf)

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