Torsion (algebra)

Torsion is the phenomenon of commutative algebra , i.e. the theory of modules over commutative rings , which fundamentally distinguishes it from the (simpler) theory of vector spaces . Torsion is related to the concept of the zero divider .

Global twist

Definitions

In its simplest form, a torsion element is an element of finite order in a group or a monoid , i.e. an element for which there is a natural number , so that (or in additive notation) applies. ${\displaystyle g}$ ${\displaystyle n}$ ${\displaystyle g^{n}=1}$ ${\displaystyle n\cdot g=0}$

For the Torsionsbegriff of commutative algebra is a (commutative) ring (with identity) and one - module . ${\displaystyle R}$ ${\displaystyle M}$ ${\displaystyle R}$

• The twist or Torsionsuntermodul of the sub-module of those elements for which the core of the figure , does not include only zero divisor. In this case it is called the torsion element . ${\displaystyle M}$ ${\displaystyle m}$ ${\displaystyle R\to M}$ ${\displaystyle r\mapsto rm}$ ${\displaystyle m}$
• Equivalently, the torsion submodule can also be used as the core of homomorphism

${\displaystyle M\to M\otimes Q}$

define if denotes the total quotient ring of . ${\displaystyle Q}$ ${\displaystyle R}$

• ${\displaystyle M}$means torsion-free if the torsional submodule is zero.
• ${\displaystyle M}$is a torsional modulus when the torsional submodule is equal . Sometimes you say briefly: "is torsion". ${\displaystyle M}$ ${\displaystyle M}$

If an Abelian group ( i.e. module), the two definitions of torsion elements agree. One then speaks of torsion (sub) groups . ${\displaystyle M}$ ${\displaystyle \mathbb {Z} }$

Simple properties

• If the torsion submodule is , then is torsion free. So there is a canonical torsional submodule and a canonical torsion-free quotient, but not vice versa. ${\displaystyle T}$ ${\displaystyle M}$ ${\displaystyle M/T}$
• The formation of the torsion sub-module is a functor ; H. is a module homomorphism, then maps the torsion sub-module of into the torsion sub-module of . Even in the case of groups, a homomorphism always maps torsion elements to torsion elements. ${\displaystyle f\colon M\to N}$ ${\displaystyle f}$ ${\displaystyle M}$ ${\displaystyle N}$
• From the alternative description of the torsion sub-module as the core of a localization, it follows immediately that the formation of the torsion sub- module is a left-exact functor.

Examples

• The torsion elements of the group include and , but their product has an infinite order. In non-Abelian groups, the torsion elements do not necessarily form a subgroup. ${\displaystyle \operatorname {SL} _{2}(\mathbb {Z} )}$ ${\displaystyle {\begin{pmatrix}0&1\\-1&0\end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}0&-1\\1&1\end{pmatrix}}}$ ${\displaystyle {\begin{pmatrix}1&1\\0&1\end{pmatrix}}}$
• Another example of this fact is the infinite dihedral group

${\displaystyle \langle x,y\mid x^{2}=y^{2}=1\rangle }$,

in which the generators are torsion elements, but not for example . ${\displaystyle xy}$

• ${\displaystyle R}$itself, or more generally a free module, is torsion-free. In particular, if there is a body , all modules are torsion-free. ${\displaystyle R}$ ${\displaystyle R}$ ${\displaystyle R}$
• ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$is a torsional modulus (over ) for every natural number . In general, for a ring and an ideal of that does not consist only of zero dividers, the module is a torsion module . ${\displaystyle \mathbb {Z} }$ ${\displaystyle n}$ ${\displaystyle R}$ ${\displaystyle J}$ ${\displaystyle R}$ ${\displaystyle R/J}$
• If a body is , then the torsional sub-module of , understood as an Abelian group or module, is equal to the group of roots of unity in . ${\displaystyle K}$ ${\displaystyle K^{\times }}$ ${\displaystyle \mathbb {Z} }$ ${\displaystyle K}$

Abelian torsion groups

• An abelian torsion group is finite if and only if it is finite.
• An Abelian torsion group is the direct sum of its -primary subgroups for each prime number , i.e. H. of the subgroups of the elements whose order is a power of . The primary subgroup is a group . ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$ ${\displaystyle p}$
• As the example of the factor group shows, the orders of the elements are generally not restricted ; the -primary subgroup already has this property. ${\displaystyle \mathbb {Q} /\mathbb {Z} }$ ${\displaystyle p}$ ${\displaystyle \mathbb {Q} _{p}/\mathbb {Z} _{p}}$
• If the order of the elements is limited, this does not mean that the group is finitely generated (and thus finite): In an infinite direct product of cyclic groups of order 2, every element (except for the neutral element) has order 2.

Torsion-free Abelian groups

• An Abelian group is torsion-free if and only if there is a total order that is compatible with the group structure .

Torsion-free modules

• If a finitely generated module is torsion-free over a main ideal ring , it is free . This is especially true for Abelian groups .
• If a finitely generated module is torsion-free over a Dedekind ring, it is projective .
• Flat modules are torsion-free. The terms “flat” and “torsion-free” even coincide with Dedekind rings (especially with main ideal rings ).

The following diagram summarizes these implications for a module over a commutative integrity ring : ${\displaystyle M}$ ${\displaystyle A}$

Torsion with respect to a ring element

Definition of the a-torsion

Let it be a commutative ring with one element and one module. In the simplest case is ; is then just an Abelian group. ${\displaystyle A}$ ${\displaystyle M}$ ${\displaystyle A}$ ${\displaystyle A=\mathbb {Z} }$ ${\displaystyle M}$

For a ring element is ${\displaystyle a\in A}$

${\displaystyle M[a]=\{m\in M\mid am=0\}\cong \mathrm {Hom} _{A}(A/Aa,M)}$

a sub-module, known as the - twist of is called. (The risk of confusion with the notation for localizations is low.) The notation is also common. ${\displaystyle a}$ ${\displaystyle M}$ ${\displaystyle M[f^{-1}]}$ ${\displaystyle {}_{a}M}$

The module

${\displaystyle M[a^{\infty }]=\bigcup _{n\geq 1}M[a^{n}]=\{m\in M\mid \exists n\colon a^{n}m=0\}=\ker(M\to M\otimes A_{a})}$

is called torsion. ${\displaystyle a^{\infty }}$

properties

• ${\displaystyle M[a]}$is naturally a module. ${\displaystyle A/a}$
• The functor is left exact (as a representable functor even exchanged with any limits); more precisely: is ${\displaystyle M\mapsto M[a]}$

${\displaystyle 0\to M'\to M\to M''\to 0}$

is an exact sequence of modules, then ${\displaystyle A}$

${\displaystyle 0\to M'[a]\to M[a]\to M''[a]\to M'/aM'\to M/aM\to M''/aM''\to 0}$

exactly, as follows directly from the snake lemma.

• The torsional sub-module of is the union of the for all non-zero divisors . ${\displaystyle M}$ ${\displaystyle M[a]}$ ${\displaystyle a\in A}$
• For ring elements is . ${\displaystyle a,b}$ ${\displaystyle b\cdot M[ab]=(bM)[a]\subseteq M[a]}$
• For an Abelian group and a prime number , the -primary part is the torsion of . ${\displaystyle M}$ ${\displaystyle p}$ ${\displaystyle M[p^{\infty }]}$ ${\displaystyle p}$ ${\displaystyle M}$

Tate module

If an Abelian group and a prime number , then the projective limit is ${\displaystyle M}$ ${\displaystyle \ell }$

${\displaystyle T_{\ell }(M)=\lim _{n}M[\ell ^{n}]}$

(the transition maps are given by multiplying by ) a module (whole - adic numbers ), which is called the - adic Tate module of (after John Tate ). By transitioning to ${\displaystyle \ell }$ ${\displaystyle \mathbb {Z} _{\ell }}$ ${\displaystyle \ell }$ ${\displaystyle \ell }$ ${\displaystyle M}$

${\displaystyle V_{\ell }(M)=T_{\ell }(M)\otimes _{\mathbb {Z} _{\ell }}\mathbb {Q} _{\ell }}$

one obtains a vector space over a field of characteristic 0; this is particularly advantageous for considerations of representation theory .

The most important example of this construction is the Tate module for an elliptic curve over a non- algebraically closed body, whose characteristic is not . The Tate module is isomorphic as a module and carries a natural operation of the Galois group . In the case of the multiplicative group , the associated Tate module is of rank 1. It is denoted by, the operation of the Galois group is carried out by the cyclotomic character . ${\displaystyle E}$ ${\displaystyle \ell }$ ${\displaystyle T_{\ell }(E)}$ ${\displaystyle \mathbb {Z} _{\ell }}$ ${\displaystyle \mathbb {Z} _{\ell }^{2}}$ ${\displaystyle \mathbb {G} _{\mathrm {m} }}$ ${\displaystyle \mathbb {Z} _{\ell }(1)}$

Generalizations

For modules, the torsional sub-module of a module is the same . The functors Tor can thus be seen as a generalization of the concept of the torsion sub-module. ${\displaystyle \mathbb {Z} }$ ${\displaystyle M}$ ${\displaystyle \operatorname {Tor} _{1}(\mathbb {Q} /\mathbb {Z} ,M)}$

literature

• David Eisenbud , Commutative algebra with a view toward algebraic geometry . Springer-Verlag, New York 1995. ISBN 0-387-94269-6 .
• Qing Liu, Algebraic Geometry and Arithmetic Curves . Oxford University Press, Oxford 2006. ISBN 0-19-920249-4 .

Individual evidence

1. ^ Nicolas Bourbaki : Algèbre (=  Éléments de mathématique ). Springer , Berlin 2007, ISBN 3-540-33849-7 , chap. 2 , p. 172 .
2. ^ David Eisenbud, Commutative algebra with a view toward algebraic geometry . Springer-Verlag, New York 1995. ISBN 0-387-94269-6 .
3. ^ Qing Liu, Algebraic Geometry and Arithmetic Curves . Oxford University Press, Oxford 2006. ISBN 0-19-920249-4 .