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.
For the Torsionsbegriff of commutative algebra is a (commutative) ring (with identity) and one - module .
- 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 .
- Equivalently, the torsion submodule can also be used as the core of homomorphism
- define if denotes the total quotient ring of .
- means torsion-free if the torsional submodule is zero.
- is a torsional modulus when the torsional submodule is equal . Sometimes you say briefly: "is torsion".
If an Abelian group ( i.e. module), the two definitions of torsion elements agree. One then speaks of torsion (sub) groups .
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.
- 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.
- 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.
- Another example of this fact is the infinite dihedral group
- ,
- in which the generators are torsion elements, but not for example .
- itself, or more generally a free module, is torsion-free. In particular, if there is a body , all modules are torsion-free.
- 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 .
- 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 .
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 .
- As the example of the factor group shows, the orders of the elements are generally not restricted ; the -primary subgroup already has this property.
- 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 :
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.
For a ring element is
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.
The module
is called torsion.
properties
- is naturally a module.
- The functor is left exact (as a representable functor even exchanged with any limits); more precisely: is
- is an exact sequence of modules, then
- exactly, as follows directly from the snake lemma.
- The torsional sub-module of is the union of the for all non-zero divisors .
- For ring elements is .
- For an Abelian group and a prime number , the -primary part is the torsion of .
Tate module
If an Abelian group and a prime number , then the projective limit is
(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
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 .
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.
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
- ^ Nicolas Bourbaki : Algèbre (= Éléments de mathématique ). Springer , Berlin 2007, ISBN 3-540-33849-7 , chap. 2 , p. 172 .
- ^ 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 .