Torsion-free Cover
   HOME

TheInfoList



OR:

In
algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathem ...
, a torsion-free module is a
module Module, modular and modularity may refer to the concept of modularity. They may also refer to: Computing and engineering * Modular design, the engineering discipline of designing complex devices using separately designed sub-components * Mo ...
over a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
such that zero is the only element
annihilated The eighth season of the television series, '' Law & Order: Special Victims Unit'' premiered September 19, 2006 and ended May 22, 2007 on NBC. The series remained in its 10pm/9c Tuesday timeslot. With the introduction of a new partner for Detecti ...
by a regular element (non zero-divisor) of the ring. In other words, a module is ''torsion free'' if its torsion submodule is reduced to its zero element. In
integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural s ...
s the regular elements of the ring are its nonzero elements, so in this case a torsion-free module is one such that zero is the only element annihilated by some non-zero element of the ring. Some authors work only over integral domains and use this condition as the definition of a torsion-free module, but this does not work well over more general rings, for if the ring contains zero-divisors then the only module satisfying this condition is the
zero module In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure, which is also an abelian group. The afore ...
.


Examples of torsion-free modules

Over a commutative ring ''R'' with
total quotient ring In abstract algebra, the total quotient ring, or total ring of fractions, is a construction that generalizes the notion of the field of fractions of an integral domain to commutative rings ''R'' that may have zero divisors. The construction embeds ...
''K'', a module ''M'' is torsion-free if and only if Tor1(''K''/''R'',''M'') vanishes. Therefore
flat modules In algebra, a flat module over a ring ''R'' is an ''R''-module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact se ...
, and in particular
free Free may refer to: Concept * Freedom, having the ability to do something, without having to obey anyone/anything * Freethought, a position that beliefs should be formed only on the basis of logic, reason, and empiricism * Emancipate, to procur ...
and projective modules, are torsion-free, but the converse need not be true. An example of a torsion-free module that is not flat is the
ideal Ideal may refer to: Philosophy * Ideal (ethics), values that one actively pursues as goals * Platonic ideal, a philosophical idea of trueness of form, associated with Plato Mathematics * Ideal (ring theory), special subsets of a ring considere ...
(''x'', ''y'') of the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variable ...
''k'' 'x'', ''y''over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
''k'', interpreted as a module over ''k'' 'x'', ''y'' Any torsionless module is a torsion-free module, but the converse is not true, as Q is a torsion-free Z-module which is ''not'' torsionless.


Structure of torsion-free modules

Over a Noetherian integral domain, torsion-free modules are the modules whose only
associated prime In abstract algebra, an associated prime of a module ''M'' over a ring ''R'' is a type of prime ideal of ''R'' that arises as an annihilator of a (prime) submodule of ''M''. The set of associated primes is usually denoted by \operatorname_R(M ...
is zero. More generally, over a Noetherian commutative ring the torsion-free modules are those modules all of whose associated primes are contained in the associated primes of the ring. Over a Noetherian
integrally closed domain In commutative algebra, an integrally closed domain ''A'' is an integral domain whose integral closure in its field of fractions is ''A'' itself. Spelled out, this means that if ''x'' is an element of the field of fractions of ''A'' which is a root ...
, any finitely-generated torsion-free module has a free submodule such that the
quotient In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
by it is isomorphic to an ideal of the ring. Over a
Dedekind domain In abstract algebra, a Dedekind domain or Dedekind ring, named after Richard Dedekind, is an integral domain in which every nonzero proper ideal factors into a product of prime ideals. It can be shown that such a factorization is then necessari ...
, a finitely-generated module is torsion-free if and only if it is projective, but is in general not free. Any such module is isomorphic to the sum of a finitely-generated free module and an ideal, and the class of the ideal is uniquely determined by the module. Over a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain in which every ideal is principal, i.e., can be generated by a single element. More generally, a principal ideal ring is a nonzero commutative ring whose ideals are princip ...
, finitely-generated modules are torsion-free if and only if they are free.


Torsion-free covers

Over an integral domain, every module ''M'' has a torsion-free cover from a torsion-free module ''F'' onto ''M'', with the properties that any other torsion-free module mapping onto ''M'' factors through ''F'', and any
endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a ...
of ''F'' over ''M'' is an automorphism of ''F''. Such a torsion-free cover of ''M'' is unique up to isomorphism. Torsion-free covers are closely related to flat covers.


Torsion-free quasicoherent sheaves

A
quasicoherent sheaf In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refe ...
''F'' over a
scheme A scheme is a systematic plan for the implementation of a certain idea. Scheme or schemer may refer to: Arts and entertainment * ''The Scheme'' (TV series), a BBC Scotland documentary series * The Scheme (band), an English pop band * ''The Schem ...
''X'' is a
sheaf Sheaf may refer to: * Sheaf (agriculture), a bundle of harvested cereal stems * Sheaf (mathematics), a mathematical tool * Sheaf toss, a Scottish sport * River Sheaf, a tributary of River Don in England * ''The Sheaf'', a student-run newspaper s ...
of \mathcal_X-modules such that for any open affine subscheme ''U'' = Spec(''R'') the
restriction Restriction, restrict or restrictor may refer to: Science and technology * restrict, a keyword in the C programming language used in pointer declarations * Restriction enzyme, a type of enzyme that cleaves genetic material Mathematics and log ...
''F'', U is associated to some module ''M'' over ''R''. The sheaf ''F'' is said to be torsion-free if all those modules ''M'' are torsion-free over their respective rings. Alternatively, ''F'' is torsion-free if and only if it has no local torsion sections..


See also

*
Torsion (algebra) In mathematics, specifically in ring theory, a torsion element is an element of a module that yields zero when multiplied by some non-zero-divisor of the ring. The torsion submodule of a module is the submodule formed by the torsion elements. ...
*
torsion-free abelian group In mathematics, specifically in abstract algebra, a torsion-free abelian group is an abelian group which has no non-trivial torsion elements; that is, a group in which the group operation is commutative and the identity element is the only ele ...
* torsion-free abelian group of rank 1; the classification theory exists for this class.


References

* * *{{Citation , author1=The Stacks Project Authors , title=The Stacks Project , url=http://stacks.math.columbia.edu/ Ring theory