HOME

TheInfoList



OR:

In
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, the Tor functors are the
derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...
s of the
tensor product of modules In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor produ ...
over a ring. Along with the
Ext functor In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic stru ...
, Tor is one of the central concepts of
homological algebra Homological algebra is the branch of mathematics that studies homology (mathematics), homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precurs ...
, in which ideas from
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
are used to construct invariants of algebraic structures. The homology of groups,
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
s, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor1 and the
torsion subgroup In the theory of abelian groups, the torsion subgroup ''AT'' of an abelian group ''A'' is the subgroup of ''A'' consisting of all elements that have finite order (the torsion elements of ''A''). An abelian group ''A'' is called a torsion group ...
of an
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
. In the special case of abelian groups, Tor was introduced by Eduard Čech (1935) and named by
Samuel Eilenberg Samuel Eilenberg (September 30, 1913 – January 30, 1998) was a Polish-American mathematician who co-founded category theory (with Saunders Mac Lane) and homological algebra. Early life and education He was born in Warsaw, Kingdom of Poland to ...
around 1950. It was first applied to the Künneth theorem and
universal coefficient theorem In algebraic topology, universal coefficient theorems establish relationships between homology groups (or cohomology groups) with different coefficients. For instance, for every topological space , its ''integral homology groups'': :H_i(X,\Z) ...
in topology. For modules over any ring, Tor was defined by
Henri Cartan Henri Paul Cartan (; 8 July 1904 – 13 August 2008) was a French mathematician who made substantial contributions to algebraic topology. He was the son of the mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of c ...
and Eilenberg in their 1956 book ''Homological Algebra''.


Definition

Let ''R'' be a ring. Write ''R''-Mod for the
category Category, plural categories, may refer to: General uses *Classification, the general act of allocating things to classes/categories Philosophy * Category of being * ''Categories'' (Aristotle) * Category (Kant) * Categories (Peirce) * Category ( ...
of left ''R''-modules and Mod-''R'' for the category of right ''R''-modules. (If ''R'' is
commutative In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Perhaps most familiar as a pr ...
, the two categories can be identified.) For a fixed left ''R''-module ''B'', let T(A) = A\otimes_R B for ''A'' in Mod-''R''. This is a
right exact functor In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much o ...
from Mod-''R'' to the
category of abelian groups In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every small abelian category can be embedded in Ab. Properties The zero object o ...
Ab, and so it has left
derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...
s L_i T. The Tor groups are the abelian groups defined by \operatorname_i^R(A,B) = (L_iT)(A), for an
integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...
''i''. By definition, this means: take any projective resolution \cdots\to P_2 \to P_1 \to P_0 \to A\to 0, and remove ''A'', and form the
chain complex In mathematics, a chain complex is an algebraic structure that consists of a sequence of abelian groups (or modules) and a sequence of homomorphisms between consecutive groups such that the image of each homomorphism is contained in the kernel o ...
: \cdots \to P_2\otimes_R B \to P_1\otimes_R B \to P_0\otimes_R B \to 0 For each integer ''i'', the group \operatorname_i^R(A,B) is the homology of this complex at position ''i''. It is zero for ''i'' negative. Moreover, \operatorname_0^R(A,B) is the
cokernel The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of . Cokernels are dual to the kernels of category theory, hence the nam ...
of the map P_1\otimes_R B \to P_0\otimes_R B, which is
isomorphic In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
to A \otimes_R B. Alternatively, one can define Tor by fixing ''A'' and taking the left derived functors of the right exact functor G(B)=A\otimes_RB. That is, tensor ''A'' with a projective resolution of ''B'' and take homology. Cartan and Eilenberg showed that these constructions are independent of the choice of projective resolution, and that both constructions yield the same Tor groups. Moreover, for a fixed ring ''R'', Tor is a functor in each variable (from ''R''-modules to abelian groups). For a commutative ring ''R'' and ''R''-modules ''A'' and ''B'', \operatorname^R_i(A,B) is an ''R''-module (using that A\otimes_RB is an ''R''-module in this case). For a non-commutative ring ''R'', \operatorname^R_i(A,B) is only an abelian group, in general. If ''R'' is an
algebra over a ring In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition ...
''S'' (which means in particular that ''S'' is commutative), then \operatorname^R_i(A,B) is at least an ''S''-module.


Properties

Here are some of the basic properties and computations of Tor groups. *Tor(''A'', ''B'') ≅ ''A'' ⊗''R'' ''B'' for any right ''R''-module ''A'' and left ''R''-module ''B''. *Tor(''A'', ''B'') = 0 for all ''i'' > 0 if either ''A'' or ''B'' is flat (for example, free) as an ''R''-module. In fact, one can compute Tor using a flat resolution of either ''A'' or ''B''; this is more general than a projective (or free) resolution. *There are converses to the previous statement: **If Tor(''A'', ''B'') = 0 for all ''B'', then ''A'' is flat (and hence Tor(''A'', ''B'') = 0 for all ''i'' > 0). **If Tor(''A'', ''B'') = 0 for all ''A'', then ''B'' is flat (and hence Tor(''A'', ''B'') = 0 for all ''i'' > 0). *By the general properties of
derived functor In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in vari ...
s, every
short exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ...
0 → ''K'' → ''L'' → ''M'' → 0 of right ''R''-modules induces a long exact sequence of the form \cdots \to \operatorname_2^R(M,B) \to \operatorname_1^R(K,B) \to \operatorname_1^R(L,B) \to \operatorname_1^R (M,B) \to K\otimes_R B\to L\otimes_R B\to M\otimes_R B\to 0, for any left ''R''-module ''B''. The analogous exact sequence also holds for Tor with respect to the second variable. *Symmetry: for a commutative ring ''R'', there is a
natural isomorphism In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natura ...
Tor(''A'', ''B'') ≅ Tor(''B'', ''A''). (For ''R'' commutative, there is no need to distinguish between left and right ''R''-modules.) *If ''R'' is a commutative ring and ''u'' in ''R'' is not a
zero divisor In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right ze ...
, then for any ''R''-module ''B'', \operatorname^R_i(R/(u),B)\cong\begin B/uB & i=0\\ B & i=1\\ 0 &\text\end where B = \ is the ''u''-torsion subgroup of ''B''. This is the explanation for the name Tor. Taking ''R'' to be the ring \Z of integers, this calculation can be used to compute \operatorname^_1(A,B) for any finitely generated abelian group ''A''. *Generalizing the previous example, one can compute Tor groups that involve the quotient of a commutative ring by any regular sequence, using the
Koszul complex In mathematics, the Koszul complex was first introduced to define a cohomology theory for Lie algebras, by Jean-Louis Koszul (see Lie algebra cohomology). It turned out to be a useful general construction in homological algebra. As a tool, its ho ...
. For example, if ''R'' is the
polynomial ring In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally also called variables) with coefficients in another ring, ...
''k'' 'x''1, ..., ''x''''n''over a field ''k'', then \operatorname_*^R(k,k) is the
exterior algebra In mathematics, the exterior algebra or Grassmann algebra of a vector space V is an associative algebra that contains V, which has a product, called exterior product or wedge product and denoted with \wedge, such that v\wedge v=0 for every vector ...
over ''k'' on ''n'' generators in Tor1. * \operatorname^_i(A,B)=0 for all ''i'' ≥ 2. The reason: every
abelian group In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commu ...
''A'' has a free resolution of length 1, since every subgroup of a
free abelian group In mathematics, a free abelian group is an abelian group with a Free module, basis. Being an abelian group means that it is a Set (mathematics), set with an addition operation (mathematics), operation that is associative, commutative, and inverti ...
is free abelian. * Generalizing the previous example, \operatorname^_i(A,B)=0 for all ''i'' ≥ 2 if R is a
principal ideal domain In mathematics, a principal ideal domain, or PID, is an integral domain (that is, a non-zero commutative ring without nonzero zero divisors) in which every ideal is principal (that is, is formed by the multiples of a single element). Some author ...
(PID). The reason: every module ''A'' over a PID has a free resolution of length 1, since every submodule of a
free module In mathematics, a free module is a module that has a ''basis'', that is, a generating set that is linearly independent. Every vector space is a free module, but, if the ring of the coefficients is not a division ring (not a field in the commu ...
over a PID is free. *For any ring ''R'', Tor preserves direct sums (possibly infinite) and
filtered colimit In category theory, filtered categories generalize the notion of directed set understood as a category (hence called a directed category; while some use directed category as a synonym for a filtered category). There is a dual notion of cofiltered c ...
s in each variable. For example, in the first variable, this says that \begin \operatorname_i^R \left (\bigoplus_ M_, N \right ) &\cong \bigoplus_ \operatorname_i^R(M_,N) \\ \operatorname_i^R \left (\varinjlim_ M_, N \right ) &\cong \varinjlim_ \operatorname_i^R(M_,N) \end *Flat base change: for a commutative flat ''R''-algebra ''T'', ''R''-modules ''A'' and ''B'', and an integer ''i'', \mathrm_i^R(A,B)\otimes_R T \cong \mathrm_i^T(A\otimes_R T,B\otimes_R T). It follows that Tor commutes with localization. That is, for a multiplicatively closed set ''S'' in ''R'', S^ \operatorname_i^R(A, B) \cong \operatorname_i^ \left (S^ A, S^ B \right ). *For a commutative ring ''R'' and commutative ''R''-algebras ''A'' and ''B'', Tor(''A'',''B'') has the structure of a
graded-commutative In Abstract algebra, algebra, a graded-commutative ring (also called a skew-commutative ring) is a graded ring that is commutative in the graded sense; that is, homogeneous elements ''x'', ''y'' satisfy :xy = (-1)^ yx, where , ''x'', and , ''y'', ...
algebra over ''R''. Moreover, elements of odd degree in the Tor algebra have square zero, and there are divided power operations on the elements of positive even degree.


Important special cases

* Group homology is defined by H_*(G,M)=\operatorname^_*(\Z, M), where ''G'' is a group, ''M'' is a representation of ''G'' over the integers, and \Z /math> is the
group ring In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the gi ...
of ''G''. *For an
algebra Algebra is a branch of mathematics that deals with abstract systems, known as algebraic structures, and the manipulation of expressions within those systems. It is a generalization of arithmetic that introduces variables and algebraic ope ...
''A'' over a field ''k'' and an ''A''-
bimodule In abstract algebra, a bimodule is an abelian group that is both a left and a right module, such that the left and right multiplications are compatible. Besides appearing naturally in many parts of mathematics, bimodules play a clarifying role, i ...
''M'',
Hochschild homology In mathematics, Hochschild homology (and cohomology) is a homology theory for associative algebras over rings. There is also a theory for Hochschild homology of certain functors. Hochschild cohomology was introduced by for algebras over a fiel ...
is defined by HH_*(A,M)=\operatorname_*^(A, M). * Lie algebra homology is defined by H_*(\mathfrak g,M)=\operatorname_*^(R,M), where \mathfrak g is a
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi ident ...
over a commutative ring ''R'', ''M'' is a \mathfrak g-module, and U\mathfrak g is the
universal enveloping algebra In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representa ...
. *For a commutative ring ''R'' with a homomorphism onto a field ''k'', \operatorname_*^R(k,k) is a graded-commutative
Hopf algebra In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously a ( unital associative) algebra and a (counital coassociative) coalgebra, with these structures' compatibility making it a bialgebra, and that moreover ...
over ''k''. (If ''R'' is a Noetherian local ring with residue field ''k'', then the dual Hopf algebra to \operatorname_*^R(k,k) is Ext(''k'',''k'').) As an algebra, \operatorname_*^R(k,k) is the free graded-commutative divided power algebra on a graded vector space π*(''R''). When ''k'' has characteristic zero, π*(''R'') can be identified with the André-Quillen homology ''D''*(''k''/''R'',''k'').Quillen (1970), section 7.


See also

*
Flat morphism In mathematics, in particular in algebraic geometry, a flat morphism ''f'' from a scheme (mathematics), scheme ''X'' to a scheme ''Y'' is a morphism such that the induced map on every Stalk (sheaf), stalk is a flat map of rings, i.e., :f_P\colon \ ...
* Serre's intersection formula * Derived tensor product * Eilenberg–Moore spectral sequence


Notes


References

* * * * * * * *


External links

*{{Citation , author1=The Stacks Project Authors , title=The Stacks Project , url=http://stacks.math.columbia.edu/ Homological algebra Binary operations Functors