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 Tor
1 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
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
. The Tor groups are the abelian groups defined by
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
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 ...
:
For each integer ''i'', the group
is the
homology of this complex at position ''i''. It is zero for ''i'' negative. Moreover,
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
, 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
.
Alternatively, one can define Tor by fixing ''A'' and taking the left derived functors of the right exact functor
. 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'',
is an ''R''-module (using that
is an ''R''-module in this case). For a non-commutative ring ''R'',
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
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
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'',
where
is the ''u''-torsion subgroup of ''B''. This is the explanation for the name Tor. Taking ''R'' to be the ring
of integers, this calculation can be used to compute
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''
1, ..., ''x''''n''">'x''1, ..., ''x''''n''over a field ''k'', then
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 Tor
1.
*
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,
for all ''i'' ≥ 2 if
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
*Flat base change: for a commutative flat ''R''-algebra ''T'', ''R''-modules ''A'' and ''B'', and an integer ''i'',
It follows that Tor commutes with
localization. That is, for a
multiplicatively closed set ''S'' in ''R'',
*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
where ''G'' is a group, ''M'' is a
representation of ''G'' over the integers, and