In mathematics, the Tor functors of homological algebra are the derived functors of the tensor product functor. They were first defined in generality to express the Künneth theorem and universal coefficient theorem in algebraic topology. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ... Homological algebra is the branch of mathematics which studies the methods of homology and cohomology in a general setting. ... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ... In mathematics, the Künneth theorem of algebraic topology describes the singular homology of the cartesian product X × Y of two topological spaces, in terms of singular homology groups Hi(X, R) and Hj(X, R). ... In mathematics, the universal coefficient theorem in algebraic topology establishes the relationship in homology theory between the integral homology of a topological space X, and its homology with coefficients in any abelian group A. It shows that the integral homology groups Hi(X,Z) do in a certain, definite sense... Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ...

Specifically, suppose R is a ring, and denote by R-Mod the category of left R-modules and by Mod-R the category of right R-modules (if R is commutative, the two categories coincide). Pick a fixed module B in R-Mod. For A in Mod-R, set T(A) = A⊗_RB. Then T is a right exact functor from Mod-R to the category of abelian groups Ab (in case R is commutative, it is a right exact functor from Mod-R to Mod-R) and its left derived functors L_nT are defined. We set In mathematics, a ring is an algebraic structure in which addition and multiplication are defined and have properties listed below. ... In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, where instead of requiring the scalars to lie in a field, the scalars may lie in an arbitrary ring. ... In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation obeys the commutative law. ... In homological algebra, an exact functor is one which preserves exact sequences. ... In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ...

i.e., we take a projective resolution In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module (that is, a module with basis vectors). ...

then chop off the last term A and tensor it with B to get the complex

and take the homology of this complex. In mathematics (especially algebraic topology and abstract algebra), homology (in Greek homos = identical) is a certain general procedure to associate a sequence of abelian groups or modules to a given mathematical object (such as a topological space or a group). ...

Properties

• For every n ≥ 1, Tor_n^R is an additive functor from Mod-R × R-Mod to Ab. In case R is commutative, we have additive functors from Mod-R × Mod-R to Mod-R.

• As is true for every family of derived functors, every short exact sequence

induces a long exact sequence of the form A preadditive category is a category that is enriched over the monoidal category of abelian groups. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ...

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• If R is commutative and r in R is not a zero divisor then

from which the terminology Tor (that is, Torsion) comes: see torsion subgroup. In abstract algebra, a non-zero element a of a ring R is a left zero divisor if there exists a non-zero b such that ab = 0. ... In group theory, the torsion subgroup of an abelian group A is the subgroup of A consisting of all elements that have finite order. ...

• In the case of abelian groups (i.e. if R is the ring of integers Z), then Tor_n^Z(A,B) = 0 for all n ≥ 2. The reason: every abelian group A has a free resolution of length 2, since subgroups of free abelian groups are free abelian. So in this important special case, the higher Tor functors are invisible.

• The Tor functors commute with arbitrary direct sums: there is a natural isomorphism

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• A module M in Mod-R is flat if and only if Tor₁^R(M, -) = 0. In this case, we even have Tor_n^R(M, -) = 0 for all n. In fact, to compute Tor_n^R(A, B), one may use a flat resolution of A or B, instead of a projective resolution (note that a projective resolution is automatically a flat resolution, but the converse isn't true, so allowing flat resolutions is more flexible).