Editing Tor functor (section)

{{Short description|Construction in homological algebra}}
In [[mathematics]], the '''Tor functors''' are the [[derived functor]]s of the [[tensor product of modules]] over a [[ring (mathematics)|ring]]. Along with the [[Ext functor]], Tor is one of the central concepts of [[homological algebra]], in which ideas from [[algebraic topology]] are used to construct invariants of algebraic structures. The [[group cohomology#Group homology|homology of groups]], [[Lie algebra homology|Lie algebra]]s, and [[Hochschild homology|associative algebras]] can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor<sub>1</sub> and the [[torsion subgroup]] of an [[abelian group]].

In the special case of abelian groups, Tor was introduced by [[Eduard Čech]] (1935) and named by [[Samuel Eilenberg]] around 1950.<ref>Weibel (1999).</ref> It was first applied to the [[Künneth theorem]] and [[universal coefficient theorem]] in topology. For modules over any ring, Tor was defined by [[Henri Cartan]] and Eilenberg in their 1956 book ''Homological Algebra''.<ref>Cartan & Eilenberg (1956), section VI.1.</ref>