Tor functor

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the Tor functors are the <a href="/facts/Derived_functor/YdjtF5ds">derived functors</a> of the <a href="/facts/Tensor_product_of_modules/K3oaFIS4">tensor product of modules</a> over a <a href="/facts/Ring_(mathematics)/JnCUXz63">ring</a>. Along with the <a href="/facts/Ext_functor/ktggDk7I">Ext functor</a>, Tor is one of the central concepts of <a href="/facts/Homological_algebra/Jfm8w52l">homological algebra</a>, in which ideas from <a href="/facts/Algebraic_topology/2wA8yrjT">algebraic topology</a> are used to construct invariants of algebraic structures. The <a href="/facts/Group_cohomology/ncLTIHj4">homology of groups</a>, <a href="/facts/Lie_algebra_homology/CHGMUaaU">Lie algebras</a>, and <a href="/facts/Hochschild_homology/0phwDIyX">associative algebras</a> can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor1 and the <a href="/facts/Torsion_subgroup/dIwLuG5j">torsion subgroup</a> of an <a href="/facts/Abelian_group/FfWwmx4R">abelian group</a>.
In the special case of abelian groups, Tor was introduced by <a href="/facts/Eduard_%25C4%258Cech/QvpLd8WY">Eduard Čech</a> (1935) and named by <a href="/facts/Samuel_Eilenberg/vs6aksKX">Samuel Eilenberg</a> around 1950. It was first applied to the <a href="/facts/K%25C3%25BCnneth_theorem/R4t0LuyI">Künneth theorem</a> and <a href="/facts/Universal_coefficient_theorem/N8ALJkMd">universal coefficient theorem</a> in topology. For modules over any ring, Tor was defined by <a href="/facts/Henri_Cartan/IGrXDS4a">Henri Cartan</a> and Eilenberg in their 1956 book Homological Algebra.

Tor functor open-in-new

Tor functor