nLab
Tor

Idea

In the context of homological algebra, the $\mathrm{Tor}$-functor is the left derived functor of the tensor product.

Together with the Ext-functor it constitutes one of the central operations of interest in homological algebra.

Details

Given a ring $R$ the bifunctor ${\otimes }_R:{\mathrm{Mod}}_R{\times }_R\mathrm{Mod}\to \mathrm{Ab}$ is right exact. Its left derived functors $\mathrm{Tor}(-,B)$ and $\mathrm{Tor}(A,-)$ with respect to one argument with fixed another, if they exist, are parts of a bifunctor $\mathrm{Tor}:{\mathrm{Mod}}_R{\times }_R\mathrm{Mod}\to \mathrm{Ab}$. If there are sufficiently many projective objects in both ${\mathrm{Mod}}_R$ and ${}_R\mathrm{Mod}$, both are equal to balancing $\mathrm{Tor}$, which is the bifunctor obtained by taking the homology of total complex of a bicomplex of $\mathrm{Hom}$-s in which $A\otimes B$ is resolved by taking a projective resolution in each argument.

References

• Charles Weibel, An introduction to homological algebra, Cambridge Studies in Adv. Math. 38, CUP 1994

• H. Cartan, S. Eilenberg, Homological algebra, Princeton Univ. Press 1956.

• M. Kashiwara and P. Schapira, Categories and Sheaves, Springer (2000)

• S. I . Gelfand, Yu. I. Manin, Methods of homological algebra