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Tor

The Tor functors Tor_n^R(M,N) are the left derived functors of the tensor product over a ring R. If P_-->M is a projective resolution, then

Tor_n^R(M,N)=H_n(P_- tensor _RN).

The zeroth group is Tor_0^R(M,N)=M tensor _RN, while the higher groups measure the failure of tensor product to preserve exact sequences. For example,

Tor_1^Z(Z/mZ,Z/nZ)=Z/gcd(m,n)Z.

See also

Chain Complex, Exact Sequence, Projective Module, Tensor Product

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References

Weibel, C. A. An Introduction to Homological Algebra. Cambridge, England: Cambridge University Press, 1994.

Cite this as:

Weisstein, Eric W. "Tor." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Tor.html

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