In homological algebra, the Tor functors 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. 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.

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  • In homological algebra, the Tor functors 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. 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 LnT are defined. We set <math>\mathrm{Tor}_n^R(A,B)=(L_nT)(A)</math> i.e. , we take a projective resolution <math>\cdots\rightarrow P_3 \rightarrow P_2 \rightarrow P_1 \rightarrow A\rightarrow 0</math> then chop off the last term A and tensor it with B to get the complex <math>\cdots \rightarrow P_3\otimes B \rightarrow P_2\otimes B \rightarrow P_1\otimes B \rightarrow 0</math> and take the homology of this complex.
  • 在交換代數中，Tor 函子是張量積的導函子。此函子起初是為了表述代數拓撲中的 Künneth 定理與普遍係數定理而定義。
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  • In homological algebra, the Tor functors 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. 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.
  • 在交換代數中，Tor 函子是張量積的導函子。此函子起初是為了表述代數拓撲中的 Künneth 定理與普遍係數定理而定義。
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  • Tor functor
  • Tor函子
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