Tor functor

Let R be a ring. Write R-Mod for the category of left R-modules and Mod-R for the category of right R-modules. (If R is commutative, the two categories can be identified.) For a fixed left R-module B, let ${\displaystyle T(A)=A\otimes _{R}B}$ for A in Mod-R. This is a right exact functor from Mod-R to the category of abelian groups Ab, and so it has left derived functors ${\displaystyle L_{i}T}$. The Tor groups are the abelian groups defined by

$${\displaystyle \operatorname {Tor} _{i}^{R}(A,B)=(L_{i}T)(A),}$$

for an integer i. By definition, this means: take any projective resolution

$${\displaystyle \cdots \to P_{2}\to P_{1}\to P_{0}\to A\to 0,}$$

and remove A, and form the chain complex:

$${\displaystyle \cdots \to P_{2}\otimes _{R}B\to P_{1}\otimes _{R}B\to P_{0}\otimes _{R}B\to 0}$$

For each integer i, the group ${\displaystyle \operatorname {Tor} _{i}^{R}(A,B)}$ is the homology of this complex at position i. It is zero for i negative. Moreover, ${\displaystyle \operatorname {Tor} _{0}^{R}(A,B)}$ is the cokernel of the map ${\displaystyle P_{1}\otimes _{R}B\to P_{0}\otimes _{R}B}$, which is isomorphic to ${\displaystyle A\otimes _{R}B}$.

Alternatively, one can define Tor by fixing A and taking the left derived functors of the right exact functor G(B) = A ⊗_R B. That is, tensor A with a projective resolution of B and take homology. Cartan and Eilenberg showed that these constructions are independent of the choice of projective resolution, and that both constructions yield the same Tor groups.[3] Moreover, for a fixed ring R, Tor is a functor in each variable (from R-modules to abelian groups).

For a commutative ring R and R-modules A and B, Tor^R
_i(A, B) is an R-module (using that A ⊗_R B is an R-module in this case). For a non-commutative ring R, Tor^R
_i(A, B) is only an abelian group, in general. If R is an algebra over a ring S (which means in particular that S is commutative), then Tor^R
_i(A, B) is at least an S-module.

Here are some of the basic properties and computations of Tor groups.[4]

• Tor^R
  ₀(A, B) ≅ A ⊗_R B for any right R-module A and left R-module B.
• Tor^R
  _i(A, B) = 0 for all i > 0 if either A or B is flat (for example, free) as an R-module. In fact, one can compute Tor using a flat resolution of either A or B; this is more general than a projective (or free) resolution.[5]
• There are converses to the previous statement:
  • If Tor^R
    ₁(A, B) = 0 for all B, then A is flat (and hence Tor^R
    _i(A, B) = 0 for all i > 0).
  • If Tor^R
    ₁(A, B) = 0 for all A, then B is flat (and hence Tor^R
    _i(A, B) = 0 for all i > 0).
• By the general properties of derived functors, every short exact sequence 0 → K → L → M → 0 of right R-modules induces a long exact sequence of the form[6]
  $${\displaystyle \cdots \to \operatorname {Tor} _{2}^{R}(M,B)\to \operatorname {Tor} _{1}^{R}(K,B)\to \operatorname {Tor} _{1}^{R}(L,B)\to \operatorname {Tor} _{1}^{R}(M,B)\to K\otimes _{R}B\to L\otimes _{R}B\to M\otimes _{R}B\to 0,}$$

  
  for any left R-module B. The analogous exact sequence also holds for Tor with respect to the second variable.
• Symmetry: for a commutative ring R, there is a natural isomorphism Tor^R
  _i(A, B) ≅ Tor^R
  _i(B, A).[7] (For R commutative, there is no need to distinguish between left and right R-modules.)
• If R is a commutative ring and u in R is not a zero divisor, then for any R-module B,
  $${\displaystyle \operatorname {Tor} _{i}^{R}(R/(u),B)\cong {\begin{cases}B/uB&i=0\\B[u]&i=1\\0&{\text{otherwise}}\end{cases}}}$$

  
  where
  $${\displaystyle B[u]=\{x\in B:ux=0\}}$$

  
  is the u-torsion subgroup of B. This is the explanation for the name Tor. Taking R to be the ring ${\displaystyle \mathbb {Z} }$ of integers, this calculation can be used to compute ${\displaystyle \operatorname {Tor} _{1}^{\mathbb {Z} }(A,B)}$ for any finitely generated abelian group A.
• Generalizing the previous example, one can compute Tor groups that involve the quotient of a commutative ring by any regular sequence, using the Koszul complex.[8] For example, if R is the polynomial ring k[x₁, ..., x_n] over a field k, then ${\displaystyle \operatorname {Tor} _{*}^{R}(k,k)}$ is the exterior algebra over k on n generators in Tor₁.
• ${\displaystyle \operatorname {Tor} _{i}^{\mathbb {Z} }(A,B)=0}$ for all i ≥ 2. The reason: every abelian group A has a free resolution of length 1, since every subgroup of a free abelian group is free abelian.
• For any ring R, Tor preserves direct sums (possibly infinite) and filtered colimits in each variable.[9] For example, in the first variable, this says that
  $${\displaystyle {\begin{aligned}\operatorname {Tor} _{i}^{R}\left(\bigoplus _{\alpha }M_{\alpha },N\right)&\cong \bigoplus _{\alpha }\operatorname {Tor} _{i}^{R}(M_{\alpha },N)\\\operatorname {Tor} _{i}^{R}\left(\varinjlim _{\alpha }M_{\alpha },N\right)&\cong \varinjlim _{\alpha }\operatorname {Tor} _{i}^{R}(M_{\alpha },N)\end{aligned}}}$$

  
• Flat base change: for a commutative flat R-algebra T, R-modules A and B, and an integer i,[10]
  $${\displaystyle \mathrm {Tor} _{i}^{R}(A,B)\otimes _{R}T\cong \mathrm {Tor} _{i}^{T}(A\otimes _{R}T,B\otimes _{R}T).}$$

  
  It follows that Tor commutes with localization. That is, for a multiplicatively closed set S in R,
  $${\displaystyle S^{-1}\operatorname {Tor} _{i}^{R}(A,B)\cong \operatorname {Tor} _{i}^{S^{-1}R}\left(S^{-1}A,S^{-1}B\right).}$$

  
• For a commutative ring R and commutative R-algebras A and B, Tor^R
  _*(A,B) has the structure of a graded-commutative algebra over R. Moreover, elements of odd degree in the Tor algebra have square zero, and there are divided power operations on the elements of positive even degree.[11]

• Group homology is defined by ${\displaystyle H_{*}(G,M)=\operatorname {Tor} _{*}^{\mathbb {Z} [G]}(\mathbb {Z} ,M),}$ where G is a group, M is a representation of G over the integers, and ${\displaystyle \mathbb {Z} [G]}$ is the group ring of G.
• For an algebra A over a field k and an A-bimodule M, Hochschild homology is defined by
  $${\displaystyle HH_{*}(A,M)=\operatorname {Tor} _{*}^{A\otimes _{k}A^{\text{op}}}(A,M).}$$

  
• Lie algebra homology is defined by ${\displaystyle H_{*}({\mathfrak {g}},M)=\operatorname {Tor} _{*}^{U{\mathfrak {g}}}(R,M)}$, where ${\displaystyle {\mathfrak {g}}}$ is a Lie algebra over a commutative ring R, M is a ${\displaystyle {\mathfrak {g}}}$-module, and ${\displaystyle U{\mathfrak {g}}}$ is the universal enveloping algebra.
• For a commutative ring R with a homomorphism onto a field k, ${\displaystyle \operatorname {Tor} _{*}^{R}(k,k)}$ is a graded-commutative Hopf algebra over k.[12] (If R is a Noetherian local ring with residue field k, then the dual Hopf algebra to ${\displaystyle \operatorname {Tor} _{*}^{R}(k,k)}$ is Ext^*
  _R(k,k).) As an algebra, ${\displaystyle \operatorname {Tor} _{*}^{R}(k,k)}$ is the free graded-commutative divided power algebra on a graded vector space π_*(R).[13] When k has characteristic zero, π_*(R) can be identified with the André-Quillen homology D_*(k/R,k).[14]

• Flat morphism
• Serre's intersection formula
• Derived tensor product
• Eilenberg–Moore spectral sequence

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