Tor functor

In mathematics, the Tor functors are the derived functors of the tensor product of modules over a ring. Along with the Ext functor, Tor is one of the central concepts of homological algebra, in which ideas from algebraic topology are used to construct invariants of algebraic structures. The homology of groups, Lie algebras, and associative algebras can all be defined in terms of Tor. The name comes from a relation between the first Tor group Tor₁ and the torsion subgroup of an abelian group.

In the special case of abelian groups, Tor was introduced by Eduard Čech (1935) and named by Samuel Eilenberg around 1950. It was first applied to the Künneth theorem and universal coefficient theorem in topology. For modules over any ring, Tor was defined by Henri Cartan and Eilenberg in their 1956 book Homological Algebra.

Definition

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 T(A) = A ⊗_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 L_iT. The Tor groups are the abelian groups defined by

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

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

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

remove the term A, and form the chain complex:

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

For each integer i, Tor^R
_i(A,B) is the homology of this complex at position i. It is zero for i negative. For example, Tor^R
₀(A,B) is the cokernel of the map P₁⊗_RB → P₀⊗_RB, which is isomorphic to A⊗_RB.

Alternatively, one can define Tor by fixing A and taking the left derived functors of the right exact functor G(B)=A⊗_RB. 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.

For a commutative ring R and R-modules A and B, Tor^R
_i(A,B) is an R-module (using that A⊗_RB is an R-module in this case). For a noncommutative 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.

Properties

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

• Tor^R
  ₀(A,B) ≅ A⊗_RB 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.
• 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

${\begin{aligned}\cdots \to \mathrm {Tor} _{2}^{R}(M,B)\to \mathrm {Tor} _{1}^{R}(K,B)&\to \mathrm {Tor} _{1}^{R}(L,B)\to \mathrm {Tor} _{1}^{R}(M,B)\\[4pt]&\to K\otimes _{R}B\to L\otimes _{R}B\to M\otimes _{R}B\to 0,\end{aligned}}$

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). (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

$\operatorname {Tor} _{i}^{R}(R/(u),B)\cong {\begin{cases}B/uB&{\text{if }}i=0\\B[u]&{\text{if }}i=1\\0&{\text{otherwise,}}\end{cases}}$

for any R-module B. Here B[u] denotes the u-torsion subgroup of B, {x ∈ B: ux = 0}; this is the explanation for the name Tor. Taking R to be the ring Z of integers, this calculation can be used to compute Tor^Z
₁(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. For example, if R is the polynomial ring k[x₁,...,x_n] over a field k, then Tor^R
  _*(k,k) is the exterior algebra over k on n generators in Tor₁.
• Tor^Z
  _i(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. For example, in the first variable, this says that

$\mathrm {Tor} _{i}^{R}(\oplus _{\alpha }M_{\alpha },N)\cong \oplus _{\alpha }\mathrm {Tor} _{i}^{R}(M_{\alpha },N)$

and

$\mathrm {Tor} _{i}^{R}(\varinjlim _{\alpha }M_{\alpha },N)\cong \varinjlim _{\alpha }\mathrm {Tor} _{i}^{R}(M_{\alpha },N).$

• For a commutative ring R, Tor commutes with localization. That is, for a multiplicatively closed set S in R, R-modules A and B, and an integer i,

$S^{-1}\operatorname {Tor} _{i}^{R}(A,B)\cong \operatorname {Tor} _{i}^{S^{-1}R}\left(S^{-1}A,S^{-1}B\right).$

Important special cases

• Group homology is defined by $H_{*}(G,M)=\operatorname {Tor} _{*}^{\mathbb {Z} [G]}(\mathbb {Z} ,M),$ where G is a group, M is a representation of G over the integers, and $\mathbb {Z} [G]$ is the group ring of G.

• For an algebra A over a ring R and an A-bimodule M, Hochschild homology is defined by $HH_{*}(A,M)=\operatorname {Tor} _{*}^{A\otimes _{R}A^{\text{op}}}(A,M)$.

• Lie algebra homology is defined by $H_{*}({\mathfrak {g}},M)=\operatorname {Tor} _{*}^{U{\mathfrak {g}}}(R,M),$ where ${\mathfrak {g}}$ is a Lie algebra over a commutative ring R, M is a ${\mathfrak {g}}$-module, and $U{\mathfrak {g}}$ is the universal enveloping algebra.

• For a commutative ring R with a homomorphism onto a field k, $\operatorname {Tor} _{*}^{R}(k,k)$ has the structure of a graded-commutative Hopf algebra over k. When k has characteristic zero, this is the free graded-commutative algebra on the André-Quillen homology $D_{*}(k/R,k).$

See also

• flat morphism
• Serre's intersection formula
• derived tensor product
• Eilenberg–Moore spectral sequence

References

• Cartan, Henri; Eilenberg, Samuel (1999) [1956], Homological algebra, Princeton: Princeton University Press, ISBN 0-691-04991-2, MR 0077480
• Čech, Eduard (1935), "Les groupes de Betti d'un complexe infini", Fundamenta Mathematicae, 25: 33–44, doi:10.4064/fm-25-1-33-44, JFM 61.0609.02
• Quillen, Daniel (1970), "On the (co-)homology of commutative rings", Applications of categorical algebra, Proc. Symp. Pure Mat., 17, American Mathematical Society, pp. 65–87, MR 0257068
• Weibel, Charles A. (1994). An introduction to homological algebra. Cambridge Studies in Advanced Mathematics. 38. Cambridge University Press. ISBN 978-0-521-55987-4. MR 1269324. OCLC 36131259.
• Weibel, Charles (1999), "History of homological algebra", History of topology (PDF), Amsterdam: North-Holland, pp. 797–836, MR 1721123