Tor

The left derived functors of tensor product; measures failure of tensor to be left exact (flatness).

Tor

Let $R$ be a ring .

To form a tensor product over $R$ in full generality, one typically takes a right $R$ -module $M$ and a left $R$ -module $N$ , producing an abelian group

M \otimes_R N,

see tensor product . (If $R$ is commutative, one may treat both as left $R$ -modules.)

Definition (via a projective resolution)

Choose a projective resolution $P_\bullet \to M$ by projective right $R$ -modules:

\cdots \to P_2 \to P_1 \to P_0 \to M \to 0.

Tensor with $N$ to obtain a chain complex $P_\bullet \otimes_R N$ , and define

\mathrm{Tor}^R_n(M,N)\;:=\; H_n(P_\bullet \otimes_R N),

where $H_n(-)$ denotes homology .

This is well-defined up to canonical isomorphism and is functorial in both variables.

Basic properties

$\mathrm{Tor}^R_0(M,N)\cong M\otimes_R N$ .
Because tensor is right exact , the derived functors $\mathrm{Tor}^R_n$ measure precisely the failure of tensor to be left exact. In particular, a module $N$ is flat iff $\mathrm{Tor}^R_1(-,N)=0$ (equivalently, iff tensoring with $N$ preserves injections).
Any short exact sequence gives rise to a long exact sequence in Tor , a special case of the long exact sequence for derived functors .

Examples

Example 1: Vector spaces over a field

If $k$ is a field and $V,W$ are $k$ -vector spaces, every $k$ -module is free (hence projective), so

\mathrm{Tor}^k_n(V,W)=0 \quad (n>0), \qquad \mathrm{Tor}^k_0(V,W)=V\otimes_k W.

Example 2: $\mathrm{Tor}^{\mathbb Z}_1(\mathbb Z/n, A)\cong A[n]$

Use the standard projective resolution of $\mathbb Z/n$ :

0 \to \mathbb Z \xrightarrow{\times n} \mathbb Z \to \mathbb Z/n \to 0.

Tensor with $A$ to get

0 \to A \xrightarrow{\times n} A \to (\mathbb Z/n)\otimes_{\mathbb Z} A \to 0.

The homology at the left term is

\mathrm{Tor}^{\mathbb Z}_1(\mathbb Z/n, A)\;\cong\;\ker(A \xrightarrow{\times n} A)\;=\;\{a\in A : na=0\}=:A[n].

Also $(\mathbb Z/n)\otimes_{\mathbb Z}A\cong A/nA$ , so $\mathrm{Tor}^{\mathbb Z}_0(\mathbb Z/n,A)\cong A/nA$ .

Example 3: $\mathrm{Tor}^{\mathbb Z}_1(\mathbb Z/n,\mathbb Z/m)\cong \mathbb Z/\gcd(n,m)$

From Example 2 with $A=\mathbb Z/m$ , the $n$ -torsion subgroup has order $\gcd(n,m)$ and is cyclic, hence

\mathrm{Tor}^{\mathbb Z}_1(\mathbb Z/n,\mathbb Z/m)\cong \mathbb Z/\gcd(n,m).

(As in the Ext computations over $\mathbb Z$ , these cyclic modules have projective dimension $1$ , so $\mathrm{Tor}^{\mathbb Z}_i(\mathbb Z/n,-)=0$ for $i\ge 2$ .)