Categories, abelian categories and cohomologies.

Yoshifumi Tsuchimoto

Tensor products and Tor.

DEFINITION 09.1   Let $A$ be an associative unital (but not necessarily commutative) ring. Let $L$ be a right $A$ -module. Let $M$ be a left $A$ -module. For any ( $\mathbb{Z}$ -)module $N$ , an map

$$\varphi: L\times M \to N$$

is called an $A$ -balanced biadditive map if

1. $\varphi(x_1+x_2,y)=\varphi(x_1,y)+\varphi(x_2,y)$      $(\forall x_1,\forall x_2\in L, \forall y\in M)$ .
2. $\varphi(x,y_1+y_2)=\varphi(x,y_1)+\varphi(x,y_2)$      $(\forall x\in L, \forall y_1,\forall y_2\in M)$ .
3. $\varphi(x a, y)=\varphi(x, a y)$      $(\forall x\in L, \forall y\in M, \forall a\in A)$ .

PROPOSITION 09.2   Let $A$ be an associative unital (but not necessarily commutative) ring. Then for any right $A$ -module $L$ and for any left $A$ -module $M$ , there exists a ( $\mathbb{Z}$ -)module $X_{L,M}$ together with a $A$ -balanced map

$$\varphi_0: L\times M \to X_{L,M}$$

which is universal amoung $A$ -balanced maps.

DEFINITION 09.3   We employ the assumption of the proposition above. By a standard argument on universal objects, we see that such object is unique up to a unique isomorphism. We call it the tensor product of $L$ and $M$ and denote it by

$$L\otimes_A M.$$

LEMMA 09.4  

1. $A\otimes_A M \cong M$ .
2. $(L_1\oplus L_2) \otimes_A M \cong (L_1 \otimes M) \oplus (L_2 \otimes_A M ).$
3. For any $M$ , $L\mapsto L \otimes_A M$ is a right exact functor.

DEFINITION 09.5   For any left $A$ -module $M$ , the left derived functor $L_j F(M)$ of $F_M=\bullet \otimes_A M$ is called the Tor functor and denoted by $\operatorname{Tor}^A_j(\bullet,M)$ .

By definition, $\operatorname{Tor}^A_j(L,M)$ may be computed by using projective resolutions of $L$ .

DEFINITION 09.6   For any group $G$ , the derived functor of a functor

$$F_G:(G-modules ) \to (modules )$$

defined by

$$M\mapsto M_G=M/(\mathbb{Z}-$$span$$\{g.m-m; g\in G, M\in M\})$$

is called the homology of $G$ with coefficients in $M$ . We denote the homology group $L_j F_G(M)$ by $H_j(G;M)$ .

LEMMA 09.7  

$$H_j(G;M)\cong \operatorname{Tor}^{\mathbb{Z}[G]}_j(\mathbb{Z}, M)$$