CATEGORIES, ABELIAN CATEGORIES AND COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

Examples of derived functors(2) Tensor products and Tor.

Definition 9.1. Let A be an associative unital (but not necessarily commutative) ring. Let L be a right A-module. Let M be a left Amodule. For any (ℤ-)module N, an map

φ : L × M → N

is called an A-balanced biadditive map if (1) φ(x_{1} + x_{2}, y) = φ(x_{1}, y) + φ(x_{2}, y) (∀x_{1}, ∀x_{2} ∈ L, ∀y ∈ M).

(2) φ(x, y_{1} + y_{2}) = φ(x, y_{1}) + φ(x, y_{2}) (∀x ∈ L, ∀y_{1}, ∀y_{2} ∈ M).

(3) φ(xa, y) = φ(x, ay) (∀x ∈ L, ∀y ∈ M, ∀a ∈ A).

Proposition 9.2. Let A be an associative unital (but not necessarily commutative) ring. Then for any right A-module L and for any left Amodule M, there exists a (ℤ-)module X_{L,M} together with a A-balanced map

φ_{0} : L × M → X_{L,M}

which is universal amoung A-balanced maps.

Definition 9.3. 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 ⊗_{A} M.

Lemma 9.4. (1) A ⊗_{A} M ≅ M.

(2) (L_{1} ⊕ L_{2}) ⊗_{A} M ≅ (L_{1} ⊗ M) ⊕ (L_{2} ⊗_{A} M).

(3) For any M, L ↦ L ⊗_{A} M is a right exact functor.

Definition 9.5. For any left A-module M, the left derived functor L_{j}F (M) of F_{M} = • ⊗_{A} M is called the Tor functor and denoted by Tor_{j}^{A}(•, M).

By definition, Tor_{j}^{A}(L, M) may be computed by using projective resolutions of L.

Definition 9.6. For any group G, the derived functor of a functor

F_{G} : (G − modules) → (modules)

defined by

M ↦ M_{G} = M/(ℤ − span{g.m − m; g ∈ G, M ∈ 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 9.7.

H_{j}(G; M) ≅ Tor_{j}^{ℤ[G]}(ℤ, M)