COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

09. Ext, Tor Let 𝒞 be an abelian category. For any object M of 𝒞, the extension

group Ext_{𝒞}^{j}(M, N) is defined to be the derived functor of the “hom” functor

N ↦ Hom_{𝒞}(M, N).

We note that the Hom functor is a “bifunctor”. We may thus consider the right derived functor of • ↦ Hom(•, N) and that of

• ↦ Hom(M, •, N). Fortunately, both coincide: The extension group Ext_{𝒞}^{•}(M, N) may be calculated by using either an injective resolution of the second variable N or a projective resoltuion of the first variable M.

Example 9.1. Let us compute the extension groups Ext_{ℤ}^{j}(ℤ/36ℤ, ℤ/108ℤ).

(1) We may compute them by using an injective resolution

0 → ℤ/108ℤ → ℚ/108ℤ → ℚ/ℤ → 0

of ℤ/108ℤ.

(2) We may compute them by using a free resolution

0 ← ℤ/36ℤ ← ℤ ← 36ℤ ← 0

of ℤ/36ℤ.

Exercise 9.1. Compute an extension group Ext^{j}(M, N) for modules M, N of your choice. (Please choose a non-trivial example).

Definition 9.2. 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 9.3. 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 9.4. 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.

YOSHIFUMI TSUCHIMOTO

Lemma 9.5. Let A be an associative unital ring. Then:

(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.

(4) For any right ideal J of A and for any A-module M, we have

(A/J) ⊗_{A} M ≅ M/J.M

In particular, if the ring A is commutative, then for any ideals I, J of A, we have

(A/I) ⊗_{A} (A/J) ≅ A/(I + J)

Definition 9.6. 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.

Exercise 9.2. Compute Tor_{j}^{ℤ}(ℤ/nℤ, ℤ/mℤ) for n, m ∈ ℤ_{>0}.