COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

07. Ext as a derived functor We recommend the book of Lang [1] as a good reference. The treatment here follows the book for the most part.

Theorem 7.1. Let 𝒞_{1} be an abelian category with enough injectives, and let F : 𝒞_{1} → 𝒞_{2} be a covariant additive left functor to another abelian category 𝒞_{2}. Then:

(1) F ≅ R^{0}F.

(2) For each short exact sequence

0 → M′ → M → M′′ → 0

and for each n ≥ 0 there is a natural homomorphism

δ^{n} : R^{n}F (M′′) → R^{n+1}F (M)

such that we obtain a long exact sequence

⋯ → R^{n}F (M′) → R^{n}F (M) → R^{n}F (M′′) →^{δ^{n}} R^{n+1}F (M′) → ….

(3) δ is natural. That means, for a morphism of short exact sequences

0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0
↓  ↓  ↓
0 ⟶ N′ ⟶ N ⟶ N′′ ⟶ 0

the δ’s give a commutative diagram:

R^{n}F (M′′) ⟶^{δ^{n}} R^{n+1}F (M′)
↓  ↓
R^{n}F (N′′) ⟶^{δ^{n}} R^{n+1}F (N′)

(4) For each injective objective object I of A and for each n > 0 we have R^{n}F (I).

The collection {R^{j}F } of functors R^{j}F is a “universal delta functor”. See [1].

Lemma 7.2. Under the assumption of the previous theorem, for any exact sequence 0 → M′ → M → M′′ → 0 of objects in 𝒞_{1}, there exists injective resolutions I_{M′}, I_{M}, I_{M′′} of M′, M, M′′ respectively and a commutative diagram

0  0  0
↓  ↓  ↓
0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0
↓  ↓  ↓
0 ⟶ I_{M′} ⟶ I_{M} ⟶ I_{M′′} ⟶ 0

YOSHIFUMI TSUCHIMOTO

such that the diagram of resolutions is exact. Thus we obtain a diagram

0  0  0
↓  ↓  ↓
0 ⟶ F (M′) ⟶ F (M) ⟶ F (M′′) ⟶ 0
↓  ↓  ↓
0 ⟶ F (I_{M′}) ⟶^{α} F (I_{M}) ⟶^{β} F (I_{M′′}) ⟶ 0

such that each row in the last line is exact.

Note that j-th cohomology of the complex F (I_{M}) (respectively, F (I_{M′}), F (I_{M′′})) gives the R^{j}F (M)(respectively, (R^{j}F (M′), R^{j}F (M′′).) Using the resolution given in the lemma above, we may prove Theorem 7.1. Let us describe the map δ in more detail when 𝒞_{2} is a category of modules by “diagram chasing”. Namely, for x ∈ R^{n}(M′′), let us show how to compute δ(x).

(1) x ∈ R^{n}(M′′) may be represented as a class [c_{x}] of a cocycle c_{x} ∈ Ker(d : F (I_{M′′}^{n}) → F (I_{M′′}^{n+1})).

(2) We take a “lift” c̃_{x} ∈ F (I_{M}^{n}) such that β^{n}(c̃_{x}) = c_{x}. Note that c̃_{x} is no longer a cocycle.

(3) Consider e_{x} = dc̃_{x} ∈ F (I^{n+1}M). It is a coboundary and we have β(e_{x}) = 0.

(4) There thus exists an element a_{x} ∈ F (I_{M′}^{n}) such that α(a_{x}) = e_{x}. a_{x} is no longer a coboundary but it is a cocycle.

(5) The cohomology class [a_{x}] of a_{x} is the required δ(x).

Such computation appears frequently when we deal with cohomologies.

Definition 7.3. Let A be a ring. Let M, N be A-modules. Then an extension of N by M is a module L with a exact sequence

(E)  0 → N →^{α} L →^{β} M → 0.

of A-modules. Let

0 → N →^{α′} L′ →^{β′} M → 0.

be another extension. Then the two extensions are said to be isomorphic if there exists a commutative diagram

0 ⟶ N ⟶^{α} L ⟶^{β} M ⟶ 0
=↓  ↓  ^{=}↓
0 ⟶ N ⟶^{α′} L′ ⟶^{β′} M ⟶ 0.

Proposition 7.4. There exists a bijection between the isomorphism classs of the extensions and elements of the Ext_{A}^{1}(M, N). The bijection is given by corresponding the extension (E) to the class δ(1_{N}) ∈ Ext^{1}(M, N) of the identity map 1_{N} by δ associated to the exact sequence (E).

See [1, XX,Exercise 27]

References

[1] S. Lang, Algebra (graduate texts in mathematics), Springer Verlag, 2002.