CATEGORIES, ABELIAN CATEGORIES AND COHOMOLOGIES.

YOSHIFUMI TSUCHIMOTO

Adjoining inverses

Definition 10.1. Let A be a commutative ring. Let S be its subset. We say that S is multiplicative if

(1) 1 ∈ S

(2) x, y ∈ S ⟹ xy ∈ S holds.

Definition 10.2. Let S be a multiplicative subset of a commutative ring A. Then we define A[S^{−1}] as

A[{X_{s}; s ∈ S}]/({sX_{s} − 1; s ∈ S})

where in the above notation X_{s} is a indeterminate prepared for each element s ∈ S.) We denote by ι_{S} a canonical map A → A[S^{−1}].

Lemma 10.3. Let S be a multiplicative subset of a commutative ring A. Then the ring B = A[S^{−1}] is characterized by the following property:

Let C be a ring, φ : A → C be a ring homomorphism such that φ(s) is invertible in C for any s ∈ S. Then there exists a unique ring homomorphism ψ = ϕ[S^{−1}] : B → C such that

φ = ψ ∘ ι_{S}

holds.

Corollary 10.4. Let S be a multiplicative subset of a commutative ring A. Let I be an ideal of A given by

I = {x ∈ I; ∃s ∈ S such that sx = 0}

Then (1) I is an ideal of A. Let us put Ā = A/I, π : A → Ā the canonical projection. Then:

(2) S̄ = π(S) is multiplicatively closed.

(3) We have

A[S^{−1}] ≅ Ā[S̄^{−1}]

(4)ιS̄ : Ā → Ā[S̄^{−1}] is injective.

Definition 10.5. Let S be a multiplicative subset of a commutative ring A. Let M be an A-module we may define S^{−1}M as

{(m/s); m ∈ M, s ∈ S}/ ∼

where the equivalence relation ∼ is defined by

(m_{1}/s_{1}) ∼ (m_{2}/s_{2}) ⟺ t(m_{1}s_{2} − m_{2}s_{1}) = 0 (∃t ∈ S).

We may introduce a S^{−1}A-module structure on S^{−1}M in an obvious manner.

S^{−1}M thus constructed satisfies an universality condition which the reader may easily guess.

YOSHIFUMI TSUCHIMOTO

Lemma 10.6. Let A be a commutative ring. Let M be an A-module. Then we have a canonical isomorphism of A_{S} module

A_{S} ⊗_{A} M ≅ M_{S}.

We may also localize categories, but we need to deal with non commutativity of composition. To simplify the situation we only deal with a localization with some nice properties as follows:

(1) (a) s, t ∈ Σ ⟹ st ∈ Σ (b) X ∈ Ob(𝒞) ⟹ 1_{X} ∈ Σ.

(2) Let X, Y, Z ∈ Ob(𝒞). Let u ∈ Hom_{𝒞}(X, Y), s ∈ Hom_{𝒞}(Z, Y) ∩ Σ. Then there exist W ∈ Ob(𝒞) and morphisms v ∈ Hom_{𝒞}(W, Z), and t ∈ Hom_{𝒞}(W, X) ∩ Σ such that the diagram

\begin{matrix}W & ⟶^{v} & Z \\ ^{t}↓ & & ^{s}↓ \\ X & ⟶^{u} & Y \end{matrix}

commutes.

In a simpler (but not rigorous) words, for each “composable s^{−1}u”, there exists v, t such s^{−1}u = vt^{−1}. Similarly, for each composable us^{−1}, there exists v, t such that us^{−1} = t^{−1}v holds.

(3) Let X, Y ∈ Ob(𝒞), u, v ∈ Hom_{𝒞}(X, Y). Then the following conditions are equivalent:

(a) There exists Y′ ∈ Ob(𝒞) and s ∈ Hom_{𝒞}(Y, Y′) ∩ Σ such that su = sv.

(b) There exists X′ ∈ Ob(𝒞) and t ∈ Hom_{𝒞}(Y, Y′) ∩ Σ such

that ut = vt.

(4) If s ∈ Σ and if su ∈ Σ then u ∈ Σ.

Lemma 10.7. Let Σ be a family of morphisms in 𝒞 which satisfies the properties above. Then one may construct a localization of 𝒞_{Σ} with respect to Σ. Furthermore, if 𝒞 is additive, then 𝒞_{Σ} is also additive.