COMMUTATIVE ALGEBRA

YOSHIFUMI TSUCHIMOTO

03. Algebraic geometry of affine schemes

Definition 3.1. For any commutative ring A, we define its spectrum as

Spec(A) = {𝔭 ⊂ A; 𝔭 is a prime ideal of A.

For any subset S of A we define

V (S) = V_{Spec A}(S) = {𝔭 ∈ Spec A; 𝔭 ⊃ S}

Then we may topologize Spec(A) in such a way that the closed sets are sets of the form V (S) for some S. Namely,

F : closed ⟺ ∃S ⊂ A(F = V (S))

We refer to the topology as the Zariski topology.

Lemma 3.2. For any ring A, the following facts holds.

(1) For any subset S of A, we have

V (S) = ⋂_{s∈S} V ({s}).

(2) For any subset S of A, let us denote by ⟨S⟩ the ideal of A generated by S. then we have

V (S) = V (⟨S⟩)

Proposition 3.3. For any ring homomorphism φ : A → B, we have a map

Spec(φ) : Spec(B) ∋ 𝔭 ↦ φ^{−1}(𝔭) ∈ Spec(A).

It is continuous with respect to the Zariski topology.

Proposition 3.4. For any ring A, the following statements hold.

(1) For any ideal I of A, let us denote by π_{I} : A → A/I the canonical projection. Then Spec(π_{I}) gives a homeomorphism between Spec(A/I) and V_{Spec A}(I).

(2) For any element s of A, let us denote by ι_{s} : A → A[s^{−1}] be the canonical map. Then Spec(ι_{s}) gives a homeomorphism between Spec(A[s^{−1}]) and ∁V_{Spec_{A}}({s}).

Proposition 3.5. Let A, B be a ring. Let φ : A → B be a ring homomoprhism. We regard B as an A module via φ. If B is a finite A-module, then

Spec(φ) : Spec(B) → Spec(A)

is a closed map with respect to the Zariski topology.

Definition 3.6. Let X be a topological space. A closed set F of X is said to be reducible if there exist closed sets F_{1} and F_{2} such that

F = F_{1} ∪ F_{2}, F_{1} ≠ F, F_{2} ≠ F

holds. F is said to be irreducible if it is not reducible.

YOSHIFUMI TSUCHIMOTO

Definition 3.7. Let I be an ideal of a ring A. Then we define its radical to be

\sqrt{I} = \{x ∈ A; ∃N ∈ ℤ_{>0} such that x^{N} ∈ I\}.

Proposition 3.8. Let A be a ring. Then;

(1) For any ideal I of A, we have V (I) = V (\sqrt{I}).

(2) For two ideals I, J of A, V (I) = V (J) holds if and only if \sqrt{I} = \sqrt{J}.

(3) For an ideal I of A, V (I) is irreducible if and only if \sqrt{I} is a prime ideal.

Definition 3.9. We define a dimension of a topological space X as a supremum of the length of sequences

Y_{1} ⊋ Y_{2} ⊋ Y_{3} ⊋ ⋯ ⊋ Y_{s}

of irreducible subsets of X.

We define the Krull dimension of a ring A as the dimension of Spec(A).