RESOLUTIONS OF SINGULARITIES.

YOSHIFUMI TSUCHIMOTO

03. Spec and Proj.

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.

Exercise 3.1. Prove that Zariski topology is indeed a topolgy. That means, the collection {V (S)} satisfies the axiom of closed sets.

Exercise 3.2. Let A be a ring. Then:

(1) Show that for any f ∈ A, D(f) = {𝔭 ∈ Spec(A); f ∉ 𝔭} is an open set of Spec(A).

(2) Show that given a point 𝔭 of Spec(A) and an open set U which contains 𝔭, we may always find an element f ∈ A such that 𝔭 ∈ D(f) ⊂ U. (In other words, {D(f)} forms an open base of 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}).

Definition 3.5. 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.

Definition 3.6. 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 3.7. 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.

It is knwon that Spec A has a structure of “locally ringed space”. A locally ringed space which locally lookes like an affine spectrum of a ring is called a scheme.

Definition 3.8. Let S = ⨁_{n∈ℕ} S_{n} be a ℕ-graded ring. We put S_{+} = ⨁_{n>0} S_{n}.

We define

Proj(S) = {𝔭 ⊂ S; 𝔭 is a homogeneous prime ideal of S, 𝔭 ⊅ S_{+}}.

It is known that Proj(S) carries a ringed space strucure on it and that it is a scheme.

Definition 3.9. Let R be a ring. Let I be an ideal of R. The scheme X̃ = Proj(S) associated to the graded ring S = ⨁_{n∈ℕ} I^{n} is called the blowing up of X with respect to I.