Commutative Algebra

Excercises 8

Let R be a graded ring. A homogeneous ideal is simply an ideal I ⊂ R which is also a graded submodule of R. Equivalently, it is an ideal generated by homogeneous elements. Equivalently, if f ∈ I and

f = f_{0} + f_{1} + … + f_{n}

is the decomposition of f into homogenous pieces in R then f_{i} ∈ I for each i. We define Proj(R) to be the set of homogenous, prime ideals 𝔭 of R such that R_{+} ⊄ 𝔭. Note that Proj(R) is a subset of Spec(R) and hence has a natural induced topology.

Let R = ⊕_{d≥0}R_{d} be a graded ring, let f ∈ R_{d} and assume that d ≥ 1. We define R_{(f)} to be the subring of R_{f} consisting of elements of the form r/f^{n} with r homogenous and deg(r) = nd. Furthermore, we define

D_{+}(f) = {𝔭 ∈ Proj(R)|f ∉ 𝔭}.

Finally, for a homogenous ideal I ⊂ R we define V_{+}(I) = V (I) ∩ Proj(R).

1. Topology on Proj(R). With notations as above:

(a) Show that D_{+}(f) is open in Proj(R), show that D_{+}(ff′) = D_{+}(f) ∩ D_{+}(f′).

(b) Let g = g_{0} + … + g_{m} be an element of R with g_{i} ∈ R_{i}. Express D(g) ∩ Proj(R) in terms of D_{+}(g_{i}), i ≥ 1 and D(g_{0}) ∩ Proj(R). No proof necessary.

(c) Let g ∈ R_{0} be a homogenous element of degree 0. Express D(g) ∩ Proj(R) in terms of D_{+}(f_{α}) for a suitable family f_{α} ∈ R of homogenous elements of positive degree.

(d) Show that the collection {D_{+}(f)} of opens forms a basis for the topology of Proj(R).

(e) Show that there is a canonical bijection D_{+}(f) → Spec(R_{(f)}).

(f) Show that the map from (e) is a homeomorphism.

(g) Give an example of an R such that Proj(R) is not quasi-compact. No proof necessary.

(i) Show that any closed subset T ⊂ Proj(R) is of the form V_{+}(I) for some homogenous ideal I ⊂ R.

Remark. There is a continuous map Proj(R) ⟶ Spec(R_{0}).

2. If R = A[X] with deg(X) = 1, show that the natural map Proj(R) → Spec(A) is a bijection and in fact a homeomorphism.

3. Blowing up: part I. In this exercise R = Bl_{I}(A) = A ⊕ I ⊕ I^{2} ⊕ . . .. Consider the natural map b : Proj(R) → Spec(A). Set U = Spec(A) − V (I). Show that

b : b^{−1}(U) ⟶ U

is a homeomorphism.

Thus we may think of U as an open subset of Proj(R). Let Z ⊂ Spec(A) be an irreducible closed subscheme with generic point ξ ∈ Z. Assume that ξ ∉ V (I), in other words Z ⊄ V (I), in other words ξ ∈ U, in other words Z ∩ U ≠ ∅. We define the strict transform Z′ of Z to be the closure of the unique point ξ′ lying above ξ. Another way to say this is that Z′ is the closure in Proj(R) of the locally closed subset Z ∩ U ⊂ U ⊂ Proj(R).

4. Blowing up: Part II. Let A = k[x, y] where k is a field, and let I = (x, y). Let R be the blow up algebra for A and I.

(a) Show that the strict transforms of Z_{1} = V ({x}) and Z_{2} = V ({y}) are disjoint.

(b) Show that the strict transforms of Z_{1} = V ({x}) and Z_{2} = V ({x−y^{2}}) are not disjoint.

(c) Find an ideal J ⊂ A such that V (J) = V (I) and such that the strict transforms of Z_{1} = V ({x}) and Z_{2} = V ({x − y^{2}}) are disjoint.

5. Let R be a graded ring.

(a) Show that Proj(R) is empty if R_{n} = (0) for all n >> 0.

(b) Show that Proj(R) is an irreducible topological space if R is a domain and R_{+} is not zero. (Recall that the empty topological space is not irreducible.)

6. Blowing up: Part III. Consider A, I and U, Z as in the definition of strict transform. Let Z = V (𝔭) for some prime ideal 𝔭. Let Ā = A/𝔭 and let Ī be the image of I in Ā.

(a) Show that there exists a surjective ring map R := Bl_{I}(A) → R̄ := BlĪ(Ā).

(b) Show that the ring map above induces a bijective map from Proj(R̄) onto the strict transform Z′ of Z. (This is not so easy. Hint: Use 5(b) above.)

(c) Conclude that the strict transform Z′ = V_{+}(P) where P ⊂ R is the homogenous ideal defined by P_{d} = I^{d} ∩ 𝔭.

(d) Suppose that Z_{1} = V (𝔭) and Z_{2} = V (𝔮) are irreducible closed subsets defined by prime ideals such that Z_{1} ⊄ Z_{2}, and Z_{2} ⊄ Z_{1}. Show that blowing up the ideal I = 𝔭 + 𝔮 separates the strict transforms of Z_{1} and Z_{2}, i.e., Z′_{1} ∩ Z′_{2} = ∅. (Hint: Consider the homogenous ideal P and Q from part (c) and consider V (P + Q).)