Math 256B. Schematic Denseness and Associated Points

Proposition. An an open subset U of a noetherian scheme X is schematically dense if and only if it contains all associated points of X.

Proof. It suffices to show this when X is an affine scheme. Let A be the affine ring of X, let (0) = 𝔮_{1} ∩ ⋯ ∩ 𝔮_{n} be a minimal primary decomposition of the ideal (0) in A, and let 𝔭_{i} = \sqrt{𝔮_{i}} for each i. Then 𝔭_{1}, …, 𝔭_{n} are the primes corresponding to the associated points of X.

First suppose that U contains all associated points of X. Let 𝔞 be the ideal corresponding to the schematic closure of U , and let f ∈ 𝔞. For all i, Spec A/𝔞 contains an open neighborhood of 𝔭_{i}; therefore (A/𝔞)_{𝔭_{i}/𝔞} = A_{𝔭_{i}} . In particular, f = 0 in A_{𝔭_{i}} , so Ann(f) meets A \ 𝔭_{i}. Thus, for all i there exists x_{i} ∈ Ann(f) such that x_{i} ∉ 𝔭_{i}. It follows by an easy exercise that there exists x ∈ Ann(f) such that x ∉ 𝔭_{1} ∪ ⋯ ∪ 𝔭_{n}. But then x is not a zero divisor, by ([L 2], X Prop. 2.9). This can happen only if f = 0 , so 𝔞 = 0 and thus U is schematically dense in X.

Conversely, suppose that U does not contain the point corresponding to 𝔭_{i}. After renumbering the indices, we may assume that i = n and that {i | 𝔭_{i} ⊉ 𝔭_{n}} = {1, …, r} for some r < n. Let

𝔞 = 𝔮_{1} ∩ ⋯ ∩ 𝔮_{r}.

Since the chosen primary decomposition was minimal, we have 𝔞 ≠ (0) . We claim that U ⊆ Spec A/𝔞. Indeed, let 𝔭 be a prime ideal corresponding to a point in U . Since U is open and since 𝔭_{n} ∉ U , we have 𝔭 ⊉ 𝔭_{n}, and therefore 𝔭 ⊉ 𝔭_{i} for all i > r. Let S = A \ 𝔭. By ([A-M], Prop. 4.8(i)) and ([A-M], Prop. 4.9), we then have

(0) = ⋂_{i=1}^{n} S^{−1}𝔮_{i} = ⋂_{i=1}^{r} S^{−1}𝔮_{i} = S^{−1}𝔞.

Thus 𝔭 ∈ Spec A/𝔞 (since S^{−1}𝔞 = (0) implies 𝔞 ⊆ 𝔭), and (A/𝔞)_{𝔭} = A_{𝔭}. This holds for all 𝔭 ∈ U , so U is an open subscheme of Spec A/𝔞. This shows that U is not schematically dense in X. □

References

[A-M] M. F. Atiyah and I. G. MacDonald, Introduction to commutative algebra, Addison-Wesley, 1969.

[L] S. Lang, Algebra, 3rd edition, Addison-Wesley, Reading, Mass., 1993.