Lemma 58.25.1. Let (A, \mathfrak m) be a Noetherian local ring. Let f \in \mathfrak m. Assume
A has a dualizing complex and is f-adically complete,
one of the following is true
A_ f is (S_2) and every irreducible component of X not contained in X_0 has dimension \geq 4, or
if \mathfrak p \not\in V(f) and V(\mathfrak p) \cap V(f) \not= \{ \mathfrak m\} , then \text{depth}(A_\mathfrak p) + \dim (A/\mathfrak p) > 3.
for every maximal ideal \mathfrak p \subset A_ f purity holds for (A_ f)_\mathfrak p, and
purity holds for A.
Then purity holds for A/fA.
Comments (0)