Abstract
We prove that if f : ( R , m ) → ( S , n ) f: (R,\mathfrak {m}) \rightarrow (S,\mathfrak {n}) is a flat local homomorphism, S / m S S/\mathfrak {m} S is Cohen-Macaulay and F F -injective, and R R and S S share a weak test element, then a tight closure analogue of the (standard) formula for depth and regular sequences across flat base change holds. As a corollary, it follows that phantom depth commutes with completion for excellent local rings. We give examples to show that the analogue does not hold for surjective base change.
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