The minimum number of generators for inseparable algebraic extensions

On Del Pezzo Fibrations in Positive Characteristic

“…The assertion (1) follows from Serre duality. We now show (2). It follows from [37, Theorem 5.4 and Remark 5.5] that X has at worst rational singularities.…”

“…It follows from [BM40, Theorem 3] that the p-degree p-deg(K) is two, i.e. [K 1/2 : K] = 4 (note that the p-degree is called the degree of imperfection in [BM40]). Hence, it is enough to show that K 1 = L. Assume that K 1 = L. Then V 1 is smooth over L by (vi).…”

Del Pezzo surfaces and Mori fiber spaces in positive characteristic

“…Let us call the integer edim(A/F ) = edim(A ⊗ F F perf ) ≥ 0 the geometric embedding dimension. Similarly, define e(A/F ) = e(A ⊗ F F perf ) ≥ 1 But pdeg(K) = pdeg(F ) = 1 according to [6], Theorem 3. So by Proposition 1.2, the algebraic scheme X is geometrically reduced over the field K.…”

“…Since pdeg(F ) ≤ 1, the finite field extension F ⊂ K is obtained by adjoining a single element α ∈ K, such that K = F (α), according to [6], Theorem 1. Let f ∈ F [T ] be the minimal polynomial of this generator α ∈ K. Then K ⊗ F E = E[T ]/(f ), thus…”

Drinfeld–Stuhler modules

“…Proof. It is enough to prove (1), (2) and (3) after extending L to its algebraic closure, so we will assume that L is algebraically closed.…”