Relative Cartier divisors and Laurent polynomial extensions

On the vanishing of relative negative K-theory

“…, where the second isomorphism by Lemma 5.2 and the third isomorphism by the exact sequence (2.2). This implies that f * O X /O S ∼ = f * O × X /O × S as a sheaves on S. Now the result follows from Lemma 5.4 of [16].…”

Relative Brauer Groups and étale cohomology

“…(2) if f : L ֒→ K is a finite field extensions and characteristic of L does not divide n then the sequence0 → K × /L × ⊗ Z/nZ → H 1 et (Spec(L), µ f n ) → n Br(K|L) → 0 is exact.Proof. (1) By Lemma 5.4 of[17], H 0 et (S, I et ) ∼ = Pic (f). The long exact sequence (6.2) implies the assertion.…”

Equivariant Picard groups and Laurent polynomials

“…One can check that I is a functor from the category of ring extensions to abelian groups. The functor I is contracted with contraction [8]). A map f : X → S of schemes is said to be faithful affine if it is affine and the structure map O S → f * O X is injective.…”

“…More generally, I can be thought as a functor from the category of faithful affine map of schemes to abelian groups. In fact, I is a contracted functor on the category of faithful affine map of schemes (see Theorem 5.2 of [8]).…”

“…By Lemma 5.8, η is a morphism of contracted functors on Sch G /k. Then ker(η) is a contracted functor with contraction L ker(η) = L ker(η X ) (see Lemma 2.2 of [8]).…”