(Pro-) Étale Cohomology 3. Exercise Sheet

(Pro-) Étale Cohomology 3. Exercise Sheet Department of Mathematics Prof. Dr. Torsten Wedhorn Timo Henkel Winter Semester 18/19 2nd November 2018 Homework Exercise H9 (Clopen subschemes) (12 points) Let X be a scheme. We define Clopen(X) := {Z ⊆ X | Z open and closed subscheme of X}. Recall that Clopen(X) is in bijection to the set of idempotent elements of OX (X). Let X → S be a morphism of schemes. We consider the functor FX / S from the category of S-schemes to the category of sets, given by FX / S(T → S) = Clopen(X ×S T). Now assume that X → S is a finite locally free morphism of schemes. Show that FX / S is representable by an affine étale S-scheme which is of finite presentation over S. Exercise H10 (Lifting criteria) (12 points) Let f : X → S be a morphism of schemes which is locally of finite presentation. Consider the following diagram of S-schemes: <ul style="display: flex;"><li style="flex:1">T0 </li><li style="flex:1">X</li></ul>(1) f<ul style="display: flex;"><li style="flex:1">T</li><li style="flex:1">S</li></ul>Let C be a class of morphisms of S-schemes. We say that C satisfies the ∃≤1-lifting property (resp. ∃!-lifting property) with respect to f , if for all morphisms T0 → T in C and for all diagrams of the form (1) there exists at most (resp. exactly) one morphism of S-schemes T → X which makes the diagram commutative. Let C1 := {f : T0 → T closed immersion of S-schemes | f is given by a locally nilpotent ideal} C2 := {f : T0 → T closed immersion of S-schemes | T is affine and T0 is given by a nilpotent ideal} C3 := {f : T0 → T closed immersion of S-schemes | T is the spectrum of a local ring and T0 is given by an ideal I with I2 = 0} Show that the following assertions are equivalent: (i) C satisfies the ∃≤1-lifting property (resp. ∃!-lifting property) with respect to f . (ii) C1 satisfies the ∃ -lifting property (resp. ∃!-lifting property) with respect to f . (iii) C23 satisfies the ∃≤1-lifting property (resp. ∃!-lifting property) with respect to f . ≤1 You can use the following fact without a proof: For an S scheme Y → S which is locally of finite presentation and a filtered system of affine S-schemes (Spec(Ai) → S)i∈I the canonical map colimi∈I (HomS(Spec(Ai), Y )) → HomS(Spec(colimi∈I Ai), Y ) is bijective. 1Exercise H11 (Vanishing differentials for weakly étale maps) (4+8 points) (a) Let R be a ring and I ⊆ R be an ideal. Show that R / I is a flat R-module if and only if for all ideals J ⊆ R we have J ∩ I = J · I. (b) Let f : X → S be a weakly étale morphism of schemes. Show that Ωf = 0. Exercise H12 (Properties of weakly étale) (6+6 points) (a) Let f : X → S be a morphism of schemes. Show that the following assertions are equivalent: (i) f is weakly étale. (ii) The map OS,f (x) → OX,x is weakly étale for every x ∈ X. (b) Let <ul style="display: flex;"><li style="flex:1">X0 </li><li style="flex:1">X</li></ul>S0<ul style="display: flex;"><li style="flex:1">f</li><li style="flex:1">f</li></ul>S0 gbe a commutative and Cartesian diagram of schemes such that g is faithfully flat. Show that f 0 is weakly étale if and only if f is weakly étale. 2

(Pro-) Étale Cohomology 3. Exercise Sheet

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