(Pro-) Étale Cohomology 3. Exercise Sheet

Department of Mathematics Winter Semester 18/19 Prof. Dr. Torsten Wedhorn 2nd November 2018 Timo Henkel

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 X (X ). Let X S be a morphism of schemes. We consider the functor FX /S fromO 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:→

T0 / X (1)

f   T / S

Let be a class of morphisms of S-schemes. We say that satisfies the 1-lifting property (resp. !-lifting property) ≤ withC respect to f , if for all morphisms T0 T in andC for all diagrams∃ of the form (1) there exists∃ at most (resp. exactly) one morphism of S-schemes T X→which makesC the diagram commutative. Let →

1 := f : T0 T closed immersion of S-schemes f is given by a locally nilpotent ideal C { → | } 2 := f : T0 T closed immersion of S-schemes T is affine and T0 is given by a nilpotent ideal C { → | } 2 3 := 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 I = 0 C { → | } Show that the following assertions are equivalent:

(i) 1 satisfies the 1-lifting property (resp. !-lifting property) with respect to f . C ∃≤ ∃ (ii) 2 satisfies the 1-lifting property (resp. !-lifting property) with respect to f . C ∃≤ ∃ (iii) 3 satisfies the 1-lifting property (resp. !-lifting property) with respect to f . C ∃≤ ∃ 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.

1 Exercise 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 S,f (x) X ,x is weakly étale for every x X . O → O ∈ (b) Let

X 0 / X

f f 0   S0 g / S

be 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.

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