---
title: "Worksheet 10 - Schemes and Scheme Morphisms"
stable_id: br-bgk-2019-w10
language: en
source_course: "Kurs:Bündel, Garben und Kohomologie (Osnabrück 2019-2020)"
source_author: "Holger Brenner"
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upstream_title: "Kurs:Bündel, Garben und Kohomologie (Osnabrück 2019-2020)/Arbeitsblatt 10"
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---

# Worksheet 10: Schemes and Scheme Morphisms {#br-bgk-2019-w10}

None of the six exercises has a public solution at the frozen revision
boundary. No solution asterisks are therefore used, and this edition
creates no new solutions.

<!-- upstream_entity: Quasiaffines Schema/Nicht affin/Aufgabe -->

## Exercise 10.1 {#br-bgk-2019-w10-ex01}

Give an example of a quasi-affine scheme that is not affine.

<!-- upstream_entity: Quasiaffines Schema/Nicht quasikompakt/Aufgabe -->

## Exercise 10.2 {#br-bgk-2019-w10-ex02}

Give an example of a quasi-affine scheme that is not quasi-compact.

<!-- upstream_entity: Lokal beringter Raum/Affiner Raum/Morphismus/Fakt/Beweis/Aufgabe -->

## Exercise 10.3 {#br-bgk-2019-w10-ex03}

Let $(X,\mathcal O_X)$ be a locally ringed space. Prove that every tuple
of functions

$$
f_1,\ldots,f_n\in\Gamma(X,\mathcal O_X)
$$

determines a unique morphism of locally ringed spaces

$$
X\longrightarrow\mathbb A^n_{\mathbb Z},
$$

which sends the variable $T_i$ of affine space to $f_i$.

<!-- upstream_entity: Schema/Globaler Schnittring/Morphismus/Affin/Aufgabe -->

## Exercise 10.4 {#br-bgk-2019-w10-ex04}

Let $(X,\mathcal O_X)$ be a scheme. Prove that $X$ is affine if and only
if the canonical morphism

$$
X\longrightarrow\operatorname{Spek}(\Gamma(X,\mathcal O_X))
$$

is an isomorphism.

<!-- upstream_entity: Mannigfaltigkeit/Globaler Schnittring/Morphismus ins Spektrum/Injektiv/Aufgabe -->

## Exercise 10.5 {#br-bgk-2019-w10-ex05}

Let $X$ be a differentiable manifold. Prove that the canonical morphism

$$
X\longrightarrow\operatorname{Spek}(C^1(X,\mathbb R))
$$

is injective.

<!-- upstream_entity: Algebrahomomorphismus/Basisschema/Morphismus/Aufgabe -->

## Exercise 10.6 {#br-bgk-2019-w10-ex06}

Let $R$ be a commutative ring, and let $A,B$ be commutative $R$-algebras.
Prove that an $R$-algebra homomorphism

$$
\varphi:A\longrightarrow B
$$

is the same data as a scheme morphism

$$
\psi:\operatorname{Spek}(B)\longrightarrow\operatorname{Spek}(A)
$$

over $\operatorname{Spek}(R)$.