Exercise 111.33.1. Find a $1$-point locally ringed space which is not a scheme.
111.33 Schemes
Let $LRS$ be the category of locally ringed spaces. An affine scheme is an object in $LRS$ isomorphic in $LRS$ to a pair of the form $(\mathop{\mathrm{Spec}}(A), {\mathcal O}_{\mathop{\mathrm{Spec}}(A)})$. A scheme is an object $(X, {\mathcal O}_ X)$ of $LRS$ such that every point $x\in X$ has an open neighbourhood $U \subset X$ such that the pair $(U, {\mathcal O}_ X|_ U)$ is an affine scheme.
Exercise 111.33.2. Suppose that $X$ is a scheme whose underlying topological space has 2 points. Show that $X$ is an affine scheme.
Exercise 111.33.3. Suppose that $X$ is a scheme whose underlying topological space is a finite discrete set. Show that $X$ is an affine scheme.
Exercise 111.33.4. Show that there exists a non-affine scheme having three points.
Exercise 111.33.5. Suppose that $X$ is a nonempty quasi-compact scheme. Show that $X$ has a closed point.
Remark 111.33.6. When $(X, {\mathcal O}_ X)$ is a ringed space and $U \subset X$ is an open subset then $(U, {\mathcal O}_ X|_ U)$ is a ringed space. Notation: ${\mathcal O}_ U = {\mathcal O}_ X|_ U$. There is a canonical morphisms of ringed spaces If $(X, {\mathcal O}_ X)$ is a locally ringed space, so is $(U, {\mathcal O}_ U)$ and $j$ is a morphism of locally ringed spaces. If $(X, {\mathcal O}_ X)$ is a scheme so is $(U, {\mathcal O}_ U)$ and $j$ is a morphism of schemes. We say that $(U, {\mathcal O}_ U)$ is an open subscheme of $(X, {\mathcal O}_ X)$ and that $j$ is an open immersion. More generally, any morphism $j' : (V, {\mathcal O}_ V) \to (X, {\mathcal O}_ X)$ that is isomorphic to a morphism $j : (U, {\mathcal O}_ U) \to (X, {\mathcal O}_ X)$ as above is called an open immersion.
\[ j : (U, {\mathcal O}_ U) \longrightarrow (X, {\mathcal O}_ X). \]
Exercise 111.33.7. Give an example of an affine scheme $(X, {\mathcal O}_ X)$ and an open $U \subset X$ such that $(U, {\mathcal O}_ X|U)$ is not an affine scheme.
Exercise 111.33.8. Given an example of a pair of affine schemes $(X, {\mathcal O}_ X)$, $(Y, {\mathcal O}_ Y)$, an open subscheme $(U, {\mathcal O}_ X|_ U)$ of $X$ and a morphism of schemes $(U, {\mathcal O}_ X|_ U) \to (Y, {\mathcal O}_ Y)$ that does not extend to a morphism of schemes $(X, {\mathcal O}_ X) \to (Y, {\mathcal O}_ Y)$.
Exercise 111.33.9. (This is pretty hard.) Given an example of a scheme $X$, and open subscheme $U \subset X$ and a closed subscheme $Z \subset U$ such that $Z$ does not extend to a closed subscheme of $X$.
Exercise 111.33.10. Give an example of a scheme $X$, a field $K$, and a morphism of ringed spaces $\mathop{\mathrm{Spec}}(K) \to X$ which is NOT a morphism of schemes.
Exercise 111.33.11. Do all the exercises in [Chapter II, H], Sections 1 and 2... Just kidding!
Definition 111.33.12. A scheme $X$ is called integral if $X$ is nonempty and for every nonempty affine open $U \subset X$ the ring $\Gamma (U, \mathcal{O}_ X) = \mathcal{O}_ X(U)$ is a domain.
Exercise 111.33.13. Give an example of a morphism of integral schemes $f : X \to Y$ such that the induced maps ${\mathcal O}_{Y, f(x)} \to {\mathcal O}_{X, x}$ are surjective for all $x\in X$, but $f$ is not a closed immersion.
Exercise 111.33.14. Give an example of a fibre product $X \times _ S Y$ such that $X$ and $Y$ are affine but $X \times _ S Y$ is not.
Remark 111.33.15. It turns out this cannot happen with $S$ separated. Do you know why?
Exercise 111.33.16. Give an example of a scheme $V$ which is integral 1-dimensional scheme of finite type over ${\mathbf Q}$ such that $\mathop{\mathrm{Spec}}({\mathbf C}) \times _{\mathop{\mathrm{Spec}}({\mathbf Q})} V$ is not integral.
Exercise 111.33.17. Give an example of a scheme $V$ which is integral 1-dimensional scheme of finite type over a field $k$ such that $\mathop{\mathrm{Spec}}(k') \times _{\mathop{\mathrm{Spec}}(k)} V$ is not reduced for some finite field extension $k'/k$.
Remark 111.33.18. If your scheme is affine then dimension is the same as the Krull dimension of the underlying ring. So you can use last semesters results to compute dimension.
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