Höchste Zeit, unsern Kurs zu ändern

Recall that a scheme is a locally ringed space ${(X, {\mathscr O}_x)}$, i.e. a topological space and a sheaf of rings, such that for any ${x \in X}$ the stalk ${{\mathscr O}_{x,X}}$ is a local ring. Recall that the stalk is ${\left.\varprojlim_{U \ni x}\right. {\mathscr O}_X(U)}$.

An affine scheme is the scheme one gets from a commutative ring ${A}$ as follows:

$\displaystyle  \begin{array}{l} X = \mathrm{Spec} A = \{\mathfrak{p} \subseteq A \textrm{, prime }\}\\ \quad \quad \quad \textrm{ with the base of topology } X_f = \{\mathfrak{p} \mid f \notin \mathfrak{p}\} \textrm{ where } f \in A\\ {\mathscr O}_X(X_f)=A_{(f)} \end{array} $

Let ${f: Y \rightarrow X}$ be a map of ringed spaces and ${{\mathscr F}}$ a sheaf on ${X}$. A pullback sheaf ${f^{-1}{\mathscr F}}$ is defined as follows:

$\displaystyle f^{-1}{\mathscr F}(U) = \varprojlim_{V \supseteq f(U)} {\mathscr F}(V) $

One has the natural map ${f^*: {\mathscr F} \rightarrow f^{-1}(F)}$. It is called inverse image map. One similarly defines ${f_*: Shv(X) \rightarrow Shv(Y)}$ and ${f^{-1}: \Gamma({\mathscr F},U) \rightarrow \Gamma(f_*({\mathscr F}),U)}$ for all ${U \subseteq X}$.

A morphism of locally ringed spaces is a morphism of underlying spaces ${f: Y \rightarrow X}$ and a morphism of sheaves ${f^*: f^{-1}{\mathscr O}_X \rightarrow {\mathscr O}_Y}$ (or ${f^*: {\mathscr O}_Y \rightarrow f_*{\mathscr O}_X}$), it is called local if the induced stalk morphisms map the maximal ideal to the maximal ideal.

Imaginaries after Poizat («Une théorie de Galois imaginaire».

• A theory is said to weakly eliminate imaginaries if every formula ${\varphi}$ has a minmal algebraically closed set of parameters (i.e. a set ${A=\mathrm{acl}(A)}$ such that ${\varphi(x,b) = \psi(x,a)}$ with ${a\in A}$).
• A theory is said to eliminate imaginaries if every formula ${\varphi}$ has the (definably closed) canonical set of parameters or canonical base, i.e. any automorphism preserves ${\varphi}$ set-wise iff it preserves ${B=\mathrm{dcl}(B)}$ point-wise.

Clearer approach to imaginaries.

• A theory ${T}$ is said to weakly eliminate imaginaries if for every ${a \in M^\mathrm{eq}}$ there exists ${b \in M^n}$ such that ${b \in \mathrm{acl}(a)}$, ${a \in \mathrm{dcl}(M)}$
• A theory ${T}$ is said to eliminate imaginaries if for every ${a \in M^\mathrm{eq}}$ there exists ${b \in M^n}$ such that ${b \in \mathrm{dcl}(a)}$, ${a \in \mathrm{dcl}(M)}$

The first is equivalent to the second (proved in Poizat).

Strong elimination of imaginaries is the weak elimination of imaginaries plus elimination of finite imaginaries, i.e. those such that the equivalence classes are finite.

Notable example of weak elimination:

(These are the notes I made for a talk at Junior Logic Seminar at Oxford.)

The category of affine schemes is the category opposite to the category of commutative rings.

Let $A$ be a ring. One associates to it a topological space called $\mathrm{Spec} A$:

$$\begin{array}{l}\mathrm{Spec} A = \{ \textrm{prime ideals of } A \} \textrm{ with closed sets } \\ \quad \quad \quad X_\mathfrak{a} = \{\mathfrak{p} \subseteq \mathfrak{a} \mid \mathfrak{p} \in \mathrm{Spec} A\} \textrm{ for prime } \mathfrak{a}\in A\end{array}$$

Clearly, the points that correspond to maximal ideals are closed. If $\mathfrak{m} \subset A$ is a maximal ideal, $A/\mathfrak{m}$ is called the residue field of $\mathfrak{m}$. In classical algebraic geometry one deals with schemes over a field. Affine scheme over a field $ k$ is $ \mathrm{Spec} A$, where $ A=k[x_1,\ldots,x_n]/\mathfrak{a}$.

Proposition. Let $ A$ be the coordinate algebra of an affine scheme over $ k$. Let $ \mathrm{Max}_k\,A$ be the subset of maximal ideals with the residue field $ k$. Then $ \mathrm{Max}_k\,A\cong V(\mathfrak{a})=\{\bar{a}\in k^n \mid\forall f(\bar{x}) \in\mathfrak{a} f(\bar{a})=0\}$. Note that in general one cannot recover $ A$ from $ \mathrm{Spec} A$: consider $ k[x]/(x^2)$.

Theorem. Let $ K \supset k$ be a field extension, and $ K$ is finitely generated over $ k$ with generators $x_1, \ldots, x_n \in K$. Then $K$ is algebraic over $k$.