I found a paper Some applications of a model theoretic fact to (semi-) algebraic geometry by Lou van den Dries. In this paper, the author uses model theoretical methods to prove the completeness of projective varieties.

So my question is "can I translate the proof into modern language in algebraic geometry?"

The central part of the proof is this lemma.

Lemma: Let $T$ be a first order theory and $\phi(C)$ an $L(T)$-formula. Suppose that for each model $K$ of $T$ and each homomorphism $f:A\rightarrow L$ where $A\subset K$ and $L$ is also model of $T$, we have: if $c\in A^m$ and $K\models \phi(c)$, then $L\models \phi(f(c))$. Then there is positive quantifier free formula $\psi(C)$ such that $T\vdash \phi(C)\leftrightarrow \psi(C)$.

The completness of projective varieties says: Let $p_1(C,X),\dots,p_k(C,X)\in \mathbb{Z}[C,X]$ be homogeneous in $X=(X_0,\dots,X_n)$. Then there are polynomials $q_1(C),\dots,q_l(C)$ in $\mathbb{Z}[C]$ such that for each algebraically closed field $K$ and $c\in K^m$ the system $p_1(c,X)=\dots=p_k(c,X)$ has a non trivial solution in $K$ if and only if $q_1(c)=\dots=q_l(c)=0$.

The proof goes as follows.

Let $\phi(C)$ be the formula $\exists X\, (p_1(C,X)=...=p_k(C,X)=0)$. Let $T$ be the theory of algebraically closed fields. We have to show that $\phi(C)$ is equivalent to a positive quantifier free formula relative to T. By the model theoretic lemma we are reduced to showing the following:

If $f:A\rightarrow L$ is a morphism of a subring $A$ of an algebraically closed field $K$ into another algebraically closed field $L$ and if the system $p_1(C,X)=\dots=p_k(C,X)=0$ has a non-trivial solution in K, then the system $p_1(C,X)=\dots=p_k(C,X)=0$ has a non-trivial solution in L.

Now by Chevalley's place extension theorem, we may as well assume that $A$ is a valuation ring of $K$. Multiplying a non-trivial solution in $K$ by a suitable constant in $A$, we may as well assume that we have $x\in A^{n+1}$ with at least one coordinate invertible in $A$. Then, applying $f$ to $p_1(C,x)=\dots=p_k(C,x)=0$, we get a solution $f(x)$ of $p_1(C,f(x))=...=p_k(C,f(x))=0$, and $f$ maps the invertible coordinate of $x$ into a zero element of $L$.

I found a paper Some applications of a model theoretic fact to (semi-) algebraic geometry by Lou van den Dries. In this paper, the author uses model theoretical methods to prove the completeness of projective varieties.

So my question is "can I translate the proof into modern language in algebraic geometry?"

The central part of the proof is this lemma.

Lemma: Let $T$ be a first order theory and $\phi(C)$ an $L(T)$-formula. Suppose that for each model $K$ of $T$ and each homomorphism $f:A\rightarrow L$ where $A\subset K$ and $L$ is also model of $T$, we have: if $c\in A^m$ and $K\models \phi(c)$, then $L\models \phi(f(c))$. Then there is positive quantifier free formula $\psi(C)$ such that $T\vdash \phi(C)\leftrightarrow \psi(C)$.

The completness of projective varieties says: Let $p_1(C,X),\dots,p_k(C,X)\in \mathbb{Z}[C,X]$ be homogeneous in $X=(X_0,\dots,X_n)$. Then there are polynomials $q_1(C),\dots,q_l(C)$ in $\mathbb{Z}[C]$ such that for each algebraically closed field $K$ and $c\in K^m$ the system $p_1(c,X)=\dots=p_k(c,X)$ has a non trivial solution in $K$ if and only if $q_1(c)=\dots=q_l(c)=0$.

The proof goes as follows.

Let $\phi(C)$ be the formula $\exists X\, (p_1(C,X)=...=p_k(C,X)=0)$. Let $T$ be the theory of algebraically closed fields. We have to show that $\phi(C)$ is equivalent to a positive quantifier free formula relative to T. By the model theoretic lemma we are reduced to showing the following:

If $f:A\rightarrow L$ is a morphism of a subring $A$ of an algebraically closed field $K$ into another algebraically closed field $L$ and if the system $p_1(C,X)=\dots=p_k(C,X)=0$ has a non-trivial solution in K, then the system $p_1(C,X)=\dots=p_k(C,X)=0$ has a non-trivial solution in L.

Now by Chevalley's place extension theorem, we may as well assume that $A$ is a valuation ring of $K$. Multiplying a non-trivial solution in $K$ by a suitable constant in $A$, we may as well assume that we have $x\in A^{n+1}$ with at least one coordinate invertible in $A$. Then, applying $f$ to $p_1(C,x)=\dots=p_k(C,x)=0$, we get a solution $f(x)$ of $p_1(C,f(x))=...=p_k(C,f(x))=0$, and $f$ maps the invertible coordinate of $x$ into a non-zero element of $L$.

I found a paper "some applications of a model theoretic fact to (semi-) algebraic geometry" by Lou van den Dries.

  In this paper, the author uses model theoretical method to prove the completness of projective variety.

So my question is "can I translate the proof into modern lanquage in algebrai geometry?".

The central part of the proof is this lemma.

Lemma

Let "$T$" be a first order theory and $\phi(C)$, an $L(T)$-formula.

  Suppose that for each model $K$ of $T$ and each homomorphism $f:A\rightarrow L$ where $A\subset K$ and $L$ is also model of $T$, we have

  if $c\in A^m$ and $K$ satisfy $\phi(c)$,then $L$ also satisfy $\phi(f(c))$

Then there is positive quantifier free formula $\psi(C)$ such that $\phi(C)$ is equivalent to $\psi(C)$ in the theory $T$.

The completness of projective variety says; let $p_1(C,X),...,p_k(C,X)\in \mathbb{Z}[C,X]$ be homogeneous in $X=(X_0\cdots X_n)$. Then there are polynomials $q_1(C),\cdots,q_l(C)$ in $\mathbb{Z}[C]$ such that for each algebraically closed field $K$ and $c\in K^m$ the system $p_1(c,X)=\cdots=p_k(c,X)$ has a non trivial solution in $K$ if and only if $q_1(c)=\cdots=q_l(c)=0$

The proof goes as follows.

Let $\phi(C)$ be the formula $\exists X (p_1(C,X)=...=p_k(C,X)=0)$.Let $T$ be the theory of algebraically closed fields. We have to show that $\phi(C)$ is equivalent to a positive quanfier free formula,relative to T. By the model theoretic lemma we are reduced to showing;

if $f:A\rightarrow L$ is a morphism of a subring $A$ of an algebraically closed field $K$ into another algebraically closed field $L$ and if the system $p_1(C,X)=...=p_k(C,X)=0$ has a non trivial in K, then the system $p_1(C,X)=...=p_k(C,X)=0$ has a non trivial solution in L.

Now by Chevalley's place extension theorem we may as well assume that $A$ is a valuation ring of $K$. Multiplying a non-trivial solution in $K$ by a suitble constant in $A$,we may as well assume that we have $x\in A^{n+1}$ with at least one coordinate invertible in $A$. Then, applying $f$ to $p_1(C,x)=...=p_k(C,x)=0$, we get a solution $f(x)$ of $p_1(C,f(x))=...=p_k(C,f(x))=0$,and $f$ maps the invertible coordinate of $x$ into a zero element of $L$.

I found a paper Some applications of a model theoretic fact to (semi-) algebraic geometry by Lou van den Dries. In this paper, the author uses model theoretical methods to prove the completeness of projective varieties.

So my question is "can I translate the proof into modern language in algebraic geometry?"

The central part of the proof is this lemma.

Lemma: Let $T$ be a first order theory and $\phi(C)$ an $L(T)$-formula. Suppose that for each model $K$ of $T$ and each homomorphism $f:A\rightarrow L$ where $A\subset K$ and $L$ is also model of $T$, we have: if $c\in A^m$ and $K\models \phi(c)$, then $L\models \phi(f(c))$. Then there is positive quantifier free formula $\psi(C)$ such that $T\vdash \phi(C)\leftrightarrow \psi(C)$.

The completness of projective varieties says: Let $p_1(C,X),\dots,p_k(C,X)\in \mathbb{Z}[C,X]$ be homogeneous in $X=(X_0,\dots,X_n)$. Then there are polynomials $q_1(C),\dots,q_l(C)$ in $\mathbb{Z}[C]$ such that for each algebraically closed field $K$ and $c\in K^m$ the system $p_1(c,X)=\dots=p_k(c,X)$ has a non trivial solution in $K$ if and only if $q_1(c)=\dots=q_l(c)=0$.

The proof goes as follows.

Let $\phi(C)$ be the formula $\exists X\, (p_1(C,X)=...=p_k(C,X)=0)$. Let $T$ be the theory of algebraically closed fields. We have to show that $\phi(C)$ is equivalent to a positive quantifier free formula relative to T. By the model theoretic lemma we are reduced to showing the following:

If $f:A\rightarrow L$ is a morphism of a subring $A$ of an algebraically closed field $K$ into another algebraically closed field $L$ and if the system $p_1(C,X)=\dots=p_k(C,X)=0$ has a non-trivial solution in K, then the system $p_1(C,X)=\dots=p_k(C,X)=0$ has a non-trivial solution in L.

Now by Chevalley's place extension theorem, we may as well assume that $A$ is a valuation ring of $K$. Multiplying a non-trivial solution in $K$ by a suitable constant in $A$, we may as well assume that we have $x\in A^{n+1}$ with at least one coordinate invertible in $A$. Then, applying $f$ to $p_1(C,x)=\dots=p_k(C,x)=0$, we get a solution $f(x)$ of $p_1(C,f(x))=...=p_k(C,f(x))=0$, and $f$ maps the invertible coordinate of $x$ into a zero element of $L$.