Let $T$ be a theory with a good notion of independence.\u00a0 Generally, we have in mind forking independence, though something weaker would be acceptable.\u00a0 In fact, for our proof, the only properties we use are that independence is invariant under replacing a set by its algebraic closure<\/p>\n
the fact that<\/p>\n
and two monotonicity properties for independence<\/p>\n
(To be honest we also use, very weakly, the hypothesis that $\\operatorname{acl}(\\varnothing) \\neq \\varnothing$.)<\/p>\n
Let $n$ be a von Neumann natural number, that is, $n = \\{0, 1, \\ldots, n-1\\}$.\u00a0 We write $\\mathcal{P}(n)^-$ for $\\mathcal{P}(n) \\smallsetminus n$.\u00a0 \u00a0Let $W \\subseteq \\mathcal{P}(n)$ be closed downwards.\u00a0 That is, if $v \\subseteq w \\in W$, then $v \\in W$.<\/p>\n
A $W$-variable assignment is an association $w \\mapsto x_w$ from $W$ to variables ranging over some (possibly infinite) product $S_w$ of sorts so that for $w \\leq v \\in W$, we require that $S_v$ is a subproduct of $S_w$ and if $\\pi_{w,v}:S_w \\to S_v$ is the natural projection, then $x_v = x_w \\circ \\pi_{w,v}$.\u00a0 \u00a0 \u00a0If $\\{i\\} \\in W$, then we write $x_{\\{i\\}} =: x_i$.\u00a0 \u00a0Note our conventions:\u00a0 $x_i$ may be naturally a tuple (even infinite) of variables.\u00a0 \u00a0In this way, if $i \\in w \\in W$, then $x_i = x_w \\circ \\pi_{w,\\{i\\}}$.<\/p>\n
By an $W$-independent system of types we mean that we are given a $W$-variable assignment and then for each $w \\in W$ a complete type $p_w$ relative to $T$ in the variable $x_w$ so that<\/p>\n
We say that $T$ has $n$-amalgamation over algebraically closed sets if for every model $M \\models T$, every $E \\subseteq M$ with $\\operatorname{acl}(E) = E$, every downward closed $W \\subseteq \\mathcal{P}(n)$, and every $W$-independent system $(p_w)_{w \\in W}$ of types relative to $\\operatorname{Th}(M_E)$, there is a $\\mathcal{P}(n)$-independent system $(q_u)_{u \\in \\mathcal{P}(n)}$ with $q_w = p_w$ for $w \\in W$.<\/p>\n
Lemma:<\/strong> A theory $T$ has independent $n$-amalgamation over algebraically closed sets if and only if it has independent $m$-amalgamation for all $m < n$ and every $\\mathcal{P}(n)^{-}$-independent system over an algebraically closed subset of a model of $T$ may be extended to a $\\mathcal{P}(n)$-independent system.<\/p>\n Proof:<\/strong>\u00a0 The left to right direction is immediate. For the other direction, we suppose that $T$\u00a0 has independent $m$-amalgamation for all $m < n$ and every $\\mathcal{P}(n)^{-}$-independent system over an algebraically closed subset of a model of $T$ may be extended to a $\\mathcal{P}(n)$-independent system, that\u00a0 $W \\subseteq \\mathcal{P}(n)$ is\u00a0 downwards closed and that\u00a0 $(p_w)_{w \\in W}$ is a $W$-independent system over some algebraically closed subset $E$ of some model of $T$.<\/p>\n If $W = \\mathcal{P}(n)$.<\/p>\n Working by induction on $|W|$, it suffices to show that we may extend the system to any given downwards closed $W’ \\supseteq W$ with $|W’ \\smallsetminus W| = 1$.\u00a0 \u00a0 Let $w \\in W’ \\smallsetminus W$.<\/p>\n Lemma:<\/strong> A theory $T$ has $n$-amalgamation over algebraically closed sets if and only if every independent system in which (3) is strengthened to (3)’: $x_w = \\operatorname{acl}(\\{x_i:i \\in w\\})$ may be extended to a $\\mathcal{P}(n)$-independent system.<\/p>\n Proof:\u00a0 Let us start with some downwards closed $W \\subseteq \\mathcal{P}(n)$ and a $W$-independent system satisfying the usual definition and show how to expand it to a $W$-independent system $(q_w)_{w \\in W}$ in the variables $(y_w)_{w \\in W}$ satisfying (3)’ so that for each $w$ the variable $x_w$ ranges over a subtuple of $y_w$ and $p_w = q_w \\upharpoonright \\mathcal{L}_{x_w}$.\u00a0 \u00a0We work by induction on the size of $B :=\u00a0 \\{ w \\in W :\u00a0 p_w \\not \\vdash x_w = \\operatorname{acl}(\\{ x_i : i \\in w \\}) \\}$.\u00a0 If $B = \\varnothing$, we are done.\u00a0 Note that if $w \\in W \\smallsetminus B$, then for every $u \\subseteq w$ we have $u \\notin B$ as well.\u00a0 Thus, if $B \\neq \\varnothing$, there is some $w \\in B$ so that $\\{ u \\in W : w \\subseteq u \\} = \\{w\\}$.\u00a0 Pick such a $w$.\u00a0 Let $a_w$ realize $p_w$ in some model $M \\models T$.\u00a0 Let $b_w$ be (an enumeration of) $\\operatorname{acl}(a_w)$.\u00a0 \u00a0Let $y_w$ be the variable context corresponding to this enumeration of $b_w$.\u00a0 \u00a0For $u \\subseteq w$, let $b_u := b_w \\cap \\operatorname{acl}(a_u)$ where $a_u$ is the restriction to $x_u$ of $b_u$ and where we choose the variable context $y_u$ for $b_u$ compatibly with $y_w$ and the pre-existing $x_u$.\u00a0 Set $q_u := \\operatorname{tp}(a_u\/E)$.\u00a0 For $u \\in W$ with $u \\not \\subseteq w$, let $y_u$ be the concatenation of $x_u$ and $y_{u \\cap w}$, identifying $x_{u \\cap w}$ with the corresponding subtuple of $y_{u \\cap w}$.\u00a0 Let $q_u := p_u \\cup q_{u \\cap w}$.\u00a0 Then by our first condition on the behavior of independence with respect to algebraic closure, $(q_u)_{u \\in W}$ is still a $W$-independent system.\u00a0 $\\Box$.<\/p>\n <\/p>\n Definition:\u00a0 For $K \\models \\text{ACFA}$ and $A \\subseteq B$ and $A \\subseteq C$, we say that $B$ is free from $C$ over $A$ if $\\operatorname{acl}(B)$ is free from $\\operatorname{acl}(C)$ over $\\operatorname{acl}(A)$ in the sense of ACF.\u00a0 Equivalently, the field $\\operatorname{acl}(B)$ is linearly disjoint from the field $\\operatorname{acl}(C)$ over the field $\\operatorname{acl}(A)$.<\/p>\n Theorem:\u00a0 For every natural number $n$, ACFA has independent $n$-amalgamation over algebraically closed sets.<\/p>\n Proof:\u00a0 Note that the case of $n = 1$ is trivial – take $p_{0}$ to be any type over $E$.\u00a0 \u00a0 \u00a0 Working by induction $n$, we may assume that we know ACFA has independent $m$-amalgamation of algebraically closed sets for $m < n$. By our lemmas, it suffices to consider $E \\subseteq \\mathbb{U} \\models \\text{ACFA}$ where $E = \\operatorname{acl}(E)$, $K$ is $|E|^+$-saturated, and $(p_w)_{w \\in \\mathcal{P}(n)^-}$ is an independent $\\mathcal{P}(n)^-$-system over $E$ in which $p_w \\vdash x_w = \\operatorname{acl}(E \\cup \\{ x_i : i \\in w\\})$.\u00a0 Let $W’ := \\{ w \\in \\mathcal{P}(n)^- : n-1 \\in w \\}$.\u00a0 \u00a0Let $a_n \\models p_n$.\u00a0 For $u \\in \\mathcal{P}(n-1)^-$, let $y_u = x_{u \\cup \\{n-1\\}}$ and set $q_u := p_{u \\cup \\{ n \\}} \\cup x_n = a_n \\in S_{y_u}(a_n)$.\u00a0 The system $(q_u)_{u \\in \\mathcal{P}(n-1)^-}$ is a $\\mathcal{P}(n-1)^-$-independent system over the algebraically closed $a_n$.\u00a0 Hence, by induction, it may be extended to a $\\mathcal{P}(n-1)$-independent system $(q_u)_{u \\in \\mathcal{P}(n-1)^-}$.\u00a0 Let $b$ realize $q_{\\{0, \\ldots, n-2\\}}$ and let $L := \\operatorname{acl}(b)$.\u00a0 (If we were to apply our lemma about requiring (3)’ to hold, then we would already have $b = L$.). Let $K$ be the compositum of $b_u := x_u(b)$ for $u \\in \\mathcal{P}(n-1)$.\u00a0 Let $M$ be (generated by) a realization of $p_{\\{0, 1, \\ldots, n-2\\}}$.\u00a0 We have a copy of $K$ embedded in $M$ so that $M = K^\\text{alg}$ (though it need not be the case that $M = K^\\text{alg}$ as a sub-difference field of $\\mathbb{U}$.<\/p>\n Because $K$ is a compositum of algebraically closed fields, and both $M$ and $L$ are perfect, check that $M \\otimes_K L$ is a domain, it suffices to show that $M \\cap L = K$.\u00a0 \u00a0Note that if we succeed in showing that $M \\otimes_K L$ is a domain, then embedding this domain (with the endomorphisms coming from $\\sigma$ on each tensor factor) into $\\mathbb{U}$ over $E$, we could complete the system by taking $a_{\\{0, \\ldots, n-1\\}}$ to be the image of\u00a0 $ a_{\\{0, \\ldots, n-2\\}} \\otimes 1 \\frown 1 \\otimes b$ and $p_{\\{0, \\ldots, n-1\\}} := \\operatorname{tp}(a_{\\{0, \\ldots, n-1\\}}\/E)$, we complete the system.<\/p>\n Let us check now that $M \\cap L = K$.\u00a0 Let $c \\in M \\cap L$.\u00a0 Any element $c$ of $L$ may be expressed as $c = \\sum_{i=1}^m \\prod_{u \\in \\mathcal{P}(n-1)^-} c_{i,u}$ where $c_{i,u} \\in \\operatorname{acl}(a_{n-1},b_w)$.\u00a0 \u00a0Let $Q_{i,w}(x,y) \\in K[x,y]$ be a polynomial so that $Q_{i,w}[x,a_{n-1}]$ is the minimal monic polynomial over $K (a_{n-1})$ of $b_{i,w}$.\u00a0 Let $\\phi(x,y) := (\\exists z_{i,u})_{i=1,u \\in \\mathcal{P}(n-1)^-}^m \\left( x = \\sum_i \\prod_w z_{i,w} \\land \\bigwedge_{i,w} Q_{i,w}(z_{i,w},y) \\land Q_{i,w}(X,y) \\not \\equiv 0 \\right)$.<\/p>\n Now, $c$ satisfies $\\phi(x,a_{n-1})$ and because $c \\in M$, which is the algebraic closure of $\\{ a_i : 0 \\leq i < n-1\\}$, $c$ is free from $a_{n-1}$ over $E$ in the sense of ACF.\u00a0 Hence,\u00a0 $\\phi(c,y)$ defines a cofinite set and there is some $a \\in E$ so that $\\phi(c,a)$ holds.\u00a0 But then $\\phi(c,a)$ says that $c$ lies in the compositum of the $b_w$ for $w \\in \\mathcal{P}(n-1)^-$, that is, in $K$. $\\Box$<\/p>\n <\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" Let $T$ be a theory with a good notion of independence.\u00a0 Generally, we have in mind forking independence, though something weaker would be acceptable.\u00a0 In fact, for our proof, the only properties we use are that independence is invariant under … Continue reading \n