The theory of difference fields admits a model companion, ACFA, often called “algebraically closed fields with a generic automorphism”.\u00a0 \u00a0 To prove this, we need to show that the class of existentially closed difference fields is first-order axiomatizable.<\/p>\n
We have already seen that every existentially closed differrnce field $(K,\\sigma)$ is algebraically closed and that the distinguished endomorphism $\\sigma$ is actually an automorphism.\u00a0 Thus, ACFA is given by at least ACF together with the sentence $\\forall y \\exists x \\sigma(x) = y$.\u00a0 \u00a0To complete the axiomatization we add the following schema of axioms:<\/p>\n
For any absolutely irreducible embedded affine algebraic variety\u00a0 $X \\subseteq \\mathbb{A}_K^n$ and irreducible subvariety $Y \\subseteq X \\times X^\\sigma$ for which both projections $Y \\to X$ and $Y \\to X^\\sigma$ are dominant there is a point $a \\in X(K)$ with $(a,\\sigma(a)) \\in Y(K)$.\u00a0 \u00a0 As we discussed when showing that $\\operatorname{Fix}(\\sigma)$ is PAC, these conditions can be expressed by a set of first-order sentences.\u00a0 \u00a0The key points are the following.<\/p>\n
We call this axiom scheme “the geometric axioms”.\u00a0 Let us note that they are equivalent to the ostensibly weaker system of axioms in which we require that $\\dim X = \\dim Y$.\u00a0 Indeed, given an instance $(X,Y)$ of the usual geometric axioms, by Bertini’s Theorem we may find an affine space $V \\subseteq \\mathbb{A}_K^n \\times \\mathbb{A}_K^n$ with $\\dim V = 2n + \\dim X – \\dim Y$ so that $Z := V \\cap Y$ is (absolutely) irreducible and both projections $Z \\to X$ and $Z \\to X^\\sigma$ are dominant.\u00a0 Applying the restricted geometric axiom to $(X,Z)$ gives $a \\in X(K)$ with $(a,\\sigma(a)) \\in Z(K) \\subseteq Y(K)$. Hence, the geometric axiom for $(X,Y)$ also holds.<\/p>\n
Let use now demonstrate that ACFA is the model companion of the theory of difference fields.<\/p>\n
Theorem:<\/strong> ACFA, the theory described above, is the theory of the class of existentially closed difference fields.\u00a0 That is, ACFA is the model companion of the theory of difference fields.<\/p>\n Proof:\u00a0 \u00a0<\/strong>First, we check that if $(K,\\sigma)$ is an existentially closed difference field, then $(K,\\sigma) \\models \\text{ACFA}$.\u00a0 We have already seen that it $K$ is algebraically closed and that $\\sigma$ is an automorphism.\u00a0 For the geometric axioms, as we have seen it suffices to consider the case where $Y \\subseteq X \\times X^\\sigma \\subseteq \\mathbb{A}_K^n \\times \\mathbb{A}_K^n$ with $\\dim Y = \\dim X$ and $X$ and $Y$ are absolutely irreducible.\u00a0 \u00a0Let us write the coordinate ring of $\\mathbb{A}_K^n \\times \\mathbb{A}^n_K$ as $K[x_1, \\ldots, x_n, y_1, \\ldots, y_n]$.\u00a0 Let $L := \\mathcal{Q}(K[x_1,\\ldots,x_n]\/I(X))^\\text{alg} = K(X)^\\text{alg}$ be the algebraic closure of the function field of $X$.\u00a0 Since $Y \\to X$ is dominant and generically finite, the extension of functions field $K(Y)\/K(X)$ is finite.\u00a0 That is, $K(Y)$ embeds into $L$ over $K(X)$.\u00a0 Let $a_i$ be the image of $x_i$ in $L$ and let $b_i$ be the image of $y_i$ in $L$ under the embedding of $K(Y)$ for $1 \\leq i \\leq n$.\u00a0 We define an extension $\\tau:K[x_1, \\ldots, x_n] \\to L$ of $\\sigma$ by $x_i \\mapsto b_i$.\u00a0 Since $Y \\to X^\\sigma$ is dominant and $\\sigma$ is an automorphism, the kernel of $\\tau$ is $I(X^\\sigma)^{\\sigma^{-1}} = I(X)$.\u00a0 Hence, $\\tau$ extends to map, which we will still call $\\tau$, $\\tau:K(X) \\to L$.\u00a0 On general grounds, every map from a field to its algebraic closure extends to an endomorphism of the algebraic closure itself.\u00a0 Hence, $\\tau$ extends to a field endomorphism $\\rho:L \\to L$.\u00a0 \u00a0This gives an extension of difference fields $(K,\\sigma) \\subseteq (L,\\rho)$ and $(L,\\rho)$ satisfies that there is a point $a \\in X(L)$ with $(a,\\sigma(a)) \\in Y(L)$.\u00a0 Hence, by existential closedness of $(K,\\sigma)$, the same is true of $(K,\\sigma)$.<\/p>\n <\/p>\n Now we need to show that every model of ACFA is existentially closed as a difference field. Let $(K,\\sigma) \\models \\text{ACFA}$ and let $(L,\\sigma)$ be an extension as a difference field.\u00a0 Consider a quantifier-free formula $\\theta(x,y)$ in the free variables $x = (x_1, \\ldots, x_n)$ and $y = (y_1, \\ldots, y_m)$.\u00a0 Suppose that $b \\in K^y$ and $a \\in L^x$ with $L \\models \\theta(a,b)$.<\/p>\n Expressing $\\theta$ in disjunctive normal form, we may write it as $$\\theta = \\bigvee_i (\\bigwedge_{j=1}^{s_i} f_{i,j}(x,y) = 0 \\land \\bigwedge_{\\ell=1}^{t_i} g_{i,\\ell}(x,y) \\neq 0)$$ where each $f_{i,j}$ and $g_{i,\\ell}$ is a difference polynomial.\u00a0 Since $L \\models \\theta(a,b)$, for some $i$, it satisfies one disjunct.\u00a0 To prove that $K \\models \\exists x \\theta(x,b)$, it suffices to show that there is a witness to that disjunct in $K$ as well.\u00a0 So, we may assume that $$\\theta = \\bigwedge_{j=1}^s f_{j}(x,y) = 0 \\land \\bigwedge_{\\ell=1}^t g_{\\ell}(x,y) \\neq 0)$$.<\/p>\n For each $\\ell$, we may add a new variable $z_\\ell$, and setting $z = (z_1, \\ldots, z_s)$, if we let $c_\\ell = \\frac{1}{g_\\ell(a,b)}$,\u00a0 and set $$\\vartheta(x,y,z) := \\bigwedge_{i=1}^s f_j(x,y) = 0 \\land \\bigwedge_{\\ell=1}^t\u00a0 (z_\\ell g_\\ell(x,y) – 1) = 0$$ then $L \\models \\vartheta(a,b,c)$ and $\\text{ACFA} \\vdash \\vartheta(x,y,z) \\to \\theta(x,y)$.\u00a0 Thus, it suffices to show that $\\vartheta(x,b,z)$ may be realized in $K$.\u00a0 That is,\u00a0 replacing $\\theta$ by $\\vartheta$, $x$ by $(x,z)$, and $a$ by $(a,c)$, we may assume that no inequalities appear in the formula.<\/p>\n A difference polynomial $f(x)$ may be expressed in the form $f(x) = g(x,\\sigma(x), \\ldots, \\sigma^m(x))$ for some ordinary polynomial $g$ in the variables $x_0, \\ldots, x_m$.\u00a0 \u00a0Then $a$ satisfies $f(x) =0$ if and only if $a’ := (a_0, a_1, \\ldots, a_{m-1}) := (a,\\sigma(a), \\ldots, \\sigma^{m-1}(a))$ satisfies the system of equations $g(x_0, \\ldots, x_{m-1}, \\sigma(x_{m-1})) = 0$ and $\\sigma(x_i) = x_{i+1}$ for $0 \\leq i < m-1$.\u00a0 We repeat this process with each equation in the formula $\\theta$.\u00a0 That is, we find $m$ big enough so that each $f_j$\u00a0 may be expresses as $f_j(x,y) = g_j(x,\\sigma(x),\\ldots,\\sigma^m(x),y,\\sigma(y), \\ldots, \\sigma^m(y))$, then if we set\u00a0 $$\\eta := \\bigwedge_j g_j(x_0, \\ldots, x_{m-1},\\sigma(x_{m-1}),y_0, \\ldots, y_{m-1},\\sigma(y_{m-1})) = 0 \\land \\bigwedge_{i=0}^{m-1} \\sigma(x_i) = x_{i+1} \\land \\bigwedge_{i=0}^{m-1} \\sigma(y_i) = y_{i+1}$$ and we set $\\widetilde{b}_i := \\sigma^i(b)$ for $0 \\leq i < m$, we have $$\\text{ACFA} \\vdash (\\exists x \\theta(x,b)) \\leftrightarrow (\\exists x_0\u00a0 \\cdots \\exists x_{m-1} \\eta(x_0, \\ldots, x_{m-1}, \\widetilde{b})$$. Thus, we replacing $\\theta$ by $\\eta$ and $b$ by $\\widetilde{b}$ , we may assume that $\\sigma$ is unnested in $\\theta$.<\/p>\n Let $X := \\text{loc}(a\/K)$ and $Y := \\text{loc}((a,\\sigma(a))\/K)$.\u00a0 Here “loc” refers to the Zariski locus, the intersection of all $K$-algebraic varieties containing $a$, respectively $(a,\\sigma(a))$.\u00a0 Since $K$ is algebraically closed, these loci are absolutely irreducible.\u00a0 \u00a0As $a$ is the generic point (relative to $K$) of $X$, $\\sigma(a)$ is generic in $X^\\sigma$.\u00a0 Thus, the projections $Y \\to X$ and $Y \\to X^\\sigma$ are dominant.\u00a0 The geometric axioms apply and we find $c \\in X(K)$ with $(c, \\sigma(c)) \\in Y(K)$.\u00a0 The point $(c,\\sigma(c))$ satisfies the same polynomial equations over $K$ that $(b,\\sigma(b))$ does.\u00a0 In particular, it satisfies all of the equations in $\\theta$.\u00a0 This yields that $K \\models \\exists x (x \\in X \\land (x,\\sigma(x)) \\in Y)$.\u00a0 \u00a0$\\Box$<\/p>\n <\/p>\n","protected":false},"excerpt":{"rendered":" The theory of difference fields admits a model companion, ACFA, often called “algebraically closed fields with a generic automorphism”.\u00a0 \u00a0 To prove this, we need to show that the class of existentially closed difference fields is first-order axiomatizable. We have … Continue reading