injectivity for almost all primes

Lemma 19   Let $k$ be a field. Assume $\phi:A_n(k)\to A_n(k)$ is injective (which is always the case if $\operatorname{char}k=0$). Then $\operatorname{rank}(\mathbb{Z}\operatorname{multideg}(\phi(A_n(k)))=2 n$.

Let $N$ be the maximum of total degrees of elements $\phi(\xi_1),\dots,\phi(\xi_n),\phi(\eta_1),\dots,\phi(\eta_n)$. Then we see that

$$\phi(A_n(k)_{\leq i})\subset A_n(k)_{\leq N i}$$

holds for any $i>0$. Thus for any $i>0$, we have

$$\operatorname{{}^aHF}_{\phi(A_n(k))} (Ni)\geq \operatorname{{}^aHF}_{A_n(k)}(i) =\binom{i+2n}{i}.$$

This together with Lemma 18 gives the result. $\qedsymbol$

Lemma 20   Let $A$ be a subalgebra of a polynomial algebra $P=k[X_1,X_2,\dots,X_m]$ over a field $k$. If there exists an positive integer $c$ such that

$$\dim (A_{\leq d}) \geq \frac{1}{c m!} d^m$$

holds for all integers $d>0$, then we have

$$[Q(P):Q(A)]\leq c$$

Suppose on the contrary that $[Q(P):Q(A)] >c$. Take elements $f_1,\dots,f_{c+1}\in Q(P)$ which are linearly independent over $Q(A)$. By multiplying a ``common denominator'', we may assume that $f_i$ are elements of $P$. Then it follows that the sum

$$f_1 A + f_2 A + \dots +f_{c+1} A$$

is direct in $P$. Let $M$ be the maximum of total degrees of $f_1,\dots,f_{c+1}$. Then the directness above implies that an inequality

$$\dim \left((f_1 A +f_2 A +\dots +f_{c+1}A)_{\leq d}\right ) \geq (c+1) \dim A_{\leq(d-M)}$$

holds for any integer $d>M$. Since the left hand side is not greater than $\dim P_{\leq d}$, we obtain the following inequation

$$\frac{d^m}{m!}+O(d^{m-1})\geq (c+1)\frac{1}{c m!}(d-M)^m$$

which leads to a contradiction when $d$ is large enough. $\qedsymbol$

Proposition 2 (injectivity for almost all primes)   Let $\mathfrak{K}$ be an algebraic number field, $\mathfrak{O}=\mathfrak{O}({\mathfrak{K}})$ be the ring of integers in $\mathfrak{K}$. suppose we are given an $\mathfrak{K}$-algebra endomorphism $\phi$ of $A_n(\mathfrak{K})$.

Then the multidegree monoid of the image $\operatorname{Image}(\phi)$ has rank $2n$. Furthermore, for almost all prime ideals $\mathfrak{p}$ of $\mathfrak{O}$, we have the following facts.

1. $\phi$ induces an $k(\mathfrak{p})$-algebra endomorphism $\phi_{\mathfrak{p}}$ of $A_n(k(\mathfrak{p}))$ (where $k(\mathfrak{p})=\mathfrak{O}/\mathfrak{p}$ is the residue field of $\mathfrak{p}$).
2. The multidegree monoid of the image $\operatorname{Image}(\phi_{\mathfrak{p}})$ has rank $2n$.
3. $\phi_\mathfrak{p}$ is injective.
4. There exists a constant $C$ such that $\operatorname{geomdeg}(\phi_\mathfrak{p})\leq C$ for all $\mathfrak{p}$.

The first statement is an easy consequence of Lemma 19. This in turn implies (2) (except for finite primes). In precise, let $x_1,x_2,\dots,x_n$ be elements in $A_n(\mathfrak{K})$ such that multi degrees of $\phi(x_1),\phi(x_2),\dots,\phi(x_{2n})$ are linearly independent. Then for almost all primes, their reductions $x_1,x_2,\dots,x_n$ are defined as elements of $A_n(k(\mathfrak{p}))$ and their multi degrees stays invariant under the reduction.

We apply Lemma 18 to see that there exists a positive real number $\epsilon$ which is independent of $\mathfrak{p}$ such that an inequality

$$\dim(\phi_\mathfrak{p}(A_n(k))_{\leq s} )\geq \epsilon s^{2n}$$

holds for any large integer $s$.

(1) is already proved in subsection 3.1. To prove (4), we denote by $k=k(\mathfrak{p})$ the quotient field with characteristic $p(>0)$. Since $A_n(k)$ is a free $Z_n(k)$-module of rank $2n$ with generators $\{\xi^I\eta^J; \vert\vert I\vert\vert _{\ell^\infty},\vert\vert J\vert\vert _{\ell^\infty}\leq p-1\}$, we have

$$\phi_\mathfrak{p}(Z_n(k)) \cdot (\sum_{\vert\vert I\vert\vert _{\... ...phi_\mathfrak{p}(\xi)^I\phi_\mathfrak{p}(\eta)^J) = \phi_\mathfrak{p}(A_n(k)).$$

Then Corollary 7 gives us an relationship of total degrees of both hands sides. Namely, there exists a positive number $N$ such that

$$\phi_\mathfrak{p}(Z_n(k))_{\leq s+N} \cdot (\sum_{\vert\vert I\ve... ...\xi)^I\phi_\mathfrak{p}(\eta)^J) \supset \phi_\mathfrak{p}(A_n(k))_{\leq s}$$

holds for any positive integer $s$.

Combining these, we obtain

$$\dim (\phi_\mathfrak{p}(Z_n(k))_{\leq s+N}) p^{2n} \geq \epsilon s^{2n}$$

Then we use Lemma20 to see that (4) is true. (One should be very careful about grading of $Z_n(k)$ here. degrees of generators $\xi_i^p, \eta_i^p$ of $Z_n(k)$ are all $p$. not $1$.) (3) follows from (4) and a consideration on transcendence degrees.

$\qedsymbol$

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