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ag.algebraic geometry – Automorphisms over finite bailiwick that don’t come from an automorphism in attribute zero

You ought to clearify your notation. Let $phi:A rightarrow B$ breathe any map of commutative unital rings and let $E:=A{e_1,..,e_n}$ breathe the free $A$-module of rank $n$.

Let us outline the group $GL_A(E)$ as follows: It is by definition (we’ve chosen a foundation for $E$) the

clique of $ntimes n$ matrices $M:=(a_{ij})$ with $a_{ij}in A$ with $det(M)in A^*$ a unit. It follows for any component $Min GL_A(E)$ there’s an inverse component $M^{-1}$ and $MM^{-1}=M^{-1}M=Id(n)$ is the id matrix, therefore $GL_A(E)$ is a gaggle. Given a matrix $Min GL_A(E)$ we might perceive $M$ as an $A$-linear automorphism

$M: Erightarrow E$

We might deem the induced map

$phi^*(M):=1otimes M: Botimes_A E rightarrow Botimes_A E$

outlined by $1otimes M(botimes e):=botimes M(e)in Botimes_A E$. If we write

$Botimes_A E cong B{e_1,..,e_n}$ it follows the matrix of $1otimes M$ is the matrix $(phi(a_{ij}))$ with $phi(a_{ij})in B$ and

$det(1otimes M):=det(phi(a_{ij}))=phi(det(a_{ij}))in B^*$ since $det(a_{ij})in A^*$ and $phi$ maps items to items.

Hene we get an induced map

$phi^*: GL_A(E) rightarrow GL_B(Botimes_A E)$.

Question: Are you asking for examples the place the map $phi^*$ will not be surjective?

Your query: “My main question is the following: is there an automorphism of $mathbb{A}^n$ defined over a finite field which is not the restriction of an automorphism defined over a field in characteristic zero? This seems hard to answer in general, but can we then replace the variety $mathbb{A}^n$ by something “not too removed from it”, defined over $mathbb{Z}$ and find some automorphisms over $mathbb{F}_2$ that do not come from automorphisms in characteristic zero?”

Example: It appears to me that if $A:=mathbb{Z}$, $B=mathbb{F}_2$ and $E:=A{e_1,e_2}$, it follows the canonical map

$GL_A(E) rightarrow GL_B(Botimes_A E)$

is surjective: You might bridle that any automorphism $Min GL(2, mathbb{F}_2)cong GL_B(Botimes_A E)$ comes from an automorphism in $GL(2, mathbb{Z}):=GL_A(E)$.

When you write $mathbb{A}^n$ this often means $mathbb{A}^n_A:=Spec(A[x_1,..,x_n]):=Spec(Sym_A^*(E^*))$ – the spectrum of a polynomial ring on $n$ variables $x_i$ over a commutative ring $A$.

There are two teams related to this status:

$GL_A(E)$ and $Aut_{A-alg}(Sym_A^*(E^*)):=Aut_{A-alg}(A[x_1,..,x_n]):=Aut_A(mathbb{A}^n_A)$

and the group $Aut_A(mathbb{A}^n_A)$ differs from $GL_A(E)$ in common.

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