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

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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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