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I am reading this book which is Qing Liu introduction to algebraic and arithmetic geometry. I am trying to understand what does it mean finitely generated over $A_0$ ? I mean $A_0$ doesn't have to do anything with $A$ ?

Do we mean here that there exists generated $\{a_1,...,a_n\}$ such that every element of A can be written as linear combination with coeffients from $\phi(A_0)$, but isn't that what finite means ?

Can someone maybe give me a proof of this proposition and explain what is going on here?

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    It means finitely generated ad an $A_0$-algebra, i.e. every element of $A$ is a polynomial in a finite number of elements of $A$ (always the same elements). – Bernard Apr 17 '17 at 19:15
  • https://math.stackexchange.com/questions/630670/prove-that-an-algebra-is-finitely-generated-iff-it-is-isomorphic-to-the-quotient – user347489 Apr 17 '17 at 19:26
  • @user347489 perfect thanks –  Apr 17 '17 at 19:37
  • @user347489 do we need integral in the hypothesis ? –  Apr 17 '17 at 19:43
  • @Adeek not AFAIK. This is a completely general definition. You can find all these things in Atiyah-Macdonald, including the fact that "finite type+integral = finite" – user347489 Apr 17 '17 at 20:08

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Here's a sketch of the proof. Try to fill in the details.

One direction follows directly from the definitions: if $A$ is a finite $A_0$-algebra which is also integral over $A_0$, integrality will force finiteness as an $A_0$-module.

Conversely, if $A$ is a finite $A_0$-module in particular it's a finite $A_0$-algebra, so all we need to show is that every element of $A$ is integral over $A_0$. Take any $a\in A$ and the $A_0$-module homomorphism $\varphi:x\mapsto a\cdot x$. By Cayley-Hamilton this $\varphi$ satisfies a monic polynomial with coefficients in $A_0$. Taking $x=1$ in this equation yields the desired equation for $a$.

This is a standard result so you should be able to find it in every commutative algebra book. A good resource online are Gathmann's notes (see Proposition 9.5 in this case).

user347489
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