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Let $A\subset\mathbb{V}^n$ and $B\subset\mathbb{V}^m$ be affine varieties. They have coordinate rings $\Gamma(V),\Gamma(W)$, e.g. $$\Gamma(V)=k[X_1,\dots,X_n]/I(V),$$ which may also be interpreted as the set of functions $A\to k$.

In his book, William Fulton states the following fact (Proposition 1, p. 18):

There is a one-to-one correspondence between homomorphisms $\Gamma(W)\to\Gamma(V)$ and polynomial maps $V\to W$.

However, I think that we should say homomorphisms which fix $k$, since all polynomial maps $V\to W$ only give such homomorphisms.

Is it true ? Or does any homomorphism $\Gamma(W)\to\Gamma(V)$ fix $k$ ?

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    $\begingroup$ By convention, a homomorphism of commutative rings sends identity to identity. $\endgroup$ – Andrew Mar 24 '13 at 21:37
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    $\begingroup$ And? Can't we combine a ring homomorphism with an automorphism of $k$? $\endgroup$ – Berci Mar 24 '13 at 21:40
  • $\begingroup$ But in the view $k[X_1,\dots,X_n]/I(V)$, this only gives that multiples of $1$ are sent to $1$ (and $0\mapsto 0$). $\endgroup$ – Klaus Mar 24 '13 at 21:41
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    $\begingroup$ In algebraic geometry, the coordinate ring of a variety over $k$ is typically thought of as a $k$-algebra, i.e. a ring equipped with a homomorphism $k\to \Gamma(V)$. The resulting diagram of $k$-algebras commutes if $k$ is fixed. $\endgroup$ – Tabes Bridges Mar 24 '13 at 21:45
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Page 17: "All rings and fields will contain $k$ as a subring. By a homomorphism $\phi: R\to S$ of such rings, we will mean a ring homomorphism such that $\phi(\lambda)=\lambda$ for all $\lambda\in k$.

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  • $\begingroup$ Thank you, shame on me for my lack of attention. $\endgroup$ – Klaus Mar 24 '13 at 21:47

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