Timeline for Zariski density of Q-bar points

5 events

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Jan 31, 2014 at 16:03	comment	added	Brian		@abz : Thanks! That's a really elegant way of seeing this.
Jan 31, 2014 at 6:53	comment	added	abz		Choose a basis $(e_i)$ for $\mathbb{C}$ over $\bar{\mathbb{Q}}$, and write $f=\sum e_i f_i$ where the $f_i$ are polynomials with coefficients in $\bar{\mathbb{Q}}$. Because $f$ vanishes on the $\bar{\mathbb{Q}}$-points of $X$, so does each $f_i$, and so lies in the (radical of) the ideal defining $X$ over $\bar{\mathbb{Q}}$. Hence vanishes on the $\mathbb{C}$ points.
Jan 31, 2014 at 6:30	review	First posts
Jan 31, 2014 at 7:46
Jan 31, 2014 at 6:21	comment	added	user76758		Let $A$ be a reduced finite generated algebra over an algebraically closed field $k$, $K/k$ an extension field. Let $X={\rm{MaxSpec}}(A)$. The map $A \rightarrow \prod_xk$ defined by $a\mapsto (a(x))_x$ is injective, so it remains injective after scalar extension to $K$. For arbitrary $k$-vector spaces, $V \otimes_k(\prod_i W_i) \rightarrow\prod_i (V\otimes_k W_i)$ is injective, even with infinitely many $W_i$'s (use direct limits in $V$ to reduce to $\dim V < \infty$, and then to $\dim_k V = 1$). Setting $V=K$ and $\{W_i\}=\{k\}_{x\in X}$, $K\otimes_kA\rightarrow \prod_x K$ is injective. QED
Jan 31, 2014 at 6:11	history	asked	Brian	CC BY-SA 3.0