Timeline for Base change of integral scheme of finite type over a field
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Feb 16, 2018 at 0:35 | comment | added | anon | The dimension of the local ring at a closed point of $X$ is constant (equal to the transcendence degree of $k(X)$ over $k$). This remains true after extension of the base field. | |
| Feb 15, 2018 at 17:45 | comment | added | Daniel Loughran | If you are willing to assume that $\kappa/k$ is Galois, then this is quite easy to visualise. The Galois group of this extension acts transitively on the irreducible components over the algebraic closure, from which it follows that they all have the same dimension. | |
| Feb 15, 2018 at 16:33 | comment | added | Jason Starr | For a field extension $k\to K$, e.g., for $K=k(X)$, the algebraic closure of $k$ in $K$ is the set of all elements of $K$ that satisfy a nontrivial polynomial equation over $k$. This is a subextension $\kappa$ of $K/k$, and it is finite over $k$ if $K/k$ is finitely generated (often proved via Noether normalization). Every irreducible component $Y$ of $X\times_{\text{Spec}\ k} \text{Spec}\ \kappa$ has pure dimension equal to $\text{dim}(X)$ (Cohen-Seidenberg). As $\kappa$ is algebraically closed in $\kappa(Y)$, $Y$ is a geometrically irreducible $\kappa$-scheme of dimension $\text{dim}(X)$. | |
| Feb 15, 2018 at 16:11 | comment | added | Davide Cesare Veniani | I don't get your reduction step: why can I suppose that $k'/k$ is finite? By 'algebraic closure' you mean 'integral closure'? | |
| Feb 15, 2018 at 15:06 | comment | added | Jason Starr | After a finite field extension, the irreducible components of the base change become geometrically irreducible, precisely, after base change to the algebraic closure $\kappa$ of $k$ in the function field $k(X)$. Thus, assume that $k'/k$ is finite. Then $X'\to X$ is finite and faithfully flat. So now you can apply the Going Up and Going Down Theorems (Cohen-Seidenberg). | |
| Feb 15, 2018 at 14:59 | history | asked | Davide Cesare Veniani | CC BY-SA 3.0 |