I have a number of questions on constructible sets. The first one is on morphisms: suppose $X$ and $Y$ are constructible sets, respectively in projective spaces $\mathbf{P}_1$ and $\mathbf{P}_2$ over a field $k$. How does one define morphisms between $X$ and $Y$ as natural generalization of morphisms between closed subsets of $\mathbf{P}_1$ and $\mathbf{P}_2$ ? Can one see $X$ and $Y$ naturally as ringed spaces, and then see the morphisms as morphisms between ringed spaces ? Thanks !
2
-
$\begingroup$ I'm not convinced such a "natural generalization" exists. Take for $X$ the union of the origin in $\mathbb{A}^1$ and its complement: as a scheme, regular functions on $X$ are $k[t^{\pm1}]\times k$, but as a constructible set, $X$ is $\mathbb{A}^1$. I think the most meaningful concept you can define are essentially "functions with a constructible graph" (but they are more general than morphisms of schemes). (contd.) $\endgroup$– Gro-TsenCommented Apr 25, 2017 at 12:18
-
$\begingroup$ The abstract nonsense formulation is probably to speak of sheaves for the $\neg\neg$ topology on the Zariski site. I'm sure the topos fans out there can explain this better than I can (and why this is naturally related to constructible sets). $\endgroup$– Gro-TsenCommented Apr 25, 2017 at 12:20
Add a comment
|