A scheme is separated if the diagonal inclusion $X to X times X$ is a closed immersion. I what to know if there is a good generalization of `separated' for algebraic stacks?
My usual stack reference, Anton Gerashchenko's stack notes, doesn't seem to provide an answer.
In a previous MO question several related notions came up. The most similar is quasi-separated where you require the diagonal to be quasi-compact. You can check wikipedia for some relevant algebraic geometry terminology. How does this compare to separatedness?
The main obstacle that I can see in defining separated for stacks is that the property of a map of schemes $X to Y$ being separated does not appear to be local in the target. Since maps between affines are separated, it seems that every map of schemes is locally separated. This means that we shouldn't expect the usual trick of replacing an algebraic stack by a scheme which covers it to work very well.
No comments:
Post a Comment