What does the following term mean?
category of geometrically reduced schemes of finite type over some field
(I know what a category and a field is but I cannot translate any of the middle bits)
Asked
Active
Viewed 976 times
2 Answers
10
You need to consult a text on scheme theory, for instance Hartshorne's Algebraic Geometry. A scheme of finite type over a field $k$ is one with a finite cover of spectra of rings of the form $k[x_1,\ldots,x_m]/I$ where $I$ is an ideal. A reduced scheme is a scheme where for each open set $U$ the ring of functions defined over $U$ has no nonzero nilpotent elements. It's geometrically reduced if it remains reduced when the base field is changed to its algebraic closure.
Robin Chapman
- 22,310
-
Thanks a lot, I'll check it out! – Sep 19 '10 at 19:42
4
It is probably worth mentioning that another name for this is the category of abstract (not necessarily separated) algebraic $k$-varieties.
Pete L. Clark
- 97,892
-
-
@Akhil: this is often the case, but not always. To my mind, once you allow non-quasi-projective varieties, you might as well allow non-separated guys as well. (By the way, some people don't require varieties to be reduced either.) – Pete L. Clark Nov 06 '10 at 02:38