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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)

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2 Answers 2

up vote 8 down vote accepted

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.

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Thanks a lot, I'll check it out! – anon Sep 19 '10 at 19:42

It is probably worth mentioning that another name for this is the category of abstract (not necessarily separated) algebraic $k$-varieties.

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Doesn't one usually require separatedness for varieties? – Akhil Mathew Nov 6 '10 at 0:35
@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 6 '10 at 2:38

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