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Say I've got a variety X (or a scheme locally of finite type) over an algebraically closed field k. Then closed points of X correspond to k-points of X. (correct?)

Let's define a geometric point of X as a morphism from an algebraically closed field into X. (thus for example the morphism from k[x] to the algebraic closure of the field of fractions of k[x] is a geometric point of the line)

If a (reasonable!) property P holds for all k-points of X does it then hold for all geometric points?

My question comes from moduli stuff. For example, if E is a flat family of sheaves on X parameterised by some base S, such that the fibre of E has some behaviour over all k-points of S, will this behaviour persist on geometric points?

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    Let $k$ be an alg. closed field. Then $X(k)$ is in bijection with the closed points of $X$. This follows from Hilbert's nullstellensatz if I'm not mistaken.2011-09-30
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    Could you give an example of your property $P$?2011-09-30
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    well, what I'm really thinking about is a family of complexes of sheaves. For example one might request that the (derived) restriction has no negative self-extensions (e.g. paper by max lieblich). Can I check this only over k-points?2011-09-30
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    I don't quite understand your situation but I can make the following probably useless statement. The image of a geom. point is a closed point and therefore a k-point. So if you're taking stalks of sheaves (in the Zariski topology) in points lying on your variety then once you've checked your property P for k-points it should also follow for geom. points.2011-09-30
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    wait a minute, isn't the example I gave in my question a case of a geometric point with non-closed image?2011-09-30
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    You're completely right. My apologies. Please ignore my comment above. I can make the following even more trivial statement though. A geom. point gives a k-point or a $K(X)$-point, where $K(X)$ is the function field $X$. (I assume here $X$ is irreducible for simplicity.) So if you verify it for these points you're good if your property P just requires you check it on the points of your variety.2011-09-30
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    On one hand my naivety is in thinking of a base scheme S as something nice, after all S is allowed to be _any_ k-scheme. On the other hand if I'm working with concrete family over some curve, say, then the question makes sense again. Thanks for your interest anyways.2011-09-30
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    You might find http://math216.wordpress.com/2011/06/13/favorite-properties-of-varieties-finite-type-k-schemes-checkable-at-closed-points/ and http://math216.wordpress.com/2011/06/10/favorite-open-or-closed-conditions-2/ helpful.2011-09-30
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    thanks for the links David.2011-10-04

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