# if a property holds on closed points of an algebraic variety, does it hold over all geometric points?

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. – Gooz Sep 30 '11 at 21:08
Could you give an example of your property $P$? – Gooz Sep 30 '11 at 21:09
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? – Jacob Bell Sep 30 '11 at 21:16
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. – Gooz Sep 30 '11 at 21:23
wait a minute, isn't the example I gave in my question a case of a geometric point with non-closed image? – Jacob Bell Sep 30 '11 at 21:26