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Matthieu Romagny
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It is a result of Serre (Morphismes universels et varietes d'albanese) that the Albanese (abelian) variety, i.e. an initial object for morphisms to (torsors over) abelian varieties, exists for any reduced scheme over a perfect field.

I have a counterexample for non-reduced schemes, but how about the perfectness of the base field? So my questions are:

(1) Where does Serre's proof brakesbreak down for non-perfect base fields?

(2) Does anyone know a counterexample over non-perfect fields?

NB: Serre does not state the hypothesis (maybe because at that time varieties were always integral over an algebraically closed field), but every author who quotes him states those hypothesis.

It is a result of Serre (Morphismes universels et varietes d'albanese) that the Albanese (abelian) variety, i.e. an initial object for morphisms to (torsors over) abelian varieties, exists for any reduced scheme over a perfect field.

I have a counterexample for non-reduced schemes, but how about the perfectness of the base field? So my questions are:

(1) Where does Serre's proof brakes down for non-perfect base fields?

(2) Does anyone know a counterexample over non-perfect fields?

NB: Serre does not state the hypothesis (maybe because at that time varieties were always integral over an algebraically closed field), but every author who quotes him states those hypothesis.

It is a result of Serre (Morphismes universels et varietes d'albanese) that the Albanese (abelian) variety, i.e. an initial object for morphisms to (torsors over) abelian varieties, exists for any reduced scheme over a perfect field.

I have a counterexample for non-reduced schemes, but how about the perfectness of the base field? So my questions are:

(1) Where does Serre's proof break down for non-perfect base fields?

(2) Does anyone know a counterexample over non-perfect fields?

NB: Serre does not state the hypothesis (maybe because at that time varieties were always integral over an algebraically closed field), but every author who quotes him states those hypothesis.

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It is a result of Serre (Morphismes universels et varietes d'albanese) that the Albanese (abelian) variety, i.e. an initial object for morphisms to (torsors over) abelian varieties, exists for any reduced scheme over a perfect field.

I have a counterexample for non-reduced schemes, but how about the perfectness of the base field? So my questions are:

(1) Where does Serre's proof brakes down for non-perfect base fields?

 

(2) Does anyone know a counterexample over non-perfect fields?

NB: Serre does not state the hypothesis (maybe because at that time varieties were always integral over an algebraically closed field), but every author who quotes him states those hypothesis.

It is a result of Serre (Morphismes universels et varietes d'albanese) that the Albanese (abelian) variety, i.e. an initial object for morphisms to (torsors over) abelian varieties, exists for any reduced scheme over a perfect field.

I have a counterexample for non-reduced schemes, but how about the perfectness of the base field? So my questions are:

(1) Where does Serre's proof brakes down for non-perfect base fields?

 

(2) Does anyone know a counterexample over non-perfect fields?

NB: Serre does not state the hypothesis (maybe because at that time varieties were always integral over an algebraically closed field), but every author who quotes him states those hypothesis.

It is a result of Serre (Morphismes universels et varietes d'albanese) that the Albanese (abelian) variety, i.e. an initial object for morphisms to (torsors over) abelian varieties, exists for any reduced scheme over a perfect field.

I have a counterexample for non-reduced schemes, but how about the perfectness of the base field? So my questions are:

(1) Where does Serre's proof brakes down for non-perfect base fields?

(2) Does anyone know a counterexample over non-perfect fields?

NB: Serre does not state the hypothesis (maybe because at that time varieties were always integral over an algebraically closed field), but every author who quotes him states those hypothesis.

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YCor
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Francesco Polizzi
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Thomas Geisser
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