A complete answer to Albanese base change for incomplete varieties

Albanese varieties provide a standard tool in algebraic geometry for converting questions about general varieties into questions about Abelian varieties. A result of Serre provides the existence of an Albanese variety for any geometrically connected and geometrically reduced scheme of finite type over a field, and a result of Grothendieck–Conrad establishes that Albanese varieties are stable under base change of field provided the scheme is, in addition, proper. A result of Raynaud shows that base change can fail for Albanese varieties without this properness hypothesis. In this paper we show that Albanese varieties of geometrically connected and geometrically reduced schemes of finite type over a field are stable under separable field extensions. We also show that the failure of base change in general is explained by the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>/</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> -image for purely inseparable extensions <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>L</mml:mi> <mml:mo>/</mml:mo> <mml:mi>K</mml:mi> </mml:mrow> </mml:math> .