André Weil completes in 1944, during his exile in São Paulo (Brazil) after fleeing occupied France, the manuscript of Foundations of Algebraic Geometry, published in 1946 by the American Mathematical Society. The work develops for the first time a rigorous algebraic geometry over fields of arbitrary characteristic, introducing the concept of an abstract variety — built via an atlas of charts, with no need to embed it in projective space — and a carefully grounded local intersection theory. Weil was compelled to develop this theoretical framework to give rigor to his own proof of the Riemann hypothesis for curves over finite fields, announced in a 1940 note. The work, considered for years one of the cornerstones of modern algebraic geometry, would be largely superseded the following decade by the development of Alexander Grothendieck's scheme theory, which would offer a more flexible and powerful framework for the same problems, including the complete proof of the Weil conjectures formulated in 1949.