Foundations of algebraic geometry
Encyclopedia
Foundations of algebraic geometry is a book by that develops algebraic geometry over fields of any characteristic. In particular it gives a careful treatment of intersection theory
by defining the local intersection multiplicity of two subvarieties.
Weil was motivated by the need for a rigorous theory of correspondences on algebraic curves in positive characteristic, which he used in his proof of the Riemann hypothesis for curves over a finite field.
Weil introduced abstract rather than projective varieties partly so that he could construct the Jacobian of a curve. (It was not known at the time that Jacobians are always projective varieties.) It was some time before anyone found any examples of complete abstract varieties that are not projective.
In the 1950s Weil's work was one of several competing attempts to provide satisfactory foundations for algebraic geometry, all of which were superseded by Grothendieck
's development of scheme
s.
Intersection theory
In mathematics, intersection theory is a branch of algebraic geometry, where subvarieties are intersected on an algebraic variety, and of algebraic topology, where intersections are computed within the cohomology ring. The theory for varieties is older, with roots in Bézout's theorem on curves and...
by defining the local intersection multiplicity of two subvarieties.
Weil was motivated by the need for a rigorous theory of correspondences on algebraic curves in positive characteristic, which he used in his proof of the Riemann hypothesis for curves over a finite field.
Weil introduced abstract rather than projective varieties partly so that he could construct the Jacobian of a curve. (It was not known at the time that Jacobians are always projective varieties.) It was some time before anyone found any examples of complete abstract varieties that are not projective.
In the 1950s Weil's work was one of several competing attempts to provide satisfactory foundations for algebraic geometry, all of which were superseded by Grothendieck
Alexander Grothendieck
Alexander Grothendieck is a mathematician and the central figure behind the creation of the modern theory of algebraic geometry. His research program vastly extended the scope of the field, incorporating major elements of commutative algebra, homological algebra, sheaf theory, and category theory...
's development of scheme
Scheme (mathematics)
In mathematics, a scheme is an important concept connecting the fields of algebraic geometry, commutative algebra and number theory. Schemes were introduced by Alexander Grothendieck so as to broaden the notion of algebraic variety; some consider schemes to be the basic object of study of modern...
s.