Riemann–Roch-type theorem

In algebraic geometry, there are various generalizations of the Riemann–Roch theorem; among the most famous is the Grothendieck–Riemann–Roch theorem, which is further generalized by the formulation due to Fulton et al. == Formulation due to Baum, Fulton and MacPherson == Let G ∗ {\displaystyle G_{*}} and A ∗ {\displaystyle A_{*}} be functors on the category C of schemes separated and locally of finite type over the base field k with proper morphisms such that G ∗ ( X ) {\displaystyle G_{*}(X)} is the Grothendieck group of coherent sheaves on X, A ∗ ( X ) {\displaystyle A_{*}(X)} is the rational Chow group of X, for each proper morphism f, G ∗ ( f ) , A ∗ ( f ) {\displaystyle G_{*}(f),A_{*}(f)} are the direct images (or push-forwards) along f.

Source: Wikipedia — Riemann–Roch-type theorem (CC BY-SA 4.0)

Riemann–Roch-type theorem

In algebraic geometry, there are various generalizations of the Riemann–Roch theorem; among the most famous is the Grothendieck–Riemann–Roch theorem, which is further generalized by the formulation due to Fulton et al. == Formulation due to Baum, Fulton and MacPherson == Let G ∗ {\displaystyle G_{*}} and A ∗ {\displaystyle A_{*}} be functors on the category C of schemes separated and locally of finite type over the base field k with proper morphisms such that G ∗ ( X ) {\displaystyle G_{*}(X)} is the Grothendieck group of coherent sheaves on X, A ∗ ( X ) {\displaystyle A_{*}(X)} is the rational Chow group of X, for each proper morphism f, G ∗ ( f ) , A ∗ ( f ) {\displaystyle G_{*}(f),A_{*}(f)} are the direct images (or push-forwards) along f.

Source: Wikipedia "Riemann–Roch-type theorem" · CC BY-SA 4.0

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