Grothendieck–Riemann–Roch theorem

A theorem relating the pushforward of a vector bundle’s Chern character to the pushforward of its Todd-class-adjusted Chern character. It generalizes the Riemann–Roch theorem to morphisms between algebraic varieties.

Chern character: A characteristic class that converts vector-bundle data into additive cohomological or Chow-theoretic data. It is the bundle invariant whose pushforward the theorem relates across the morphism.
Proper morphism: A morphism of schemes that is universally closed, separated, and of finite type. Properness is the standard setting in which the theorem defines pushforwards.
Bernhard Riemann: A German mathematician whose work helped establish the theory of complex functions and Riemann surfaces. His work on surfaces supplied the historical setting for the classical formula generalized here.
Euler characteristic: The alternating sum of dimensions of cohomology groups, or its corresponding topological invariant. Taking the map to a point turns the theorem into a formula for Euler characteristics.
Todd class: A characteristic class constructed from Chern classes that appears in Riemann–Roch formulas. It supplies the geometric correction multiplying the Chern character on the source.
Virtual tangent bundle: A formal difference of vector bundles representing tangent data when an ordinary tangent bundle is unavailable. For smooth maps, its Todd class provides the correction in the pushforward formula.
Gustav Roch: A German mathematician known for the theorem on meromorphic functions now bearing his name with Riemann. The classical Riemann–Roch formula is the starting point for the theorem’s lineage.
Hirzebruch–Riemann–Roch formula: A characteristic-class formula computing the Euler characteristic of a vector bundle on a smooth projective variety. It is a direct specialization of the theorem to the structural map to a point.
Chow group: A group of algebraic cycles modulo rational equivalence, graded by codimension or dimension. The theorem can be formulated as an identity between classes in Chow groups.
Projection formula: An identity allowing a class pulled back from the target to pass through a pushforward. It governs how the theorem’s characteristic-class factors interact with pushforward.

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