Decomposition theorem for semi-simples
... We first show how to deduce the D(Y, C)-version of Theorem 2.1.2 from Theorem 2.1.1. Then we show how the D(Y, C)-version implies formally the D(Y, Q)-version. The reader should have no difficulty in replacing Q with any field of characteristic zero and proving the same result. 2.2. Proof of Theorem ...
... We first show how to deduce the D(Y, C)-version of Theorem 2.1.2 from Theorem 2.1.1. Then we show how the D(Y, C)-version implies formally the D(Y, Q)-version. The reader should have no difficulty in replacing Q with any field of characteristic zero and proving the same result. 2.2. Proof of Theorem ...
Gauss` Theorem Egregium, Gauss-Bonnet etc. We know that for a
... of faces. This shows that the LHS does not depend on the particular way the surface is embedded in R3 and the RHS does not depend on the triangulation: it is a topological invariant of the surface. 4. Classification of flat surfaces Let S be a surface which is locally isometric to the plane. Gauss’ ...
... of faces. This shows that the LHS does not depend on the particular way the surface is embedded in R3 and the RHS does not depend on the triangulation: it is a topological invariant of the surface. 4. Classification of flat surfaces Let S be a surface which is locally isometric to the plane. Gauss’ ...
Riemann–Roch theorem
The Riemann–Roch theorem is an important theorem in mathematics, specifically in complex analysis and algebraic geometry, for the computation of the dimension of the space of meromorphic functions with prescribed zeroes and allowed poles. It relates the complex analysis of a connected compact Riemann surface with the surface's purely topological genus g, in a way that can be carried over into purely algebraic settings.Initially proved as Riemann's inequality by Riemann (1857), the theorem reached its definitive form for Riemann surfaces after work of Riemann's short-lived student Gustav Roch (1865). It was later generalized to algebraic curves, to higher-dimensional varieties and beyond.