Download Abelian varieties by Mumford. PDF

By Mumford.

ISBN-10: 0387112901

ISBN-13: 9780387112909

Show description

Read or Download Abelian varieties PDF

Best algebraic geometry books

The Geometry of Moduli Spaces of Sheaves

Now again in print, this very hot booklet has been up to date to mirror contemporary advances within the thought of semistable coherent sheaves and their moduli areas, which come with moduli areas in optimistic attribute, moduli areas of significant bundles and of complexes, Hilbert schemes of issues on surfaces, derived different types of coherent sheaves, and moduli areas of sheaves on Calabi-Yau threefolds.

Spaces of Homotopy Self-Equivalences: A Survey

This survey covers teams of homotopy self-equivalence sessions of topological areas, and the homotopy kind of areas of homotopy self-equivalences. For manifolds, the total workforce of equivalences and the mapping type workforce are in comparison, as are the corresponding areas. incorporated are tools of calculation, a number of calculations, finite iteration effects, Whitehead torsion and different components.

Galois Theory of Difference Equations

This publication lays the algebraic foundations of a Galois idea of linear distinction equations and indicates its courting to the analytic challenge of discovering meromorphic features asymptotic to formal recommendations of distinction equations. Classically, this latter query was once attacked through Birkhoff and Tritzinsky and the current paintings corrects and tremendously generalizes their contributions.

Additional resources for Abelian varieties

Sample text

Assume that A is universally catenary. Then A/P is universally catenary so we can assume that A is an integral domain and that B contains A. We have that B = A[x1 , . . , xn ]/I where A[x1 , . . , xn ] is a polynomial ring over A and I a prime ideal in A[x1 , . . , xn ]. Let Q be a prime ideal in B. Then we have that Q = R/I, where R is a prime ideal in A[x1 , . . , xn ]. However the ring A[x1 , . . ) that we have ht Q = ht R − ht I. 4) for polynomial rings we have that ht R + td. κ(P ) κ(R) = ht P + td.

To prove the Lemma it suffices to show that the map A /IA ⊗A/I T → T obtained from the left vertical map of the latter diagram is surjective. 12 Reduction to noetherian rings 3 of A/I-modules. Tensorize the last two sequences with A /IA over A/I. We obtain two exact sequences 0 → A /IA ⊗A/I Q → A /IA ⊗A L0 → A /IA ⊗A M → 0 and A /IA ⊗A/I T → A /IA ⊗A L1 → A /IA ⊗A Q → 0 of A /IA -modules, where ethe first is exact since M/IM is a flat A/I-module by assumption. Consequently we obtain a commutative diagram B ⊗A/I T −−−→ B ⊗A L1 −−−→ B ⊗A L0 −−−→ B ⊗A M −−−→ 0         0 −−−→ T −−−→ B ⊗A L1 −−−→ B ⊗A Lo −−−→ B ⊗A M −−−→ 0 of B = A /IA -modules, where the three right vertical maps are isomorphisms.

Assume that F is flat over A. Since BQ is flat over B the functor that sends an AP -module N to BQ ⊗B (N ⊗A F ) is exact. However BQ ⊗B (F ⊗A N ) = FQ ⊗A N = FQ ⊗AP N . Consequently the functor that sends N to FQ ⊗AP N is exact, that is, the AP -module FQ is flat. 6 Flatness 6 with P = ϕ−1 (Q). The functor that sends an A-module N ot the AP -module NP is exact. Consequently the functor that sends N to FQ ⊗AP NP is exact. However, we have that FQ ⊗AP NP = FQ ⊗AP (AP ⊗A N ) = FQ ⊗A N . Hence the functor that sends an A–module N to FQ ⊗A N is exact.

Download PDF sample

Abelian varieties by Mumford.


by Mark
4.2

Rated 4.18 of 5 – based on 37 votes