Download e-book for iPad: Algebraic Geometry 1: From Algebraic Varieties to Schemes by Kenji Ueno

By Kenji Ueno

ISBN-10: 0821808621

ISBN-13: 9780821808627

This can be the 1st of 3 volumes on algebraic geometry. the second one quantity, Algebraic Geometry 2: Sheaves and Cohomology, is out there from the AMS as quantity 197 within the Translations of Mathematical Monographs sequence.

Early within the twentieth century, algebraic geometry underwent an important overhaul, as mathematicians, significantly Zariski, brought a far better emphasis on algebra and rigor into the topic. This was once by means of one other primary swap within the Sixties with Grothendieck's creation of schemes. this day, such a lot algebraic geometers are well-versed within the language of schemes, yet many newbies are nonetheless at the beginning hesitant approximately them. Ueno's ebook offers an inviting creation to the idea, which should still triumph over this kind of obstacle to studying this wealthy topic.

The booklet starts with an outline of the normal idea of algebraic forms. Then, sheaves are brought and studied, utilizing as few necessities as attainable. as soon as sheaf concept has been good understood, the next move is to determine that an affine scheme could be outlined by way of a sheaf over the leading spectrum of a hoop. through learning algebraic forms over a box, Ueno demonstrates how the suggestion of schemes is critical in algebraic geometry.

This first quantity provides a definition of schemes and describes a few of their trouble-free homes. it really is then attainable, with just a little extra paintings, to find their usefulness. additional homes of schemes should be mentioned within the moment quantity.

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Additional info for Algebraic Geometry 1: From Algebraic Varieties to Schemes

Example text

Un ] with n generators, choose α α = max{αij , βij }, then we have uα i uj = 0 for all i = j. Consider the k-algebras i,j k[u1 + · · · + un−1 + un ] ⊆ k[u1 + · · · + un−1 , un ] ⊆ k[u1 , . . , un−1 , un ]. We claim that (1) k[u1 , . . , un−1 , un ] is integral over k[u1 + · · · + un−1 , un ], and (2) k[u1 + · · · + un−1 , un ] is integral over k[u1 + · · · + un−1 + un ]. α For (1), since uα i uj = 0 for all i = j, by the induction hypothesis, we have that k[u1 , . . , un−1 ] is integral over k[u1 + u2 + · · · + un−1 ] and so k[u1 , .

It can be checked directly t hat II ∗ (I ∗ )2 . Thus, I is not a reduction of I ∗ . On the other hand, we know that K1 = (x4 + y 4 , x2 y + xy 2 ) is a reduction of I, also through a direct computation: K1 I 2 = I 3 . We observe that x4 , y 4 are in Γ(x4 + y 4 ) and the term x2 y +xy 2 is a combination of two monomials corresponding to two points on the left hand side of the line through (4, 0) and (0, 4). ” However, the authors are not able to verify this statement. 7, K2 = (x4 +xy 2 , x2 y+y 4 ) is a minimal reduction of I ∗ while K1 is not a reduction of I ∗ .

I be the ideal generated by fi = All the monomials occurring in the fi for all i together generate a monomial ideal, denoted I ∗ . A simple exercise shows that such monomial ideal is independent from the choices of the generating set {fi }. 1). We focus on finding a sufficient condition under which I becomes a reduction of I ∗ . In such case, the integral closure and, if I is m-primary, the Hilbert-Samuel multiplicity of I can be obtained straightforwardly from the graph of I ∗ as stated in the previous paragraph.

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Algebraic Geometry 1: From Algebraic Varieties to Schemes by Kenji Ueno


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