By William Fulton

Preface

Third Preface, 2008

This textual content has been out of print for numerous years, with the writer retaining copyrights.

Since I proceed to listen to from younger algebraic geometers who used this as

their first textual content, i'm completely satisfied now to make this version on hand for gratis to anyone

interested. i'm such a lot thankful to Kwankyu Lee for creating a cautious LaTeX version,

which used to be the root of this variation; thank you additionally to Eugene Eisenstein for aid with

the graphics.

As in 1989, i've got controlled to withstand making sweeping alterations. I thank all who

have despatched corrections to prior models, specifically Grzegorz Bobi´nski for the most

recent and thorough checklist. it's inevitable that this conversion has brought some

new mistakes, and that i and destiny readers could be thankful if you happen to will ship any blunders you

find to me at wfulton@umich.edu.

Second Preface, 1989

When this ebook first seemed, there have been few texts to be had to a amateur in modern

algebraic geometry. in view that then many introductory treatises have seemed, including

excellent texts through Shafarevich,Mumford,Hartshorne, Griffiths-Harris, Kunz,

Clemens, Iitaka, Brieskorn-Knörrer, and Arbarello-Cornalba-Griffiths-Harris.

The previous twenty years have additionally obvious a great deal of progress in our understanding

of the subjects lined during this textual content: linear sequence on curves, intersection thought, and

the Riemann-Roch challenge. it's been tempting to rewrite the publication to mirror this

progress, however it doesn't appear attainable to take action with no forsaking its elementary

character and destroying its unique objective: to introduce scholars with a bit algebra

background to a couple of the tips of algebraic geometry and to assist them gain

some appreciation either for algebraic geometry and for origins and purposes of

many of the notions of commutative algebra. If operating during the e-book and its

exercises is helping arrange a reader for any of the texts pointed out above, that would be an

added benefit.

PREFACE

First Preface, 1969

Although algebraic geometry is a hugely built and thriving box of mathematics,

it is notoriously tough for the newbie to make his method into the subject.

There are a number of texts on an undergraduate point that supply an exceptional therapy of

the classical thought of airplane curves, yet those don't organize the scholar adequately

for smooth algebraic geometry. however, so much books with a contemporary approach

demand massive history in algebra and topology, usually the equivalent

of a 12 months or extra of graduate examine. the purpose of those notes is to improve the

theory of algebraic curves from the point of view of recent algebraic geometry, but

without over the top prerequisites.

We have assumed that the reader knows a few uncomplicated homes of rings,

ideals, and polynomials, akin to is usually lined in a one-semester path in modern

algebra; extra commutative algebra is constructed in later sections. Chapter

1 starts off with a precis of the evidence we want from algebra. the remainder of the chapter

is occupied with uncomplicated homes of affine algebraic units; we have now given Zariski’s

proof of the $64000 Nullstellensatz.

The coordinate ring, functionality box, and native earrings of an affine style are studied

in bankruptcy 2. As in any glossy remedy of algebraic geometry, they play a fundamental

role in our instruction. the overall research of affine and projective varieties

is endured in Chapters four and six, yet in simple terms so far as worthwhile for our examine of curves.

Chapter three considers affine aircraft curves. The classical definition of the multiplicity

of some degree on a curve is proven to rely purely at the neighborhood ring of the curve at the

point. The intersection variety of aircraft curves at some extent is characterised via its

properties, and a definition when it comes to a undeniable residue classification ring of an area ring is

shown to have those homes. Bézout’s Theorem and Max Noether’s Fundamental

Theorem are the topic of bankruptcy five. (Anyone acquainted with the cohomology of

projective forms will realize that this cohomology is implicit in our proofs.)

In bankruptcy 7 the nonsingular version of a curve is developed through blowing

up issues, and the correspondence among algebraic functionality fields on one

variable and nonsingular projective curves is validated. within the concluding chapter

the algebraic technique of Chevalley is mixed with the geometric reasoning of

Brill and Noether to turn out the Riemann-Roch Theorem.

These notes are from a path taught to Juniors at Brandeis college in 1967–

68. The path used to be repeated (assuming the entire algebra) to a gaggle of graduate students

during the extensive week on the finish of the Spring semester. we've retained

an crucial characteristic of those classes by way of together with a number of hundred difficulties. The results

of the starred difficulties are used freely within the textual content, whereas the others variety from

exercises to functions and extensions of the theory.

From bankruptcy three on, ok denotes a hard and fast algebraically closed box. each time convenient

(including with no remark a few of the difficulties) we have now assumed ok to

be of attribute 0. The minor changes essential to expand the idea to

arbitrary attribute are mentioned in an appendix.

Thanks are because of Richard Weiss, a pupil within the direction, for sharing the task

of writing the notes. He corrected many error and more suitable the readability of the text.

Professor PaulMonsky supplied numerous important feedback as I taught the course.

“Je n’ai jamais été assez loin pour bien sentir l’application de l’algèbre à los angeles géométrie.

Je n’ai mois element cette manière d’opérer sans voir ce qu’on fait, et il me sembloit que

résoudre un probleme de géométrie par les équations, c’étoit jouer un air en tournant

une manivelle. l. a. most appropriate fois que je trouvai par le calcul que le carré d’un

binôme étoit composé du carré de chacune de ses events, et du double produit de

l’une par l’autre, malgré l. a. justesse de ma multiplication, je n’en voulus rien croire

jusqu’à ce que j’eusse fai los angeles determine. Ce n’étoit pas que je n’eusse un grand goût pour

l’algèbre en n’y considérant que los angeles quantité abstraite; mais appliquée a l’étendue, je

voulois voir l’opération sur les lignes; autrement je n’y comprenois plus rien.”

Les Confessions de J.-J. Rousseau

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**Example text**

Show that the resulting sequence 0 −→ N /P −→ M /P −→ M /N −→ 0 is exact (“Second Noether Isomorphism Theorem”). (c) Let U ⊂ W ⊂ V be vector spaces, with V /U finite-dimensional. Then dimV /U = dimV /W + dimW /U . (d) If J ⊂ I are ideals in a ring R, there is a natural exact sequence of R-modules: 0 −→ I /J −→ R/J −→ R/I −→ 0. (e) If O is a local ring with maximal ideal m, there is a natural exact sequence of O -modules 0 −→ mn /mn+1 −→ O /mn+1 −→ O /mn −→ 0. 30. Then mn /mn+1 is an R-module, and so also a k-module, since k ⊂ R.

For convenience we usually concentrate on Un+1 . Let H∞ = Pn Un+1 = {[x 1 : . . : x n+1 ] | x n+1 = 0}; H∞ is often called the hyperplane at infinity. The correspondence [x 1 : . . : x n : 0] ↔ [x 1 : . . : x n ] shows that H∞ may be identified with Pn−1 . Thus Pn = Un+1 ∪ H∞ is the union of an affine n-space and a set that gives all directions in affine n-space. Examples. (0) P0 (k) is a point. 1 (1) P (k) = {[x : 1] | x ∈ k} ∪ {[1 : 0]}. P1 (k) is the affine line plus one point at infinity.

Show that k[[X ]] is a DVR with uniformizing parameter X . Its quotient field is denoted k((X )). 32. 30. Any z ∈ R then determines a power series λi X i , if λ0 , λ1 , . . 30(b). (a) Show that the map z → λi X i is a one-to-one ring homomorphism of R into k[[X ]]. We often write z = λi t i , and call this the power series expansion of z in terms of t . (b) Show that the homomorphism extends to a homomorphism of K into k((X )), and that the order function on k((X )) restricts to that on K . 24, t = X .