14. Further Reading

Back

In this course, we have associated an affine algebraic variety to any affine algebra

over a field k. For many reasons, for example, in order to be able to study the reduction

of varieties to characteristic p _= 0, Grothendieck realized that it is important to

attach a geometric object to every commutative ring. Unfortunately, A _→ specmA

is not functorial in this generality:if α: A → B is a homomorphism of rings, then

α−1(m) for m maximal need not be maximal — consider for example the inclusion

Z 8→ Q. Thus he was forced to replace specm(A) with spec(A), the set of all prime

ideals in A. He then attaches an affine scheme Spec(A) to each ring A, and defines a

scheme to be a locally ringed space that admits an open covering by affine schemes.

There is a natural functor V _→ V ∗ from the category of varieties over k to the

category of absolutely reduced schemes of finite-type over k, which is an equivalence of

categories. To construct V ∗ from V , one only has to add one point for each irreducible

closed subvariety of V. Then U _→ U∗ is a bijection from the set of open subsets of

V to the set of open subsets of V ∗. Moreover, Γ(U∗,OV ∗) = Γ(U,OV ) for each open

subset U of V . Therefore the topologies and sheaves on V and V ∗ are the same —

only the underlying sets differ.

Every aspiring algebraic and (especially) arithmetic geometer needs to learn the

basic theory of schemes, and for this I recommend reading Chapters II and III of

Hartshorne 1997.

Among the books listed below, I especially recommend Shafarevich 1994 — it is

very easy to read, and is generally more elementary than these notes, but covers more

ground (being much longer).

Commutative Algebra

Atiyah, M.F and MacDonald, I.G., Introduction to Commutative Algebra, Addison-

Wesley 1969. This is the most useful short text. It extracts the essence of a good

part of Bourbaki 1961–83.

Bourbaki, N., Alg`ebre Commutative, Chap. 1–7, Hermann, 1961–65; Chap 8–9, Masson,

1983. Very clearly written, but it is a reference book, not a text book.

Eisenbud, D., Commutative Algebra, Springer, 1995. The emphasis is on motivation.

Nagata, M., Local Rings, Wiley, 1962. Contains much important material, but it is

concise to the point of being almost unreadable.

Reid, M., Undergraduate Commutative Algebra, Cambridge 1995. According to the author,

it covers roughly the same material as Chapters 1–8 of Atiyah and MacDonald

1969, but is cheaper, has more pictures, and is considerably more opinionated. (However,

Chapters 10 and 11 of Atiyah and MacDonald 1969 contain crucial material.)

Serre:A lg`ebre Locale, Multiplicit´es, Lecture Notes in Math. 11, Springer, 1957/58

(third edition 1975).

Zariski, O., and Samuel, P., Commutative Algebra, Vol. I 1958, Vol II 1960, van Nostrand.

Very detailed and well organized.

Elementary Algebraic Geometry

Reid, M., Undergraduate Algebraic Geometry. A brief, elementary introduction. The

final section contains an interesting, but idiosyncratic, account of algebraic geometry

in the twentieth century.

154 Algebraic Geometry: 14. Further Reading

Abhyankar, S., Algebraic Geometry for Scientists and Engineers, AMS, 1990. Mainly

curves, from a very explicit and down-to-earth point of view.

Computational Algebraic Geometry

Cox, D., Little, J., O’Shea, D., Ideals, Varieties, and Algorithms, Springer, 1992. This

gives an algorithmic approach to algebraic geometry, which makes everything very

down-to-earth and computational, but the cost is that the book doesn’t get very far

in 500pp.

Subvarieties of Projective Space

Shafarevich, I., Basic Algebraic Geometry, Book 1, Springer, 1994. Very easy to read.

Harris, Joe:A lgebraic Geometry: A first course, Springer, 1992. The emphasis is on

examples.

Algebraic Geometry over the Complex Numbers

Griffiths, P., and Harris, J., Principles of Algebraic Geometry, Wiley, 1978. A comprehensive

study of subvarieties of complex projective space using heavily analytic

methods.

Mumford, D., Algebraic Geometry I:C omplex Projective Varieties. The approach is

mainly algebraic, but the complex topology is exploited at crucial points.

Shafarevich, I., Basic Algebraic Geometry, Book 3, Springer, 1994.

Abstract Algebraic Varieties

Dieudonn´e, J., Cours de G´eometrie Alg´ebrique, 2, PUF, 1974. A brief introduction to

abstract algebraic varieties over algebraically closed fields.

Kempf, G., Algebraic Varieties, Cambridge, 1993. Similar approach to these notes, but

is more concisely written, and includes two sections on the cohomology of coherent

sheaves.

Kunz, E., Introduction to Commutative Algebra and Algebraic Geometry, Birkha¨user,

1985. Similar approach to these notes, but includes more commutative algebra and

has a long chapter discussing how many equations it takes to describe an algebraic

variety.

Mumford, D. Introduction to Algebraic Geometry, Harvard notes, 1966. Notes of a

course written (as I recall) by W. Waterhouse. Apart from the original treatise

(Grothendieck and Dieudonn´e 1960–67), this was the first place one could learn the

new approach to algebraic geometry. The first chapter is on varieties, and last two

on schemes.

Mumford, David:The Red Book of Varieties and Schemes, Lecture Notes in Math.

1358, Springer, 1988. Reprint of Mumford 1966.

Schemes

Eisenbud, D., and Harris, J., Schemes:the language of modern algebraic geometry,

Wadsworth, 1992. A brief elementary introduction to scheme theory.

Grothendieck, A., and Dieudonn´e, J., El´ements de G´eom´etrie Alg´ebrique. Publ. Math.

IHES 1960–1967. This was intended to cover everything in algebraic geometry in 13

massive books, that is, it was supposed to do for algebraic geometry what Euclid’s

“Elements” did for geometry. Unlike the earlier Elements, it was abandoned after 4

books. It is an extremely useful reference.

Algebraic Geometry: 14. Further Reading 155

Hartshorne, R., Algebraic Geometry, Springer 1977. Chapters II and III give an excellent

account of scheme theory and cohomology, so good in fact, that no one seems willing

to write a competitor. The first chapter on varieties is very sketchy.

Iitaka, S. Algebraic Geometry:an introduction to birational geometry of algebraic varieties,

Springer, 1982. Not as well-written as Hartshorne 1977, but it is more elementary,

and it covers some topics that Hartshorne doesn’t.

Shafarevich, I., Basic Algebraic Geometry, Book 2, Springer, 1994. A brief introduction

to schemes and abstract varieties.

History

Dieudonn´e, J., History of Algebraic Geometry, Wadsworth, 1985.

Of Historical Interest

Hodge, W., and Pedoe, D., Methods of Algebraic Geometry, Cambridge, 1947–54.

Lang, S., Introduction to Algebraic Geometry, Interscience, 1958. An introduction to

Weil 1946.

Weil, A., Foundations of Algebraic Geometry, AMS, 1946; Revised edition 1962. This is

whereWeil laid the foundations for his work on abelian varieties and jacobian varieties

over arbitrary fields, and his proof of the analogue of the Riemann hypothesis for

curves and abelian varieties. Unfortunately, not only does its language differ from

the current language of algebraic geometry, but it is incompatible with it.

There is also a recent book by Kenji Ueno, which I haven’t seen.

J.S. Milne, Mathematics Department, University of Michigan, Ann Arbor, MI 48109.

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