''Algebraic Geometry'' is an
algebraic geometry
Algebraic geometry is a branch of mathematics which uses abstract algebraic techniques, mainly from commutative algebra, to solve geometry, geometrical problems. Classically, it studies zero of a function, zeros of multivariate polynomials; th ...
textbook
A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions, but also of learners ( ...
written by
Robin Hartshorne
__NOTOC__
Robin Cope Hartshorne ( ; born March 15, 1938) is an American mathematician who is known for his work in algebraic geometry.
Career
Hartshorne was a Putnam Fellow in Fall 1958 while he was an undergraduate at Harvard University (under ...
and published by
Springer-Verlag
Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
Originally founded in 1842 in ...
in 1977.
Importance
It was the first extended treatment of
scheme theory
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations and define the same algebraic variety but different s ...
written as a text intended to be accessible to graduate students, and is considered to be the standard reference.
[ ''...has endured as the best one-volume treatment of this essential set of tools. Everyone in algebraic geometry eventually studies this book''.] This book was cited when Hartshorne was awarded the
Leroy P. Steele Prize for mathematical exposition in 1979.
Contents
The first chapter, titled "Varieties", deals with the classical algebraic geometry of
varieties over
algebraically closed field
In mathematics, a field is algebraically closed if every non-constant polynomial in (the univariate polynomial ring with coefficients in ) has a root in . In other words, a field is algebraically closed if the fundamental theorem of algebra ...
s. This chapter uses many classical results in
commutative algebra
Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideal (ring theory), ideals, and module (mathematics), modules over such rings. Both algebraic geometry and algebraic number theo ...
, including
Hilbert's Nullstellensatz
In mathematics, Hilbert's Nullstellensatz (German for "theorem of zeros", or more literally, "zero-locus-theorem") is a theorem that establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic ge ...
, with the books by Atiyah–Macdonald, Matsumura, and Zariski–Samuel as usual references. The second and the third chapters, "Schemes" and "Cohomology", form the technical heart of the book. The last two chapters, "Curves" and "Surfaces", respectively explore the geometry of 1- and 2-dimensional objects, using the tools developed in the chapters 2 and 3.
Notes
References
*
* {{citation, last=Shatz, first=Stephen S., title=Review: Robin Hartshorne, ''Algebraic geometry'', journal=
Bull. Amer. Math. Soc. (N.S.), volume=1, issue=3, year=1979, pages=553–560, url=http://projecteuclid.org/euclid.bams/1183544340, doi = 10.1090/S0273-0979-1979-14618-4, doi-access=free
Graduate Texts in Mathematics
1977 non-fiction books
Algebraic geometry
Mathematics textbooks
Monographs