Hartshorne's Algebraic Geometry
Encyclopedia
Algebraic Geometry is an influential algebraic geometry
textbook
written by Robin Hartshorne
and published by Springer-Verlag in 1977. It was the first extended treatment of scheme theory written as a text intended to be accessible to graduate students.
The first chapter, titled "Varieties", deals with the classical algebraic geometry of varieties
over algebraically closed fields. This chapter uses many classical results in commutative algebra
, including Hilbert's Nullstellensatz
, with the books by Atiyah–Macdonald, Matsumura, and Zariski–Samuel as usual references. The second and the third chapters, "Schemes" and "Cohomology", form a technical heart of the book. The last two chapters, "Curves" and "Surfaces", respectively explore the geometry of 1-dimensional and 2-dimensional objects, using the tools developed in the Chapters 2 and 3.
Algebraic geometry
Algebraic geometry is a branch of mathematics which combines techniques of abstract algebra, especially commutative algebra, with the language and the problems of geometry. It occupies a central place in modern mathematics and has multiple conceptual connections with such diverse fields as complex...
textbook
Textbook
A textbook or coursebook is a manual of instruction in any branch of study. Textbooks are produced according to the demands of educational institutions...
written by Robin Hartshorne
Robin Hartshorne
Robin Cope Hartshorne is an American mathematician. Hartshorne is an algebraic geometer who studied with Zariski, Mumford, J.-P. Serre and Grothendieck....
and published by Springer-Verlag in 1977. It was the first extended treatment of scheme theory written as a text intended to be accessible to graduate students.
The first chapter, titled "Varieties", deals with the classical algebraic geometry of varieties
Algebraic variety
In mathematics, an algebraic variety is the set of solutions of a system of polynomial equations. Algebraic varieties are one of the central objects of study in algebraic geometry...
over algebraically closed fields. This chapter uses many classical results in commutative algebra
Commutative algebra
Commutative algebra is the branch of abstract algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra...
, including Hilbert's Nullstellensatz
Hilbert's Nullstellensatz
Hilbert's Nullstellensatz is a theorem which establishes a fundamental relationship between geometry and algebra. This relationship is the basis of algebraic geometry, an important branch of mathematics. It relates algebraic sets to ideals in polynomial rings over algebraically closed fields...
, with the books by Atiyah–Macdonald, Matsumura, and Zariski–Samuel as usual references. The second and the third chapters, "Schemes" and "Cohomology", form a technical heart of the book. The last two chapters, "Curves" and "Surfaces", respectively explore the geometry of 1-dimensional and 2-dimensional objects, using the tools developed in the Chapters 2 and 3.