| Algebraic Geometry I: Schemes: With Examples and Exercises Corr. 2020 Edition Contributor(s): Görtz, Ulrich (Author), Wedhorn, Torsten (Author) |
|
 |
ISBN: 3658307323 ISBN-13: 9783658307325 Publisher: Springer Spektrum OUR PRICE: $85.49 Product Type: Paperback Published: July 2020 |
| Additional Information |
| BISAC Categories: - Mathematics | Geometry - Algebraic |
| Physical Information: 1.28" H x 6.69" W x 9.61" (2.20 lbs) 626 pages |
| Descriptions, Reviews, Etc. |
| Publisher Description: Algebraic geometry has its origin in the study of systems of polynomial equations f (x, . . ., x )=0, 1 1 n . . . f (x, . . ., x )=0. r 1 n Here the f ? k[X, . . ., X ] are polynomials in n variables with coe?cients in a ?eld k. i 1 n n ThesetofsolutionsisasubsetV(f, . . ., f)ofk . Polynomialequationsareomnipresent 1 r inandoutsidemathematics, andhavebeenstudiedsinceantiquity. Thefocusofalgebraic geometry is studying the geometric structure of their solution sets. n If the polynomials f are linear, then V(f, . . ., f ) is a subvector space of k. Its i 1 r "size" is measured by its dimension and it can be described as the kernel of the linear n r map k ? k, x=(x, . . ., x ) ? (f (x), . . ., f (x)). 1 n 1 r For arbitrary polynomials, V(f, . . ., f ) is in general not a subvector space. To study 1 r it, one uses the close connection of geometry and algebra which is a key property of algebraic geometry, and whose ?rst manifestation is the following: If g = g f +. . . g f 1 1 r r is a linear combination of the f (with coe?cients g ? k[T, . . ., T ]), then we have i i 1 n V(f, . . ., f)= V(g, f, . . ., f ). Thus the set of solutions depends only on the ideal 1 r 1 r a? k[T, . . ., T ] generated by the f . |