Algebraic Geometry I: Schemes, With Examples And Exercises (Vieweg Advanced Lectures In Mathematics)

By Ulrich GortzUlrich G. RtzTorsten Wedhorn, et al.

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 .

• Format: Paperback
• Language: English
• ISBN: 3834806765
• ISBN13: 9783834806765
• Release Date: June 2010
• Publisher: Vieweg+teubner Verlag
• Length: 615 Pages
• Weight: 2.16 lbs.
• Dimensions: 1.3" x 6.7" x 9.6"