Computational Algebraic Geometry
The aim of this book is to present an understandable introduction to classical questions of commutative algebra and algebraic geometry using new computational tools (computer algebra package Macaulay 2). The book starts with basics of commutative algebra, followed by the definition and simple properties of projective spaces and projective varieties, graded rings and modules and the Hilbert function and series. Free resolutions and regular sequences of modules are introduced and Groebner bases and syzygies are used. The next section is devoted to algebraic topology and combinatorics, namely simplicial complexes and homology and related algebraic theory (localization, functors, and tensor products with applications to geometry of points in projective space). More facts on chain complexes are presented in the second part of the book (derived functors, Tor and Ext functors, the Hilbert syzygy theorem being an easy exercise). In the last part, a quick introduction to sheaves, Čech cohomology and divisors on algebraic curves is given. There is also a description of the Riemann-Roch theorem and its applications. More advanced topics (projective dimensions of modules, Cohen-Macaulay modules) are also studied. In the appendices, basic facts from algebra and complex functions theory are summarized. The author presents the book as an advertisement for other, more advanced texts. It is a very good introduction to this circle of ideas and it will undoubtedly attract the interest of students to the field.