![]() ![]() Weekly problem sets will be posted online. It can be downloaded free of charge at the link above. Plane Algebraic Curves by Gerd Fischer (AMS bookstore)įulton is the book closest to the content of the course that is available.Algebraic Curves by William Fulton (pdf).In addition to the notes, we also recommend the following references: Please let us know if you find any typos. Notes for the course by Michael Artin (errata).We will follow the notes written for the course: It would be reasonable to spend a whole semester on plane The geometry of varieties of any dimension. They provide a good introduction, and as we will see, they govern Cohomology is introduced only for modules (aka quasicoherent sheaves).Theorems may not be stated or proved in their most general form.We work exclusively with quasiprojective varieties over the field of complex numbers.To help make the material accessible, we’ve made some simplifying restrictions: The most important are: Though algebraic geometry is usually taught assuming familiarity with commutative algebra, we won’tĪssume things beyond 18.702 (Algebra II) are known, and we will keep commutative algebra at a minimum. For this, it is essential that you becomeįamiliar with cohomology. Geometry, and you should be able to read some papers in the subject. Chapter III of Hartshorne's Algebraic Geometry is dedicated to the cohomology of coherent sheaves on (noetherian) schemes.When you have completed this course, you will be well prepared for a graduate course in algebraic (For example, they allow their rings to be non-commutative.) The last chapters of Lang's Algebra also cover some homological algebra. The first few chapters of Cartan and Eilenberg's Homological Algebra give a good introduction to the general theory but is strictly more than what is needed for the purposes of algebraic geometry. For this, some background in homological algebra is required unfortunately, homological algebra is not quite within the scope of commutative algebra so even Eisenbud treats it very briefly. ![]() Perhaps the most important piece of technology in modern algebraic geometry is sheaf cohomology. Also worth mentioning is Eisenbud and Harris's Geometry of Schemes, which is a very readable text about the geometric intuition behind the definitions of scheme theory. ![]() Volume II of Shafarevich's Basic Algebraic Geometry also discusses some scheme theory. The theory of affine schemes is already very rich – hence the 800 pages in Eisenbud's Commutative Algebra! For general scheme theory, the standard reference is Chapter II of Hartshorne's Algebraic Geometry, but Vakil's online notes are probably much more readable. A scheme is a space which is locally isomorphic to an affine scheme, and an affine scheme is essentially the same thing as a commutative ring. Modern algebraic geometry begins with the study of schemes, and there it is important to have a thorough understanding of localisation, local rings, and modules over them. Reid's Undergraduate Algebraic Geometry, Chapter I of Hartshorne's Algebraic Geometry and Volume I of Shafarevich's Basic Algebraic Geometry all cover material of this kind. Classical algebraic geometry, in the sense of the study of quasi-projective (irreducible) varieties over an algebraically closed field, can be studied without too much background in commutative algebra (especially if you are willing to ignore dimension theory). That said, it is not necessary to learn all of Eisenbud's Commutative Algebra before starting algebraic geometry. (The structure sheaf $\mathscr_X$ in the topos.) If you are willing to restrict yourself to smooth complex varieties then it is possible to use mainly complex-analytic methods, but otherwise there has to be some input from commutative algebra. Indeed, in a very precise sense, a scheme can be thought of as a generalised local ring. The trouble with algebraic geometry is that it is, in its modern form, essentially just generalised commutative algebra. ![]() Reid's Undergraduate Algebraic Geometry requires very very little commutative algebra if I remember correctly, what it assumes is so basic that it is more or less what Eisenbud assumes in his Commutative Algebra! ![]()
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