MATH 538

The course will concentrate on the Theory of Schemes, as developed by Grothendieck. The text is "Commutative Algebra and Algebraic Geometry", by Seigfried Bosch; but we will cover many topics differently, and in some cases with very different proofs, than appear in the text. Hartshorne's text, "Algebraic Geometry", covers a lot of material -- but requiring that all schemes and rings be Noetherian. We will not require Noetherian assumptions for most things. (I've occasionally actually needed to work with such schemes.) Bosch's book is a bit pedantic; but it is complete -- everything is proved, mostly in the fullest generality. Hartshorne's book skips many proofs, instead giving you references to other texts (such as Atiya-MacDonald's Cmv Alg, and a text by Matsamura), while Bosch's book is self contained (and very large). (Note: Bosch's book, and Hartshorne's book, are both available on-line to all students, faculty and staff at UR.)

We will start by studying algebraic varieties over a field, stating and completely proving many different forms of Hilbert's Nullstellensatz, starting with Zariski's version of the Nullstellensatz (often called 'Zariski's Lemma'.)

We'll study algebraic varieties over fields only very briefly; since these are of course a special case of schemes of finite type over a field -- which in turn is a special case of schemes over a base scheme.

Of course, we'll also have to cover the theory of sheaves of rings over a topological space; local ringed spaces; and local morphisms of such, in order to define the category of schemes.

At the bottom of this page, you will see a link to Serre's paper "Algebraic Coherent Sheaves". What Serre calls a "sheaf" in that paper is what we call an "etale space" today; Serre's equivalent definition of a sheaf is what we call a sheaf today. Following Grothendieck (I think unwisely in this case), Hartshorne doesn't discuss etale spaces (except in excercises) in his text. In my opinion, the first chapter of Serre's "Algebraic Coherent Sheaves" is one of the cleanest and briefest developments of the subject; you'll perhaps find that first chapter useful, which is why I've placed it on this course home page.

Early in the course, we'll establish that the functor Spec, from the category of rings (=commutative rings with identity) into the category of schemes, is the adjoint functor of the functor that maps each scheme into its ring of global sections. An important theorem is the theorem that the cohomology of a quasicoherent sheaf over an affine scheme vanishes in dimension > 0. This was proved by Serre (using results of Leray) for algebraic varieties, and generalized to schemes by Grothendiek. A much simpler elementary proof was done in 1980 by George Kempf in the Rocky mountain Journal of Mathematics, Vol 10, no 3, Summer 1980; I'll present that proof early in the term. (And I'll use this and the Leray spectral sequence to show how to easily compute the cohomology of a quasicoherent sheaf on any quasicompact quasiseparated scheme -- and also on those that are not quasiseparated.)

(Also, we'll be devloping some commutative algebra early in the term -- this is necessary for algebraic geometry. We'll study Nakayama's Lemma, the radical and Jacobson radical of a ring, etc.)