| Wednesday January 11 | |
| 9:00 | Milan Lopuhaä |
| 10:00 | Joey van der Leer Duran |
| 11:00 | Joost Nuiten |
| Friday January 13 | |
| 9:00 | Maxim Mornev |
| 10:00 | Johan Commelin |
| 11:00 | Maarten Derickx |
| 13:00 | Arie Blom |
| 14:00 | Milo Boogaard |
| Monday January 23 | |
| 9:00 | Valentin Zakharevich |
| 10:00 | Simen Bruinsma (afgemeld) |
| 11:00 | Joachim de Ronde |
| 12:00 | Daniël van Dijk |
| Tuesday January 24(Groningen) | |
| 14:45 | Paul Helminck |
| Thursday January 26 | |
| 16:00 | Arie Blom (herkansing) |
| Friday February 10 | |
| 10:00 | Simen Bruinsma |
| 11:00 | Daniël van Dijk (herkansing) |
For next time: read II.2 of Hartshorne until the rings get graded, that is, up to example 2.3.6, to see if you recognise what I have told. And read II.1, and try to do some exercises (this is very important).
For next time: read the rest of II.2. Describe all points of Spec(Z) and of Spec(Z[x]) (the affine line over Z). For example, the maximal ideals can be generated by 2 elements, the prime ideals p with V(p) of dimension one by 1 element, and there is a unique generic point: the zero ideal. Also think in the same way of Spec(k[x]) for k a field. Try to do exercise II.2.3.
For next time: read ahead in section II.3, and read somewhere (Lang's algebra?) about the tensor product of modules and rings because we need this for the fibered product, and about Yoneda's lemma.
A morphism of schemes f: X --> Y is locally of finite presentation if for every x in X there are open affine neighborhoods U of x and V of fx with fU contained in V and O(U) of finite presentation as O(V)-algebra (it is given by finitely many generators and finitely many relations).
A morphism of schemes f: X --> Y is called of finite presentation if for every y in Y there is an affine open neighborhood V of y such that f^{-1}V can be covered with finitely many open affines U_i such that (1) for every i O(U_i) is finitely presented as O(V)-algebra, and (2) for every i and j the intersection of U_i and U_j is quasi-compact.
In this situation, each U_i\cap U_j can be covered with finitely many open affines U_{i,j,k}. The morphism f^{-1} V --> V can be obtained from the collection of morphisms O(U_i) --> O(U_{i,j,k}) of O(V)-algebras by glueing (see Lemma 19.14.1 of [Stacks]).
For X a scheme, U and V open affines and x in U and V, there is an open affine W contained in U and in V and containing x, such that W is principal open in U and in V.
Open and closed immersions had already been discussed, so we skipped them. The notion of reduced induced scheme structure on a closed subset Y of a scheme X was very briefly discussed but is supposed to be read for next week.
We started with the notion of fibred product in a category C, calling them Cartesian diagrams. Fibered products, if they exist, are unique up to unique isomorphism. Existence and description in (Sets) and (Top) were given. For manifolds there are problems, but not if f: X --> S is a submersion. Existence of all fibred products in (Sch) (Theorem II3.3) was discussed, and the proof only in the case where X, Y and S are affine (Spec of the tensor product of the rings). The tensor product of B=A[{x_i : i in I}]/({f_j : j in J}) over A with C is C[{x_i : i in I}]/({f_j : j in J}). In general, XxSY is covered by open affines of the form UxWV with U, V and W open affines in X, Y and S, with fU and gV contained in W. It is a good idea to read the proof of Theorem II3.3, but as the fibred product is characterised by a universal property, one can work with it just knowing that it exists.
I did Exercise II.3.10. The steps were the following. Replace Y by an open affine U containing y, and X by f^{-1}V; one shows that that square is Cartesian. Then one base changes U <-- Spec(O_{Y,y}), and one replaces f^{-1}U by an open affine V containing x. Finally, one does the base change by Spec(O_{Y,y}) <-- Spec(\kappa(y)). One uses that for A-->B in (Ring) and S a multiplicative subset the square with S^{-1}A and S^{-1}B is co-Cartesian. And similarly for A-->B and A-->A/I and B/IB. ETc...
It was shown that the underlying set set(X x_S Y) is the set of (x,s,y,p) with x in X, s in S and y in Y with fx=gy, and p a prime ideal of the tensor product of \kappa(x) and \kappa(y) over \kappa(s). A very funny example was discussed: the tensor product of C(x) and C(y) over C. The tensor product is the localisation of C[x,y] by the multiplicative system of fg with f and g nonzero, f in C[x] and g in C[y]. As a consequence, the set of the fibered product is the generic point (zero ideal) and the generic points of all irreducible closed curves in Spec(C[x,y]) that are not fibers for one of the two projections. The ring is a Dedekind domain.... It is not of finite type over C.
I did exercise II.3.4. Or rather, I planned to do it and I skipped it because of time... But I did exercise II.3.8, even a bit more general. Let X be an integral scheme, \eta its generic point and K the local ring of O_X at \eta: the function field of X. Then for all non-empty open U in X, O(U) is a subring of K, so that there is never ambiguity about maps between the O(U). Let K-->L be an algebraic extension of fields. Then for each affine open U, non-empty, we let O(U) be the integral closure of O(U) in L. It was shown that the Spec(O(U)) can be glued, naturally. Consider a U and and a V. Cover the intersection with open affines D_U(f)=D_V(g), and use that integral closure commutes with localisation. As an application: for k a field take X the projective line over k (well, once we have it), and L a finite extension of the function field k(x), for example k(x)[y]/(f) with f=y^n+f_{n-1}y^{n-1}+\cdots+f_0 irreducible. Then we obtain projective curves.
We read Chapter 4 in [Stacks] up to 4.3. I mentioned the example of the category Open(X) of open subsets of X in (Top). A contravariant functor from Open(X) to (Sets) is a presheaf of Sets on X. This provides many examples of categories, functors and morphisms of functors. The notion of representable functors was discussed, and Yoneda's lemma. The point of this is that giving a morphism from X to Y in a category C, is equivalent with giving, for all T in C, a map from X(T) to Y(T), functorial in T.
Example 1. Let X be the spec of Z[x,y]/(x^6+y^6-z^6), and let Y be the spec of Z[x,y]/(x^2+y^2-z^2). Then for A a ring X(A) is the set of triples (a,b,c) of elements of A such that a^6+b^6=c^6, and Y(A) is the set of triples (a,b,c) of elements of A such that a^2+b^2=c^2. Hence a morphism of schemes from X to Y is given by sending, for all A, (a,b,c) to (a^3,b^3,c^3).
Example 2. Let n be a positive integer, and GL_n the functor from (Ring) to (Grp) given by sending A to GL_n(A), the automorphism group of the free A-module A^n, or, if you want, the group of invertible n by n matrices with coefficients in A. Then the functor, still denoted GL_n, gotten by composing GL_n with the forgetful functor from (Grp) to (Set) is represented by the spectrum of the ring H, with H the polynomial ring over Z with variables x_{i,j} and y_{i,j} (i and j in {1,...,n}), divided by the n^2 equations given by the matrix identity xy=1. Then the spectrum of H\otimes H represents the functor sending A to GL_n(A)xGL_n(A), and therefore the multiplication maps from GL_n(A)xGL_n(A) to GL_n(A) for varying A correspond to a morphism of schemes from Spec(H\otimes H) to Spec(H), that is, to a ring morphism from H to H\otimes H. One can write out such morphisms, but it is easier to think about all this in terms of the group structures on all GL_n(A), hence in terms of the functor GL_n itself.
The question was asked: what is P^n(A)? Please think about this for next time.
For A an arbitrary ring and M an A-module, we showed that Z-gradings of M are in bijection with actions of the multiplicative group Gm,A on M. See the scanned notes for more details.
We considered the case S=Z[x0,...,xn], giving Pn. We proved, using Nakayama's lemma, that Pn is universally closed over Spec(Z).
For more details see the scanned notes.
Suggested exercises: [H] II.4.1 (do II.3.5 first), II.4.2, II.4.8.
The notion of quasi-coherent O-modules on schemes was defined. The category of them is denoted QCoh(O). On an affine scheme X these are the O-modules attached to O(X)-modules; M gives the sheaf M~; the functors M|-->M~ and F|-->F(X) form an equivalence of categories (X is still affine!) between O(X)-mod and QCoh(O), respecting many structures on both sides.
For f:X-->Y in (Top) we discussed pushforward of sheaves, and for ringed spaces pushforward of O-modules. Then pullback of presheaves (hence also colimits and limits), the sheaf associated to a presheaf and finally pullback of sheaves. Adjointness between pushforward and pullback was stated.
For more details see the scanned notes.
Suggested exercises: [H] II.1.14, II.1.17, II.1.18, II.1.19.
Suggested exercises: II.5.1., II.5.2, II.5.3, II.5.6.a--c, II.5.7.
We described the functor of points of projective spaces in the category (LRS), now that we have locally free modules of rank one at our disposal. Finally, the Picard group of a ringed space was defined.
Suggested exercises: II.5.8, II.5.16, II.7.2, II.7.3.
The notes also contain the beginning of lecture 12: some examples of morphisms to Pn. Please read this before the lecture, and try (again?) exercises II.7.2, II.7.3.
For next week, read thoroughly sections III.1 and III.2 of [H], included all that was not treated in the lecture.