Math 256 Homework Set 7
1. (a) Find an example of a morphism X Y and a point z X Y X such that
p1 (z) = p2 (z), but z is not on the diagonal.
(b) Prove that in the above situation, the point z is not in the closure of the diagonal.
(c) Let X be a scheme.
Math 256 Homework Set 3
On this homework, assume that all (pre)sheaves take values in the category of sets, abelian
groups, or rings. More generally everything will hold for sheaves with values in a category
K which has all projective limits and ltered in
Math 256 Homework Set 6
1. (a) Show that a nite disjoint union of ane schemes X = X1 Xn is ane and
describe its coordinate ring in terms of those of the Xi .
(b) Show that an innite disjoint union of (non-empty) ane schemes is not ane in
2. (a) L
Math 256 Homework Set 2
On this homework set k always stands for an algebraically closed eld.
1. Prove that every regular function dened on the whole of projective space Pn (k) is
constant. For this you may assume the theorem, which we will prove later, t
Math 256 Homework Set 4
1. Verify in detail the gluing construction (EGA 0, 4.1.7) for ringed spaces, as follows.
(a) Suppose given a collection of sets X , and for every two indices , a subset V X
and a bijection : V V , satisfying the gluing conditions:
Math 256 Homework Set 5
1. Let X be a scheme. By denition (EGA I, 4.1.3), a subscheme of X is a closed
subscheme of an open subscheme; and by (EGA I, 4.1.6), any open subscheme of a closed
subscheme is a subscheme. However, it is not true in general that
NOTES ON DERIVED CATEGORIES AND DERIVED FUNCTORS
You might want to consult these general references for more information:
1. R. Hartshorne, Residues and Duality, Springer Lecture Notes 20 (1966), is a standard
2. C. Weibe
NOTES ON SHEAF COHOMOLOGY FOR SCHEMES
1. Cech resolutions
Let M be a sheaf of abelian groups on a topological space X. Given U X open, we
write jU : U X for the inclusion. Then we have a canonical functorial sheaf homomorphism
M (jU ) jU M
Math 256 Homework Set 8
1. Let U0 , . . . , Un be the standard covering of Pn by open anes. Describe the inclusion
ji : An Ui Pn in terms of the the functor on schemes over k represented by Pn . What
line bundle L on An and n + 1 global sections g
Math 256 Homework Set 1
1. In class we discussed the parametrization of the curve C = V (y 2 x3 ) via the morphism
: A1 C sending the point (t) to (t2 , t3 ), and showed that is bijective but not an
isomorphism. Now prove that the subring k[t2 , t3 ] of