Math. 632. Homework 2
1. Let k=F(t) be a purely transcendental extension of a field Fand Kbe an extension of k
obtained by adjoining a p-th root of t. Compute the Galois group scheme of the extension K/k.
2. Let Fbe a functor from the category of algebras over a ring Rto the category of sets. Assume
Fis representable by an affine scheme of finite type over R. Show that the functor F0defined by
F0(K) = F(K[[T]]) (check that this defines a functor) is representable by a scheme.
3. Show that X= Spec k[X, Y ]/(X , Y 3) is contained in a closed reduced subscheme of A2
kof
the form Spec(k[X, Y ]/(f(X, Y )), where f(x, y ) = 0 is a nonsingular curve. Show that for X=
Spec k[X, Y ]/(X2, Y 2, X Y ) this is not true.
4. Let k/F be a finite extension of fields. consider the functor which assigns to a F-algebra Kthe
group of invertible elements of k⊗FK. Show that this functor is representable by an affine group
scheme. Find it explicitly in the case F=Rand k=C. Show that its real points is the group C∗
and its complex points is the group C∗×C∗. What is its Lie algebra scheme?
5. Give an example of a sheaf of ideals on a scheme Xwhich is not a quasi-coherent sheaf.
6. Prove that assigning to a finitely generated projective module Mover a domain Athe vector
bundle V(M) = Spec S(M∗)) establishes a one-to-one correspondence between vector bundles
on SpecAand finitely generated projective A-modules. Show that this defines an isomorphism of
categories of projective bundles over Spec A(a subcategory of the category of schemes over SpecA)
and the category of finitely generated projective modules (a full subcategory of the category of
modules over A).
7. Let Xbe the open subscheme of An+1
R= Spec R[T0, . . . , Tn] whose complement is the closed
point (T1, . . . , Tn). Show that Xadmits a morphism Pn
Rand an open embedding (as Pn-schemes)
into a line bundle over Pn(may do it in the case n= 1).
8. Let Sbe a scheme and X=An
S→Sbe the affine space over S. Show that the canonical
homomorphism Pic S→Pic Xis an isomorphism.
9. Give an example of a morphism X→Ssuch that it is locally isomorphic to a vector bundle
over Sbut not a vector bundle (i.e the transition isomorhisms are not linear).
10. Let Kbe a finite extension of Qand Obe a normal subring of Kwith fraction field K(a
maximal order in K). Show that Pic(O) is a finite group and each element can be represented by
a fractional ideal in K(an O-submodule of K). Show that Pic Ois trivial when K=Qor Q(i)
and give an example when it is not trivial.
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