Download Exercises. VII A- Let A be a ring and L a locally free A

Exercises. VII
A- Let A be a ring and L a locally free A-module of rank one (i.e.,
there exists f1 , . . . , fr ∈ A generating the ideal A and such that Lf '
Af (as an Af -module)). Consider the ring
A[L] = A ⊕ L ⊕ (L ⊗ L) ⊕ · · · ⊕ (L⊗n ) ⊕ . . .
with multiplication induced by (x1 ⊗ · · · ⊗ xm ).(y1 ⊗ · · · ⊗ yn ) = x1 ⊗
· · · ⊗ xm ⊗ y1 ⊗ · · · ⊗ yn . Show that A[L] is a graded commutative
ring. (Hint: use the fact that L is locally of rank one.) Show that
Proj(A[L]) is canonically isomorphic to Spec(A). (Hint: Remark first
that the question is local over Spec(A) to reduce to the case where L
is free of rank one.)
B- Let k be a field. We denote by Pn the scheme Proj(k[t0 , . . . , tn ])
where k[t0 , . . . , tn ] is graded by the degree of the monomials (i.e., deg(ti ) =
1). We denote by OPn (r) the quasi-coherent module associated to
the graded module k[t0 , . . . , tn ](r). Show that OPn (r) is locally free
of rank one. Show that there is a natural isomorphism of k-vector
spaces k[t0 , . . . , tn ]r → OPn (r)(Pn ). Show that the obvious morphism
(OPn )n+1 → OPn (1)
P
which sends (a0 , . . . , an ) to ni=0 ai ti is surjective (we say that OPn (1)
is generated by its global sections). Deduce that for r ≥ 0, OPn (r) is
generated by its global sections.
C- Let A = ⊕n≥0 An be an N-graded ring. Show that there exists a
natural morphism of schemes
p : Spec(A)\Z(A+ ) → Proj(A)
such that for all f ∈ A+ homogenous, we have a cartesian square
/
Spec(Af )
Spec(A)
p
/
Spec((Af )0 )
Proj(A).
Show that p is surjective.
Assume that the ideal A+ is generated by A1 . Let x be a point of
Proj(A). Show that the fiber p−1 (x) is isomorphic to Spec(κ(x)[t, t−1 ]).
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