(c)Karen E. Smith 2018 UM Math Dept
licensed under a Creative Commons
By-NC-SA 4.0 International License.
Second Worksheet on the Localization
(1) Fix a ring R, multiplicative set U⊂Rand ideal I⊂R. Let Udenote the image of Uunder the
natural quotient map R→R/I.
(a) Describe the canonical isomorphism U−1R/IU−1R∼
=U−1R/I both explicitly (saying what
elements correspond) and using the universal properties.
(b) For the natural quotient map R→R/I, review the induced map of Spec R/I →Spec R.
Why is it injective and what is the image?
(c) For the natural localization map R→U−1R, review the induced map of Spec U−1R→Spec
R. Why is it injective and what is the image?
(d) For the natural localization map R→RP, review the induced map of Spec RP→Spec R.
Why is it injective and what is the image?
(e) For the natural localization map R→R[1
f], explain why the induced map of spectra can be
thought of as the inclusion of a basic open set D(f)⊂Spec R.
(f) Consider the natural map R→U−1R/I U −1R, which can be viewed either as a localiza-
tion followed by a quotient map or vice versa. Discuss the induced map of Spectra Spec
U−1R/IU−1R→Spec R? Is it injective? What is the image?
(g) For P∈Spec R, define the residue field of Pto be k(P) = RP/P RP. There is a canonical
map R→k(P) (why?). Describe the induced map of Spectra.
Theorem: Let f:Y→Xbe the map of spectra induced by the ring homomophisms Rφ
→S.
For each P∈X, the fiber f−1(P)⊂Ycan be identified with the space Spec k(P)⊗RS.
(2) Fibers. Let f:Y→Xbe the map of spectra induced by each of the ring homomophisms
Rφ
→Sbelow. Use the Theorem to (or otherwise) describe the fibers over each point. Note that
there are different kinds of points to consider.
(a) Z,→Q.
(b) C[x],→C[x, y].
(c) Z,→Z[x].
(d) R[x],→C[x].
(3) Let f:Y→Xbe the map of spectra induced by a ring homomophisms Rφ
→S.
(a) Show that for any P∈X, the closure of Pin Spec Ris V(P).
(b) Show that for any P∈X, Spec S/P S can be identified with {Q∈Spec S|f(Q)⊃P}.
(c) For any P∈X, let Ube the image of R\Punder the map Rφ
→S. Prove that Uis a
multiplicative set in S. Now show there is a natural bijection between Spec U−1Sand the
set {Q∈Spec S|f(Q)⊂P}.
(d) Show that for any P∈X, the fiber over Pis homeomorphic to Spec U−1S/P U−1S(with U
as in (c)). That is, the fiber over Pis Spec S⊗Rk(P) where k(P) is the residue field of P.
(4) For the map Z,→Z[x, y]/hy2−x3−5x+ 30i, discuss the induced map of Spectra, including
fibers. Are any of the fibers dramatically different from other? In what ways? What is the
typical fiber like? What about the ”generic fiber” (the one over the generic, or dense point of
Spec Z.)
