math 216: foundations of algebraic geometry - Stanford University

24 Math 216: Foundations of Algebraic Geometry
warning: If p is a prime ideal, then Ap means you’re allowed to divide by elements
not in p. However, if f ∈ A, Af means you’re allowed to divide by f. This can be
confusing. For example, if (f) is a prime ideal, then Af = A (f).)
Warning: sometimes localization is first introduced in the special case where A
is an integral domain and 0 /∈ S. In that case, A ↩→ S −1 A, but this isn’t always true,
as shown by the following exercise. (But we will see that noninjective localizations
needn’t be pathological, and we can sometimes understand them geometrically,
see Exercise 4.2.K.)
2.3.C. EXERCISE. Show that A → S −1 A is injective if and only if S contains no
zerodivisors. (A zerodivisor of a ring A is an element a such that there is a nonzero
element b with ab = 0. The other elements of A are called non-zerodivisors. For
example, an invertible element is never a zerodivisor. Counter-intuitively, 0 is a
zerodivisor in every ring but the 0-ring.)
If A is an integral domain and S = A−{0}, then S −1 A is called the fraction field
of A, which we denote K(A). The previous exercise shows that A is a subring of its
fraction field K(A). We now return to the case where A is a general (commutative)
ring.
2.3.D. EXERCISE. Verify that A → S −1 A satisfies the following universal property:
S −1 A is initial among A-algebras B where every element of S is sent to an invertible
element in B. (Recall: the data of “an A-algebra B” and “a ring map A → B”
are the same.) Translation: any map A → B where every element of S is sent to an
invertible element must factor uniquely through A → S −1 A. Another translation:
a ring map out of S −1 A is the same thing as a ring map from A that sends every
element of S to an invertible element. Furthermore, an S −1 A-module is the same
thing as an A-module for which s × · : M → M is an A-module isomorphism for
all s ∈ S.
In fact, it is cleaner to define A → S −1 A by the universal property, and to
show that it exists, and to use the universal property to check various properties
S −1 A has. Let’s get some practice with this by defining localizations of modules
by universal property. Suppose M is an A-module. We define the A-module map
φ : M → S −1 M as being initial among A-module maps M → N such that elements
of S are invertible in N (s × · : N → N is an isomorphism for all s ∈ S). More
precisely, any such map α : M → N factors uniquely through φ:
M φ
S
α
−1M ∃!
N
(Translation: M → S −1 M is universal (initial) among A-module maps from M to
modules that are actually S −1 A-modules. Can you make this precise by defining
clearly the objects and morphisms in this category?)
Notice: (i) this determines φ : M → S −1 M up to unique isomorphism (you
should think through what this means); (ii) we are defining not only S −1 M, but
also the map φ at the same time; and (iii) essentially by definition the A-module
structure on S −1 M extends to an S −1 A-module structure.