5.2. Localizations
An epi f: Y > X is called essential
provided that for any map g: Z > Y, if fg is epic
then g is epic.
Proposition 5.2.1. Suppose f:
Y > X is a coflat map and X is integral.
(a} If f is an essential epi then Y is integral.
(b} If f is a bimorphism then Y is integral.
Proof. Let p: P(X) > X be
the generic residue of the integral object X, which is an epi coflat
fraction by (5.1.2.c). Let (g:
Z > Y, h: Z > P(X)) be the pullback of (f:
Y > X, p: P(X) > X).
(a) Assume f is an essential epi. Since f and p
are coflat and epic, g and h are epic. Thus Z is noninitial,
which has a simple strong subobject q: Q > Z (4.2.2.d).
Since P is simple, hq: Q > P(X) is epic.
Since p is epic, phq = fgq is also epic. Since f is
an essential epi, gq: Q > Y is epic. Then Y, as
a quotient of a simple object, is integral.
(b) This follows from (a) as any coflat bimorphism is an essential epi.
by (1.4.4.e).
Proposition 5.2.2. Suppose f:
Y > X is any map. The image of Spec(f) is {V Î
Spec(X): f^{+1}(f^{1}(V))
= V}.
Proof. Suppose V = Spec(f)(W) = f^{+1}(W)
for a prime W of Y. Then f^{+1}f^{1}(V)
is a subobject of X contained in V. But W
f^{1}(V) implies that V = f^{+1}(W)
f^{+1}f^{1}(V). Thus V = f^{+1}f^{1}(V).
Conversely, assume V = f^{+1}f^{1}(V)
holds. Consider the pullback (r: Z > f^{1}(V),
s: Z > P(V) of the maps (u: f^{1}(V)
> V, p: P(V) > V). Since u is epic and p is
coflat epic, s: Z > p(V)
is epic, thus Z is noninitial, which has a simple subobject q:
Q > Z. But psq = urq and the left side is epic, thus the right
side urq is epic. Let W = (rq)^{+1}(Q).
Then u^{+1}(W) = V. But u is the restriction
of f on f^{1}(V). This shows that f^{+1}(W)
= V, i.e. V is lying in the image of Spec(f).
Proposition 5.2.3. Suppose f:
Y > X is a coflat map.
(a} The image of f is {V Î
Spec(X): f^{1}(V) is noninitial}.
(b} If V is in the image of Spec(f) and W
is a integral strong subobject containing V, then W is in
the image (i.e. the image of Spec(f) is closed under generalizations).
Proof. (a) If V is in the image then by (5.2.2)
f^{+1}f^{1}(V) = V, so f^{1}(V)
is noninitial.
Conversely, assume f^{1}(V) is noninitial.
The induced map t: f^{1}(V) > V
is the pullback of f along V > X, thus is also coflat.
Since V is integral and f^{1}(V) is noninitial,
t is epic by (3.2.6.a). Thus f^{+1}f^{1}(V)
= V, which implies that V is in the image of Spec(f)
by (5.2.2).
(b) If V Í W are two primes
of X and V is in the image of Spec(f), then
by (a) f^{1}(V) is noninitial. This implies that
f^{1}(W) is noninitial, so W is in the image
of Spec(f) by (a).
Proposition 5.2.4. Suppose Y
is a local object with the simple prime M and f: Y >
X is a coflat mono. Then Spec(f) induces a homeomorphism
between Spec(Y) and the subspaces of Spec(X)
consisting of primes which contains M.
Proof. Since f is a coflat mono, Spec(f)
is a topological embedding by (3.6.9.a).
Since any prime V of Y contains M, we have f^{+1}(V)
f^{+1}(M)
M. Conversely, if W is a prime of X which contains M,
then it contains f^{+1}(M). Thus by (5.2.3.b)
it is in the image of Spec(f).
Proposition 5.2.5. Suppose f:
Y > X is a coflat local map of local objects. Then Spec(f):
Spec(Y) > Spec(X) is surjective.
Proof. By (3.4.7) the simple prime
P of X is in the image of f. Since any prime of X
contains P, by (5.2.3.b) it is in the
image of f.
Suppose V is an integral subobject of an object X. Denote
by X_{V} the intersection of all the analytic subobjects
of X which is not disjoint with V.
Proposition 5.2.6. Assume V is
an integral subobject of X with the generic residue P(V).
(a) X_{V} is the intersection of all the analytic subobjects
of X containing P(V).
(b) X_{V} is a local object whose simple prime
contains P(V).
(c) X_{V} is the intersection of all the fractions of
X containing P(V).
(d) If V is a prime then P(V) = X_{V}
V.
Proof. (a) Since V is integral and the simple object
P(V) is the intersection of noninitial fractions of V,
an analytic subobject of X is not disjoint with V iff it
contains P(V).
(b) X_{V} contains P(V) by (a), thus is
noninitial. We prove that any noninitial strong subobject S of
X_{V} contains P(V), which would implies that
the intersection P'(V) of all the noninitial strong subobjects
of X_{V} is noninitial, and therefore X_{V}
is local whose simple prime P'(V) contains P(V).
If this is not the case, then the simple object P(V) is disjoint
with S. Since X_{V} is a coflat subobject of X,
S is induced by a strong subobject S' such that S
= S' X_{V}.
Then P(V) is disjoint with S'. Since A
is locally disjunctable, we can find a disjunctable strong subobject T
of X such that S'
T and P(V) is not contained in T. Then the
simple object P(V) is disjoint with the strong subobject
T, so P(V)
T^{c}, thus T^{c} is not disjoint with V,
so X_{V} as the intersection of such analytic subobjects
is contained in T^{c}, which is absurd as S is in
X_{V} but not in T^{c} as S
S' T. This shows that
any noninitial strong subobject S of X_{V} contains
P(V), so X_{V} is local.
(c) As P(V) ® X_{V}
is local (thus quasilocal by (3.3.4.a)
and X_{V} > X is a faction, this pair of maps is
the quasilocalfraction factorization of the inclusion P(V)
> X. By (4.2.9) X_{V}
is the intersection of fractions of X containing P(V).
(d) Clearly P(V) is contained in X_{V}
V, thus X_{V}
V is noninitial, which is a fraction of V as X_{V}
is a fraction of X. It suffices to prove that X_{V}
V is simple. Consider a proper strong subobject S of X_{V}
V. Since X_{V}
V is a coflat subobject of V, S is induced by a proper
strong subobject S' of V, i.e. S = (X_{V}
V) S'. Since
V is a proper strong subobject of X, S' is also a
proper strong subobject of X. Since A
is locally disjunctable, we can find a disjunctable strong subobject T
of X containing S' but not V. Then T
V is a proper subobject of V, and T^{c}
V = (T V)^{c}.
Since V is reduced and T
V is proper, (T
V)^{c}is a noninitial analytic subobject of V.
Since by definition X_{V}
T^{c} V
= (T V)^{c},
X_{V} is disjoint with T
V. As S S'
T V, we see that X_{V}
is disjoint with S. Since X_{V} is noninitial
containing S, this means that S is initial. We have
proved that any proper strong subobject S of X_{V}
V is noninitial. Thus X_{V}
V is simple.
Proposition 5.2.7. If P is a
simple subobject of X then X_{P} is a local object,
and X_{P }is the intersection of all the analytic
subobjects (or fractions) of X which contains P.
Proof. The assertion follows from (5.2.6).
Definition 5.2.8. (a) A fraction Y
> X with a local domain
Y is called a localization.
(b) Suppose V is an integral subobject of an object X.
With the notations of (5.2.6), the localization l_{V}:
X_{V} > X is called the localization
of X at V.
Remark 5.2.9. Suppose
is a small strong cogenerating set formed by finitely copresentable
objects. If W is a principal
regular subobject of an object X, we say that W^{c}
is a principal
analytic subobject of X. Similar to (5.2.6)
and (5.2.7) one can show that for any integral subobject
V of X, the localization X_{V } of
X at V is the intersection of principal
analytic subobjects of X which is not disjoint with V (the
key step is (5.2.6.c) to prove that the intersection
is a local object).
Proposition 5.2.10. Suppose f:
Y > X is a localization and P is the simple prime
of the local object Y and V = f^{+1}(P).
Then Y is the localization of X at the prime V.
Proof. Since f is coflat, P = f^{1}f^{+1}(P)
= Y f^{+1}(P)
implies that P is a epic simple fraction, thus P is the generic
residue of the prime V = f^{+1}(P). Since
the inclusion P > Y is local and f: Y >
X is a fraction, this pair of maps is the quasilocalfraction factorization
of P > X. But the inclusion P > X_{V}
is also local and the inclusion X_{V} > X
is a fraction, we obtain another pair of maps which is also a quasilocalfraction
factorization of the inclusion P > X. By the uniqueness
of such a factorization we see that Y = X_{V} as
subobjects of X.
Proposition 5.2.11. Let f: Y
> X be a map. Suppose W
Spec(Y) and V = f^{+1}(W). Then
there is a unique local map f_{W}: Y_{W}
> X_{V} of local objects such that the following diagram
commutes:
Proof. Since X_{V} is the intersection of analytic
subobjects of X which is not disjoint with V, f^{1}(X_{V})
contains Y_{W}, which is the intersection of analytic subobjects
of Y that is not disjoint with W. Thus the composition W
> X of the inclusion W > Y and f factors
through the inclusion X_{V} > X in a unique map
f_{W}: Y_{W} > X_{V}. Clearly
f sends Y_{W} > W = P(W) into
X_{V} > V = P(V), where P(W)
and P(V) are the generic residues of W and V
respectively. Thus f_{W} is local by (3.4.7.b).
With the notation of (5.2.11) we say that X_{V} is the local
image of the localization Y_{W}. of Y (under
f).
Definition 5.2.12. Let f: Y
> X be a map. We say that the goingup
theorem holds for f if the following condition
is satisfied:
For any V, V'
Spec(X) such that V
V', and for any W
Spec(Y) with f^{+1}(W) = V,
there exists W' Spec(Y)
with f^{+1}(W') = V' such that W
W'.
Now we can prove a generalization of the dual version of the goingdown
theorem for a flat homomorphism of commutative rings (cf. [Matsumura
1980, p.33, Theorem 4]):
Proposition 5.2.13. (Going Up Theorem)
The goingup theorem holds for any coflat map f: Y > X.
Proof. Suppose V, V'
Spec(X) such that V
V', and W Spec(B)
with f^{+1}(W) = V. Consider the localization
l_{W}: Y_{W} > Y, which is coflat. Then
fl_{W}: Y_{W} > X is coflat. By (5.2.3.b)
V' is in the image of fl_{W}. Thus there is a prime
Q of Y_{W} such that fl_{W}^{+1}(Q)
= V. Let W' = l_{W}^{+1}(Q).
Then W' contains W and f(W') = W as
desired (alternatively the theorem follows from the fact that the induced
local map Y_{W} > X_{V} is local, thus
the mapping Spec(Y_{W}) > Spec(X_{V})
is surjective, and the fact that the mappings Spec(Y_{W})
> Spec(Y) and Spec(X_{V}) > X
are embeddings.).
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