5.7. Faithfully Coflat Maps
Consider any category A with finite limits. Recall that any faithful
functor reflects epis and monos.
Definition 5.7.1. (a) We say a map f: Y >
X is faithful if the pullback functor
F:
A/X > A/Y along f is faithful.
(b) A fatihful and coflat map is called a faithfully
coflat map.
Proposition 5.7.2. The following coniditons are equivalent for
a map f: Y > X:
(a) f is faithful.
(b) F refltects epis.
(c) f is a universal epi.
Proof. (a) => (b) because any faithful functor reflects epis.
(b) => (a): Let (g, h) be a pair of parallel maps in
A/X
such that F(g) = F(h). Let
q be the
equalizer of (g, h). Then F(q) is the equalizer
of (F(g), F(h)). Hence F(q) is
an isomorphism. Therefore the regular mono q is an epi, hence is
an isomorphism. Consequently g = h. Thus F is faithful.
It follows that (b) implies (a).
(b) => (c): Let u: Z > X be an arbitrary map;
let (v: E > Y, h: E > Z)
be the product of (f, u), and let (w: V >
E,
k: V > E) be the pullback of (h,
h).
Then k is a retraction, thus a (regular) epi. The image of
the maps h: (W, uh) > (Z,
u)
is precisely the map k: (V, vk) > (W,
v).
Since F reflects epis, h is epic. As a result
f is
a universal epi.
(c) => (b). Let g: (U, ug) > (Z, u) be a
map in A/X such that F(g) is is epic. Let (h:
E
> Z. v: E > Y) be the pullback of (u,
f) and let (w: V > U, k:
V
> E) be the pullback of (g, h). Then h is
epic and F(g) is the map k: (V, kv)
> (W, v). Thus gw = hk is epic and hence
g
is epic, so that F reflects epic.
Proposition 5.7.3. (a) If f is a universal regular epi
then the pullback functor F: A/X > A/Y
along f reflects isomorphism.
(b) If the pullback functor F: A/X > A/Y
along a map f reflects regular epis then f is a universal
regular epi.
(c) If f is coflat and the pullback functor F: A/X
> A/Y along f reflects isomorphism then F
reflects regular epis.
(d) If f is coflat and the pullback functor F: A/X
> A/Y along f reflects isomorphism then f
is a universal regular epi.
Proof. (a) Let g: (U, ug) > (Z,
u)
be a map in A/X such that F(g) is an isomorphism.
Let (h: E > Z. v: E > Y)
be the pullback of (u, f) and let (w:
V >
U, k: V > E) be the pullback of (g,
h). Since f is universal regular epic, h and w
are regular epi, and F(g) is the map k: (V,
kv) > (W, v), so k is an isomorphism. It
follows that gw = hk is regular epic, thus g is a
regular epi. Since F is faithful according to (5.7.1), it reflects
mono. Since F(g) is an isomorphism, g is also a mono.
Thus g is an isomorphism. As a result F reflects isomorphism.
(b) Similar to the proof that (5.7.1.b) implies (5.7.1.c).
(c) Let g be a map in A/X such that F(g)
is a regular epi. Let (m, n) be the kernel pair of g,
let e be the coequalizer of (m, n), and let h
be the map such that he = g. Then (F(m), F(n))
is the kernel pair of F(g) which is the coequalizer of (F(m),
F(n)).
Since f is coflat, F preserves epis, so F(e)
is epi. The relatio F(h)F(e) = F(g)
implies that F(e) F(g).
Since F(g) is the coequalizer of (F(m), F(n))
and F(e)F(m) = F(e)F(n),
F(e)
factors through F(g), so F(e) F(g).
Consequently F(g) = F(e), so F(h)
is an isomorphism. Therefore h is an isomorphism and hence g
is a regular epi.
(d) follows from (b) and (c).
Corollary 5.7.4. (a) Any universal epi and univeral regular epi
in an analyitc category is unipotent and faithful.
(b) Any faithful map in an analytic category is unipotent.
Proof. (a) Any universal epi is faithful by (5.7.1). The pullback
of any universal epi f along any noninital map is epic, thus noninitial,
so f is unipotent.
(b) Any faithful map is univeral epi by (5.7.1), so it is unipotent
by (a).
Now consider a coherent analytic category A.
Proposition 5.7.5. (a) If f is unipotent then Spec(f):
Spec(Y)
> Spec(X) is surjective.
(b) If f is coflat and Spec(f) is surjective then
f
is unipotent.
(c) If f is coflat and any minimal prime of X is contained
in the image of Spec(f) then f is unipotent and Spec(f)
is surjective.
(d) If f is a coflat local map of local objects then f
is unipotent and Spec(f) is surjective.
Proof. (a) Consider a prime V of X with the generic
residue t: P(V) > X. Let (u: W
> Y, v: W > P(V)) be the pullback
of (f, t). Since f is unipotent, W is noninitial.
So W has a prime k: K > W. Now fuk
= tvk is epic as t and vk are so. So f^{+1}(uk)^{+1}(W))
= (fuk)^{+1}(W) = (tvk)^{+1}(W)
= t^{+1}((vk)^{+1}(W)) = t^{+1}(P(V))
= V. Therefore V is in the image of Spec(f).
(b) Consider a map t: P > X where P is
a simple object. Let V = t^{+1}(P). Since
Spec(f)
is surjective, f^{+1}(f^{1}(V)) =
V, and f^{+1}(f^{1}(V)) >
V is coflat as f is coflat. Since P > V is
epic, its pullback along f^{+1}(f^{1}(V))
> V is noninitial. Thus t is not disjoint with f,
so f is unipotent.
(c) f is surjective by the Going Up Theorem (5.2.13), so it
is unipotent by (b).
(d) follows from (c).
Definition 5.7.6. A coherent analytic cateory is called faithfully
strict if any coflat unipotent (i.e. coflat surjective) imap
is regular epic.
Proposition 5.7.7. Any faithfully strict coherent analytic category
is strict.
Proof. Since any analyitc cover contains a finite cover, we only
need to consider a finite analyitc cover {U_{i}} on an object
X.
The induced map U_{i}
> X is coflat and unipotent, thus a regular epi.
Proposition 5.7.8. Suppose A is faithfully strict.
(a) Any surjective coflat map is universal regular epi.
(b) A coflat map f is faithful iff Spec(f) is
surjective.
(c) Any coflat local map of local objects is faithful.
Proof. (a) Any pullback of a unipotent coflat map f is
unipotent coflat, thus regular epic. Therefore f is universal regular epic.
(b) By (5.7.4.b) any coflat faithful map is unipotent, thus surjective
by (5.7.5.a). Conversely if Spec(f) is surjective it is a
universal regular epi by (a), thus faithful by (5.7.2).
(c) follows from (5.7.5.d).
Example 5.7.8.1. The opposite of the category of commutative
rings is a faithfully strict coherent analytic category.
