Locally Finitely Presented Categories of Sheaves of Modules Prest, Mike and Ralph, Alexandra ...

17 Pages·2017·0.29 MB·English

Locally Finitely Presented Categories of Sheaves
of Modules
Prest, Mike and Ralph, Alexandra
2010
MIMS EPrint: 2010.21
Manchester Institute for Mathematical Sciences
School of Mathematics
The University of Manchester
Reports available from: http://eprints.maths.manchester.ac.uk/
And by contacting: The MIMS Secretary
School of Mathematics
The University of Manchester
Manchester, M13 9PL, UK
ISSN 1749-9097
Locally Finitely Presented Categories of Sheaves
of Modules
Mike Prest and Alexandra Ralph,
School of Mathematics
Alan Turing Building
University of Manchester
Manchester M13 9PL
UK
first author: [email protected]
February 15, 2010
1 Introduction
We know what is meant by an element of a module. What should we mean by
anelementofasheafofmodules? Oneansweristhatitissimplyasection(over
an open set). If, however, one takes the algebraic view that an element should
be of finitary character and hence should belong to a sum of subobjects iff it
belongs to some finite subsum, then this is not a good answer: sections are, in
general, of infinitary character.
Oneofouroriginalmotivationswastodevelopsomemodeltheoryforsheaves
ofmodules,atleasttoseeunderwhatconditionsonaringedspaceareasonable
modeltheoryforsheavesofmodulesmaybedeveloped(forthissee[?],also[12]).
Here we mean the usual model theory which is based on elements (of finitary
character). Through this we were lead to the problem of determining when a
sheafofmodulesisfinitelygeneratedorfinitelypresentedintheusualalgebraic
sense and to the problem of determining when the category of O -modules,
X
where O is a ringed space, is locally finitely presented. This condition on a
X
category is equivalent to its objects being determined by their elements.
Sofaraswecoulddetermine,thesequestionsinfullgeneralityhadnothith-
erto been addressed. Presheaves are genuinely algebraic objects and categories
ofpresheavesarealwayslocallyfinitelypresented. Butthesheafpropertyisnot
an algebraic one and does not fit well with notions like “finitely presented” and
“finitely generated” unless the base space has strong compactness properties
(such as is usually the case for those spaces considered in algebraic geometry
and analysis, where finiteness conditions on sheaves are of central importance).
Our other initial motivation was to investigate the representation of R-
modules, where R is any ring, as sheaves over a certain ringed space which
originally arose in the model theory of modules. This ringed space is the sheaf
of locally definable scalars over the rep-Zariski (=dual-Ziegler) spectrum of R
(see, e.g. [12][13]). Thisspectrumhasbad(separationandcompactness)prop-
erties compared with the spaces usually considered in algebraic geometry and
1
analysis. But our interest in this space explains why we consider ringed spaces
in full generality (arbitrary spaces and arbitrary rings with 1).
Our main result (3.5) is that if X has a basis of compact open sets and if
O is any sheaf of rings over X then the category Mod-O of sheaves of O -
X X X
modules is locally finitely presented, with the jO , where U ranges over any
! U
basisofcompactopensets,formingageneratingsetoffinitelypresentedobjects.
Here jO is the extension by 0 of the restriction, O , of O to U. Indeed,
! U U X
over any ringed space, jO is finitely presented iff the open set U is compact
! U
(3.7). The dependence of this result on X but not on the sheaf O prompted
X
ustoask(seetheearlierversions, [14], ofthispaper)whethertheanswertothe
question also depends only on X; that is, does Mod-O being locally finitely
X
presented depend only on the underlying space X? This question is still open
but the independence, given X, of 3.5 from the choice of sheaf O is explained
X
in a paper [3] by Bridge: there it is shown that if (C,J) is a Grothendieck site
suchthatthetoposofsheavesofsetsover(C,J)islocallyfinitelypresentedand,
if R is any ring object in that topos, then the category of R-modules will be
locally finitely presented. Taking the site to be the poset of open subsets of X,
regardedasacategoryintheusualway,withtheusualnotionofcovering,gives
3.5. In the original version of this paper we also commented that it seemed our
results would generalise to locales; that also is covered by Bridge’s result.
We also investigate the weaker condition that Mod-O be locally finitely
X
generated and we begin by showing that if F is a finitely generated sheaf then
the support of F is compact (4.5). We also prove that if K is a locally closed
subset of X then jO is finitely generated iff K is compact (4.6). Using this
! K
we obtain a necessary, but not sufficient, condition for Mod-O to be locally
X
finitely generated. Namely, if Mod-O is locally finitely generated then for
X
every x∈X and every open neighbourhood, U, of x there is a compact locally
closed set K with x ∈ K ⊆ U (4.8). We give examples which show that the
property of local finite generation depends on the structure sheaf, not just on
the space: if X is the (closed) unit interval in R with the usual topology and if
O is the sheaf of continuous functions on X then Mod-O is locally finitely
X X
generated whereas, if we let O(cid:48) be the constant sheaf on the same space then
X
Mod-O is not locally finitely generated (4.9, 4.10). We also show that, in the
X
firstcase,althoughMod-O islocallyfinitelygenerated,itisnotlocallyfinitely
X
presented (5.5); for that we develop a criterion (5.3, 5.4) for jO to be finitely
! K
presented over Hausdorff locally compact spaces.
AdirectionthatwehavenotpursuedhereisthatofreplacingMod-O with
X
the full subcategory of quasicoherent sheaves or of sheaves with quasicoherent
cohomology.
Although we have not been able to answer in full the question with which
we started, Bridge’s result, as well as recent interest (see [16]) in this topic,
has prompted the preparation of this (somewhat shorter) version of [14] for
publication.
2 Some general constructions and results on sheaves
First, some basic definitions and notation.
Let O be a sheaf of rings (all our rings will be associative with an identity
X
1 (cid:54)= 0). We denote by PreMod-O the category of presheaves over X which
X
2
are O -pre-modules. That is, M ∈PreMod-O means that M is a presheaf of
X X
abeliangroupssuchthat,foreachopensetU ⊆X,M(U)isarightR -module,
U
where we set R = O (U), and such that, for every inclusion V ⊆ U ⊆ X
U X
of open subsets of X, the restriction map, resM : M(U) −→ M(V), is a
U,V
homomorphism of R -modules, where we regard M(V) as an R -module via
U U
resOX : R −→ R . The full subcategory of sheaves, that is, of O -modules
U,V U V X
is denoted Mod-O . Then (see [2, Sections I.3, II.4]) both PreMod-O and
X X
Mod-O are Grothendieck abelian categories.
X
An object C of a category C is finitely presented (fp) if the representable
functor(C,−):C −→AbcommuteswithdirectlimitsinC. IfCisGrothendieck
abelian,thenitissufficienttocheckthatforeverydirectedsystem((D ) ,(g :
λ λ λµ
D −→ D ) ) in C, with limit (D,(g : D −→ D) ), every f ∈ (C,D)
λ µ λ<µ λ∞ λ λ
factorsthroughsomeg . AcategoryCisfinitelyaccessibleifthefullsubcat-
λ∞
egory,Cfp,offinitelypresentedobjectsisskeletallysmallandifeveryobjectofC
is a direct limit of finitely presented objects; if C also is complete (equivalently,
[1, 2.47], cocomplete) then C is said to be locally finitely presented (lfp).
Abeliancategorieswhicharefinitelyaccessiblehence,[6,2.4],Grothendieckand
lfp are in many ways as well-behaved as categories of modules over rings. In
particular, objects of C are determined by their “elements” (morphisms from
finitely presented objects) and these “elements” have finitary character (as op-
posed to what one has for merely presentable categories). Such categories have
a good model theory and they admit a useful embedding into a related func-
tor category (see e.g., [8], [10], [11], [12]). The category PreMod-O is locally
X
finitely presented (see [5, p. 7] for example), indeed it is a variety of finitary
many-sorted algebras in the sense of [1, Section 3A], but Mod-O need not be
X
(see 4.10, 5.5).
Next, we recall, following [9] (also see [7], [17]) some standard functors on
categoriesofsheaves. LetY ⊆X andletF ∈Mod-O . Letj :Y −→X denote
X
the inclusion. The sheaf j∗F is defined by: for any open subset U of Y we set
j∗F.U to be the set of those s such that s is a set-theoretic section of the stalk
space of F over the set U such that for all y ∈U there exists V ⊆X open with
y ∈ V ∩Y ⊆ U and there exists t ∈ FV such that for all z ∈ V ∩Y we have
s =t . That is, sections of j∗F locally look like sections of F. If Y is open in
z z
X then j∗F.U =FU for U ⊆Y open.
One may check that j∗F is, indeed, a sheaf (exercise in [9, p. 65] or [17,
p. 58]). It is also denoted F | or F (cid:22)Y and called the restriction of F to Y.
Y
If F ∈ Mod-O then j∗F ∈ Mod-O where we let O denote O | (cf. p.
X Y Y X Y
110 of [7] where the notation j−1 is used for what we have denoted j∗: here the
structure sheaf over a subspace is always that induced by the structure sheaf of
the whole space, so the j∗/j−1 distinction does not arise).
Fact 2.1. (e.g. [9, p. 97]) j∗ : Mod-O −→ Mod-O is exact and is left
X Y
adjoint to the left exact functor j : Mod-O −→ Mod-O which is given by
∗ Y X
j G.U = G(U ∩Y) for G ∈ Mod-O and U open in X (the direct image
∗ Y
functor): (j∗F,G)(cid:39)(F,j G) for F ∈Mod-O , G∈Mod-O .
∗ X Y
Let K ⊆X be locally closed (the intersection of an open set with a closed
set); denote the inclusion by j : K −→ X, and let G ∈ Mod-O . Define the
K
sheaf jG on X, the extension of G by zero by: jG.U = {s ∈ G(U ∩K) :
! !
supp(s) is closed in U}. This is a sheaf and jG ∈ Mod-O (for an alternative
! X
3
description, see [17, p. 63/4]). Recall that the support of a section s ∈ FU is
supp(s) = {x ∈ X : s (cid:54)= 0} where s is the germ of s at x; this is a closed
x x
subset of U.
Fact 2.2. (e.g. [9, p. 106/7]) j : Mod-O −→ Mod-O is an exact functor
! K X
whichisanequivalencebetweenMod-O andthecategoryofO -moduleswhich
K X
have all stalks over X\K equal to 0.
Now, given F ∈ Mod-O and K ⊆ X locally closed, let FK be the sheaf
X
(clearly it is a sheaf) given by FKU = {s ∈ FU : supp(s) ⊆ K}. So FK is a
subsheaf of F. Set j!F =j∗FK ∈Mod-O ; sections of j!F are locally given by
K
sections of F with support contained in K.
Fact 2.3. (e.g. [9, p. 108/9]) The functor j! : Mod-O −→ Mod-O is left
X K
exact and is right adjoint to the functor j : (jG,F)(cid:39)(G,j!F).
! !
Fact 2.4. (e.g. [9, p. 109])
If K =C is closed then j =j.
∗ !
If K =U is open then j∗ =j!.
Always j!F ≤j∗F.
Fact 2.5. (e.g. [9, p. 110], [17, 3.8.11]) If U ⊆ X is open, C = X \U is its
complement and F ∈Mod-O then there is an exact sequence
X
0−→j(F | )−→F −→i(i∗F)=i (i∗F)−→0
! U ! ∗
where j : U −→ X and i : C −→ X are the inclusions, where the first map is
the natural inclusion and where F −→ (ii∗F) is the identity if x ∈ C and is
x ! x
0 otherwise (see [9, p. 97, 4.3]).
In particular we have an exact sequence
0−→jO −→O −→iO −→0.
! U X ! C
It follows that if C is closed, K is locally closed and C ⊆K ⊆X then there
is an exact sequence
0−→i(cid:48)O −→O −→iO −→0
! K\C K ! C
where i : C −→ K and i(cid:48) : K \C −→ K are the inclusions. Then, since j is
!
exact, where j :K −→X is the inclusion, we have
0−→ji(cid:48)O −→jO −→jiO −→0
! ! K\C ! K ! ! C
that is,
0−→(ji(cid:48))O −→jO −→(ji)O −→0
! K\C ! K ! C
where ji(cid:48) :K\C −→X, j :K −→X and ji:C −→X are the inclusions.
Lemma 2.6. Let C ⊆X be closed and let U =X \C. The canonical sequence
0 −→ jO −→ O −→ iO −→ 0 is split iff C is open. (Here and elsewhere
! U X ! C
letters such as j,i will denote the obvious inclusions.)
Proof. Supposethatwehaveg :iO −→O splittingthecanonicalsurjection
! C X
(cid:40)
1 ∈O if x∈C
f :O −→iO .Settings=g1 ,wehaves = x X,x . Then
X ! C C x
0 otherwise
(cid:40)
0 if x∈C
(1 −s) = , so supp(1 −s)=X\C and, since the support of
X x X
1 if x∈/ C
x
any section must be closed, we deduce that C is open in X.
Conversely, if C is open in X then the inclusion iO −→O clearly splits
! C X
f, as required. (cid:3)
4
Since, if K ⊆X is locally closed, the functor j is exact, where j :K −→X
!
is the inclusion, we have the more general statement.
Lemma 2.7. Let C ⊆K ⊆X with C closed and K locally closed, i:C −→K
and j : K −→ X the inclusions. Then the canonical epimorphism jO −→
! K
(ji)O (from the exact sequence before 2.6) is split iff C is open in K.
! C
Proof. If C is open in K then we have a split exact sequence as in 2.6 with K
replacing X so then apply j.
!
Conversely,ifjO −→jiO issplitthenapply(j)∗,noting(see[9,II.6.4])
! K ! ! C
that (j)∗j =Id, and then apply 2.6. (cid:3)
!
The dual statement follows immediately.
Lemma 2.8. Let U ⊆ K ⊆ X with U open and K locally closed, i : U −→ X
and j : K −→ X the inclusions. Then the canonical monomorphism iO −→
! U
jO is split iff U is closed in K.
! K
(cid:84)
Lemma 2.9. Let C = C be closed sets, where the intersection is directed
λ λ
(i.e. ∀λ,µ∃ν C ⊆ C ∩C ). Let ((jO ) ,(g : jO −→ jO ) )
ν λ µ ! Cλ λ λµ ! Cλ ! Cµ Cλ⊇Cµ
be the corresponding directed system of epimorphisms (we use j generically
to denote inclusions). Then lim (jO ) = jO and each limit map g :
−→λ ! Cλ ! C λ∞
jO −→jO is an epimorphism.
! Cλ ! C
Proof. Let G=limjO with limit maps g :jO −→G.
−→ ! Cλ λ∞ ! Cλ
ForeachλtheinclusionC −→C givesrise,bythecanonicalexactsequence
λ
(2.5), to an epimorphism h :jO −→jO and these are compatible, so we
λ ! Cλ ! C
have a unique induced map h:G−→jO with hg =h for all λ. We show
! C λ∞ λ
that h is an isomorphism - so it is sufficient to show that h is an isomorphism
at each stalk.
Let x ∈ X. Then G = lim GU = lim lim G U (the presheaf and
x −→x∈U −→x∈U−→λ λ
(cid:40)
O if x∈C
sheaflimitsagreeonstalks)=lim lim G U =lim (G ) = x .
−→λ−→x∈U λ −→λ λ x 0 if x∈/ C
So G has the same stalks as jO and since all maps in the system are,
! C
at the level of stalks, either the identity or zero, we can check that h =
x
(cid:40)
id:O −→O if x∈C
x x . So G −→ jO is an isomorphism. Also, we
! C
0 otherwise
have seen that each g is stalkwise surjective and hence is an epimorphism.
λ∞
(cid:3)
3 The category Mod-O : local finite presenta-
X
tion
Every category Mod-O is locally presentable because it is a Grothendieck
X
category (e.g. [4, 3.4.2, 3.4.16]); we would like to determine when Mod-O is
X
locally finitely presented.
Remark 3.1. (cf. [2, p. 260]) The sheaves jO , with U ⊆ X open, together
! U
generate Mod-O . This follows, for instance, from the corresponding result
X
(see [2, Section I.3]) for presheaves by localising/sheafifying (which preserves
generating sets) to the category of sheaves.
5
Proposition 3.2. If U is a basis of open sets for X then the jO for U ∈ U
! U
together generate Mod-O .
X
Proof. Let F ∈ Mod-O , let x ∈ X and take a ∈ F . Since F = lim FU
X x x −→x∈U
thereisU =U(x,a)openandb=b ∈FU suchthatthecanonicalmapfrom
U,a
FU to F takes b to a. Without loss of generality U ∈ U. Define f(cid:48) : O −→
x U
F (cid:22) U by 1 ∈ O U (cid:55)→ b ∈ FU (by linearity and restriction this defines a
U U,a
presheaf morphism, which is enough).
Since j is left adjoint to (−) there is, corresponding to f(cid:48) ∈ (O ,F (cid:22) U),
! U U
a morphism f ∈(jO ,F) with (f ) :1 (cid:55)→b . Note that (f ) :
x,U,a ! U x,U,a U U U,a x,U,a x
(jO ) −→F maps 1 to a∈F .
! U x (cid:76) x (cid:76) OX,x (cid:76) (cid:76)x
Hence f : jO −→ F is an epimorphism
x∈X a∈Fx x,U,a x a ! U=U(x,a)
on stalks and hence is an epimorphism in Mod-O .
X
(cid:3)
Proposition 3.3. Suppose lim G = G in Mod-O and suppose that U ⊆
−→λ λ X
X is compact open. Then the canonical map g : lim (G U) −→ GU is an
−→λ λ
isomorphism.
Proof. The sheaf G is the sheafification of the presheaf direct limit, G(cid:48), of the
G and this is given, for V ⊆ X open, by G(cid:48)V = lim(G V). So the propo-
λ −→ λ
sition asserts that G(cid:48) and G agree at compact open sets. Let g : G(cid:48) −→ G
be the sheafification map in the category PreMod-O . We show that g is an
X U
isomorphism.
Let s ∈ G(cid:48)U and suppose that g s = 0. Then there must be an open cover
U
{U } ofU suchthat,foralli,wehaveresG(cid:48) s=0.SinceU iscompactwemay
i i U,Ui
take the cover to be finite: U ,...,U say. Then, by definition of the restriction
1 n
mapsinthelimit,foreachithereareλ anda ∈G U suchthat(g(cid:48) ) (a )=
i i λi λi,∞ U i
s(whereg(cid:48) :G −→G(cid:48) isthecanonicalmap)and(g(cid:48) ) resGλi (a )=0.
λi,∞ λi λi,∞ Ui U,Ui i
SincetherearejustfinitelymanyU wemaytakeλwithλ≥λ ,...,λ andalso
i 1 n
suchthat(g ) (a )=(g ) (a )=b,say,foralli,j (sincethea allmapto
λi,λ U i λj,λ U j i
thesameelementinthelimit)and,furthermore,suchthat(g ) resGλi (a )=
λi,λ Ui U,Ui i
0 for all i (since each resGλi a maps to 0 in the presheaf limit). But then
U,Ui i
resGλ (b) = (g ) resGλi (a ) = 0 for each i and hence, since G is a sheaf,
b=U0,U.iThereforλei,λs=Ui(g(cid:48)U,U)i (ib)=0 as required. λ
λ,∞ U
To see that g is onto, take any t ∈ GU. For each x ∈ U there is an
U
open neighbourhood U of x and t(x)∈G(cid:48)(U ) such that g t(x)=resG (t).
x x Ux UUx
Since U is compact we may take finitely many open sets U ,...,U say, with
1 n
corresponding t ∈G(cid:48)U , which cover U.
i i
For each i there is λ and s ∈ G U with g(cid:48) s = t . Since there are
i i λi i λi∞ i i
only finitely many λ we may take λ ≥ λ ,...,λ and we may suppose that
i 1 n
s ∈G U for each i.
i λ i
We have, furthermore, that for each pair, i,j, of indices, resG(cid:48) (t ) =
Ui,Ui∩Uj i
resG(cid:48) (t ) and hence there is µ ≥ λ such that g(cid:48) resGλ (s ) =
Uj,Ui∩Uj j ij λµij Ui,Ui∩Uj i
g(cid:48) resGλ (s ). So, choosing µ≥µ for each i,j and setting s(cid:48) =g (s )
λµij Uj,Ui∩Uj j ij i λµ i
wemaysupposethatforalli,j wehaveresGµ (s(cid:48))=resGµ (s(cid:48)). Since
Ui,Ui∩Uj i Uj,Ui∩Uj j
G is a sheaf, there is s ∈ G U such that resGµ (s) = s for each i = 1,...,n.
µ µ U,Ui i
6
Then (g(cid:48) ) (s) ∈ G(cid:48)U with (since G is separated) g (cid:0)(g(cid:48) ) (s)(cid:1) = t, as
µ∞ U U µ∞ U
required. (cid:3)
It follows that if X is a noetherian space, that is, if every open subset of X
is compact, and if (G ) is a directed system in Mod-O then the direct limit,
λ λ X
limG ,computedinPreMod-O isasheafandhenceequalslimG ,computed
−→ λ X −→ λ
in Mod-O .
X
From the adjunction (jO ,F)(cid:39)(O ,F (cid:22)U)(cid:39)Γ(F (cid:22)U)(cid:39)FU for U ⊆X
! U U
open we deduce that Γ (−) (cid:39) (jO ,−) as functors on Mod-O . Here Γ
U ! U X
denotes the global section functor, F (cid:55)→FX, and Γ is the functor F (cid:55)→FU.
U
Corollary 3.4. If U ⊆ X is compact open then jO is a finitely presented
! U
object of Mod-O .
X
Proof. Let (G ) be a directed system in Mod-O with direct limit G. Then,
λ λ X
as just noted, (jO ,G) (cid:39) GU and lim (jO ,G ) (cid:39) lim(G U) - and these
coincide by 3.3, a!s rUequired. (cid:3) −→λ ! U λ −→ λ
With 3.2 this gives our first result.
Theorem 3.5. [15], [14] If X has a basis of compact open sets then Mod-O
X
is locally finitely presented, with the jO for U ⊆X compact open (or with the
! U
U from any basis of such sets) as a generating set of finitely presented objects
of Mod-O .
X
Corollary 3.6. If X is locally noetherian and O is any sheaf of rings on X
X
then Mod-O is locally finitely presented.
X
We also have the converse to 3.4.
Proposition 3.7. If U is open then jO is finitely presented in Mod-O iff
! U X
U is compact.
Proof. If U is compact then we have 3.4, so suppose that U is not compact
- say {U } is an open cover with no finite subcover. We may suppose that
i i
{U } isclosedunderfiniteunion. SetG =jO ; theseformadirectedsystem
i i i ! Ui
under inclusion (see 2.5), so set G = limjO . Then G = jO since from the
−→ ! Ui ! U
canonical inclusions jO −→ jO we obtain a map G = limjO −→ jO
! Ui ! U −→ ! Ui ! U
which locally, and hence stalkwise, is an isomorphism and which is, therefore,
an isomorphism.
SotheidentitymapofGwouldfactorthroughsomejO ifG=jO were
! Ui ! U
finitely presented - but since U (cid:54)= U there can be no such factorisation (recall
i
that if x∈/ U then (jO ) =0). (cid:3)
i ! Ui x
Example 3.8. If F ∈ Mod-O is finitely presented and U ⊆ X is open then
X
F | might not even be finitely generated. Let X = [0,1] and take U = (0,1).
U
TakeO tobethesheafificationoftheconstantpresheafk wherek isanychosen
X
ring. By 3.4, O is a finitely presented sheaf but, by 3.7, O is not, because U
X U
is not compact. Indeed, O is not even finitely generated, as one sees by writing
U
O as the sum over n≥1 of the sheaves jO .
U ! (1,1−1)
n n
7
4 The category Mod-O : local finite generation
X
Recall that an object F in an abelian category is finitely generated if, when-
ever F = (cid:80) F for some subobjects F , we have F = (cid:80)n F for some
λ λ λ i=1 λi
λ ,...,λ . If the category is locally finitely presented then it is equivalent that
1 n
F be the image of a finitely presented object.
Suppose that F ∈ Mod-O and let s ∈ FX. Define a subpresheaf (cid:104)s(cid:105)0 of
X
F by setting: (cid:104)s(cid:105)0U = resF (s).R (recall that R = O U) and with the
X,U U U X
restrictionmapscomingfromF.Thisisaseparatedpresheaf; let(cid:104)s(cid:105)denotethe
sheafification of (cid:104)s(cid:105)0 - a subsheaf of F.
Moregenerally, ifU ⊆X isopenands∈FU thenwedefinethesubsheafof
F generatedbystobej(cid:104)s(cid:105),wherej :U −→X istheinclusionand(cid:104)s(cid:105)≤F |
! U
is defined as above. Recall (2.5) that there is an inclusion j(F | ) −→ F and
! U
so, since j is (left) exact, j(cid:104)s(cid:105) is indeed a subsheaf of F. Although we call it
! !
the sheaf generated by s it need not be a finitely generated sheaf - unless U
is compact s might not be a “finitary element” of F. Here, though, is some
justification for the terminology.
Lemma 4.1. Let F ∈ Mod-O , let U ⊆ X be open and let s ∈ FU. Suppose
X
that G≤F is a subsheaf such that s∈GU. Then j(cid:104)s(cid:105)≤G.
!
Proof. It is immediate from the definition that (cid:104)s(cid:105)0 is a subpresheaf of G and
U
hence that (cid:104)s(cid:105)≤G . Therefore j(cid:104)s(cid:105)≤j(G )≤G. (cid:3)
U ! ! U
(cid:80)
Lemma 4.2. Let F ∈ Mod-O . Then F = {j(cid:104)s(cid:105) : U ⊆ X is open and s ∈
X !
FU} (we write j for (j ) where j :U −→X is the inclusion).
! U ! U
Proof. Bytheremarksabove,F containstherighthandside. Conversely,given
U ⊆ X open and s(cid:48) ∈ FU we have s(cid:48) ∈ j(cid:104)s(cid:48)(cid:105).U (since s(cid:48) ∈ (cid:104)s(cid:48)(cid:105)0U) so s(cid:48) ∈
!
((cid:80) (cid:80) j(cid:104)s(cid:105)).U as required. (cid:3)
U s∈FU !
NotethatinfinitesumsinMod-O areobtainedbyfirstformingthepresheaf
X
sum (that is, the algebraic sum of U-sections at each open U ⊆ X) and then
sheafifying. We say that a sheaf F is finitely generated if whenever F =
(cid:80)
F with the F subsheaves of F, then there are λ ,...,λ such that F =
λ λ λ 1 n
F + ··· + F . It is immediate that any finitely presented sheaf is finitely
λ1 λn
generated.
Lemma 4.3. If F ∈ Mod-O , s ∈ FX and (V ) are open sets with V =
X λ λ
(cid:83) V ⊇supp(s) then (cid:104)s(cid:105)=(cid:80) j(cid:104)resF s(cid:105).
λ λ λ ! X,Vλ
Proof. Arguingasabove, therighthandside, Gsay, isasubpresheafof(cid:104)s(cid:105).We
have, for each λ, a section in GV which agrees with s on V and hence, since
λ λ
(cid:104)s(cid:105) is a sheaf, we deduce that resF s∈GV and hence, since G is a sheaf, that
X,V
s∈GX. Therefore, by 4.1, (cid:104)s(cid:105)≤G, as required. (cid:3)
We define the support of a (pre)sheaf F to be the union, supp(F) = {x ∈
X :F (cid:54)=0}, of supports of sections of F.
x
Lemma 4.4. Suppose that F ∈ Mod-O is finitely generated. Then there are
X
open subsets U ,...,U of X and s ∈ F(U ) such that supp(F) = (cid:83)nsupp(s ).
1 n i i 1 i
In particular, supp(F) is a locally closed subset of X.
(cid:80)
If F = j(cid:104)s (cid:105) where s ∈F(U ) then we may take the U and s from this
i ! i i i i i
representation.
8
Proof. Take a representation of F as given (we know there is such by 4.2).
Since F is finitely generated we have F =(cid:80)nj(cid:104)s (cid:105), say. Certainly supp(F)⊇
1 ! i
(cid:83)nsupp(s ).
1 i
Conversely, if x ∈ supp(F) then there is an open set V containing x and
t ∈ FV such that t (cid:54)= 0. We have FV = (cid:80)nj(cid:104)s (cid:105).V. If a ∈ j(cid:104)s (cid:105).V then
x 1 ! i ! i
supp(a) ⊆ supp(s ) (using the definition of (cid:104)s(cid:105)0). But then t ∈ (cid:80)nj(cid:104)s (cid:105).V
i 1 ! i
implies x∈supp(s ) for some i, as required.
i
Finally, supp(F) is locally closed since it is a finite union of closed subsets
of open sets. (cid:3)
Proposition 4.5. Let F ∈ Mod-O be finitely generated. Then supp(F) is
X
compact.
Proof. We know that F =(cid:80)nj(cid:104)s (cid:105) for some s ∈F(U ) for some open U and
1 ! i i i i
then, by the lemma above, supp(F)=(cid:83)nsupp(s ).
1 i
Suppose that supp(F) is not compact. Then there are open subsets V
λ
(cid:83)
of X such that supp(F) ⊆ V = V but no finite number of these cover
λ λ
supp(F). For each i = 1,...,n the V ∩U cover V ∩U and so, by 4.3, F =
λ i i
(cid:80)n (cid:80) (j )(cid:104)resF (s )(cid:105) (ignoring those where V ∩U =∅).
i=1 λ Vλ∩Ui,X ! Ui,Vλ∩Ui i λ i
Since F is finitely generated there is a finite subsum
F = (cid:80)m (j )(cid:104)resF (s )(cid:105). By 4.4 above we have supp(F) =
(cid:83)msupkp=(s1 )Vλ∩k∩VUik,⊆X(cid:83)!mVUik,-Vλcko∩nUtrikadiicktion, as required. (cid:3)
1 ik λk 1 λk
Proposition 4.6. Let K ⊆X be locally closed. Then jO is finitely generated
! K
iff K is compact.
Proof. If jO is finitely generated then, by 4.5, K is locally closed.
! K
For the converse, suppose that K is locally closed and compact. We have
K = U ∩ C for some open U and closed C. Let U ⊆ U be open. Then
0 0 0
jO .U = {s ∈ O (K ∩ U) : supp(s) is closed in U} is generated by 1
! K K K∩U
since supp(1 )=K∩U =C∩U is closed in U. Hence jO =(cid:104)1 (cid:105).
K∩U ! K K
(cid:80)
Now suppose that jO = F for some subsheaves F . We may suppose
! K λ λ λ
(cid:80)
that the sum is directed. Let x ∈ K. Since F is the sheafification of the
λ λ
presheaf sum of the F there is an open set U containing x (without loss of
λ x
generality U ⊆ U ) such that 1 (∈ jO .U ) belongs to the presheaf sum
x 0 Ux∩K ! K x
of the F U and hence (since the sum is directed) belongs to F U for some λ.
λ x λ x
As x varies over K we get a cover (U ) and so, by compactness, some
x x∈K
finite subset, U ,...,U , covers K. Set U = U ∪...∪U ⊆ U . The sum
x1 xn 1 x1 xn 0
is directed so we may choose λ such that 1 ∈ F (U ) for i = 1,...,n
K∩Uxi λ xi
and hence such that 1 = 1 ∈ F (U ) (since F is a sheaf). Therefore
K K∩U1 λ 1 λ
jO =(cid:104)1 (cid:105)≤F , as required. (cid:3)
! K K λ
Proposition 4.7. Let U ⊆ X. Suppose that there is a finitely generated sheaf
F ∈ Mod-O such that there is a non-zero homomorphism f : F −→ jO .
X ! U
ThenU containsacompactlocallyclosedset. Ifx∈X issuchthatthemorphism
of stalks f : F −→ O is non-zero then this compact locally closed set may
x x X,x
be taken to contain x.
Proof. Let F(cid:48) = im(f). Being an image of a finitely generated object, F(cid:48) is
finitely generated. Since F(cid:48) is a non-zero subfunctor of jO , we have ∅ =(cid:54)
! U
9