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FLABBY ENVELOPES OF SHEAVESEdgar Enochs a & Luis Oyonarte ba Department of Mathematics , University of Kentucky , Lexington, Kentucky, 40506-0027,U.S.A.b Department Álgebra y Análisis Matemático , Universidad de Almería , Almería, 04120,SpainPublished online: 20 Aug 2006.

To cite this article: Edgar Enochs & Luis Oyonarte (2001) FLABBY ENVELOPES OF SHEAVES, Communications in Algebra, 29:8,3449-3457, DOI: 10.1081/AGB-100105031

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FLABBY ENVELOPES OF SHEAVES

Edgar Enochs1 and Luis Oyonarte2

1Department of Mathematics, University of Kentucky,Lexington, Kentucky 40506-0027

E-mail: enochs@ms.uky.edu2Department Algebra y Analisis Matematico,Universidad de Almerıa, 04120 Almerıa, Spain

E-mail: loyonart@ualm.es

1. PRELIMINARIES

Given a class of objects F in a category C, and an object C 2 ObðCÞ, anF -precover of C is a morphism f : F ! C for an object F 2 F , such that thesequence HomðF 0;FÞ ! HomðF 0;CÞ is surjective for all F 0 2 F . If moreoverf � j ¼ f for j : F ! F implies that j is an automorphism, then f is said tobe an F -cover. The dual notions are those of F -preenvelope and F -envel-ope. It is not hard to see that if F -covers (F -envelopes) exists, they areunique up to isomorphism.

We recall that given a class of objects of an abelian category C, theorthogonal class of F , ?F , is the class of all those objects X such thatExtðX ;FÞ ¼ 0 8F 2 F . In a similar manner, F? is the class of objects X

such that ExtðF;X Þ ¼ 0 8F 2 F . A morphism f : F ! C ( f : C ! F) withF 2 F is said to be a special F -precover (F -preenvelope) whenKerð f Þ 2 F? (Cokerð f Þ 2 ?F ) and when f is an epimorphism (mono-morphism). It is immediate that a special F -precover (F -preenvelope) isindeed an F -precover (F -preenvelope).

If F is a class of objects of an abelian category C, the pair of classesðF ;F?Þ is said to be cogenerated by a set whenever there exists a set X of

3449

Copyright # 2001 by Marcel Dekker, Inc. www.dekker.com

COMMUNICATIONS IN ALGEBRA, 29(8), 3449–3457 (2001)

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objects of C such that A 2 F? , ExtðF;AÞ ¼ 0 8F 2 X . A pair of classesðF ;GÞ is said to have enough injectives (projectives) if for every object C

of C there exists an exact sequence 0! C ! A! B! 0 (0! A!B! C ! 0) with A 2 G and B 2 F . Finally, ðF ;GÞ is said to be a cotorsiontheory if F? ¼ G and ?G ¼ F . For further information on these topicssee [9].

We recall that a continuous chain of subobjects of a given object X ofan abelian category, is a set of subobjects of X , fXa; a < lg (for someordinal number l), such that Xa is a subobject of Xb for all a � b < l, andthat Xg ¼

Pa<g Xa whenever g < l is a limit ordinal.

Throughout all the paper we will study sheaves of modules on atopological space X , so, from now on, by the word (pre)sheaf on X (orsimply (pre)sheaf), we will mean a (pre)sheaf of modules on X . The set of allopen subsets of X will be denoted by O, and the letter R will be used toappeal a ring. All rings considered will be associative with unity, and theword module will mean a unital left R-module unless otherwise specified.For any set X , the cardinality of X will be expressed by the symbol jX j. Wewill let F be the class of all flabby (pre)sheaves, and we will say flabbypreenvelopes to mean F -preenvelopes.

Given any (pre)sheaf F, the restriction maps of F, FðV Þ ! FðUÞwhenever U � V are elements of O, will be denoted by FU ;V , or by the usualrU ;V if it is clear which (pre)sheaf is involved. If F and G are two (pre)-sheaves and f : F ! G is a morphism of (pre)sheaves, we will denote byf ðV Þ : FðV Þ ! GðV Þ the corresponding homomorphisms of modules for allV 2 O, and by fx : Fx ! Gx the corresponding homomorphism between thestalks for all x 2 X .

An object X of an abelian category is said to be @-generated (where @is an infinite cardinal number) if for any exact sequence

‘i2I Ai ! X ! 0

there exists J � I , jJ j � @ such that‘

i2J Ai ! X ! 0 is also exact. @-gen-erated objects in abelian categories with exact direct limits have beencharacterized by B. Osofsky in [8, Proposition 2.1(I)]. It is not hard to seethat, in these categories, an object X is @-generated if and only if, wheneverX ¼

Pi2I Xi, there exists J � I , jJ j � @ such that X ¼

Pi2J Xi.

The next two results were given in [1]. We include them here forcompleteness.

Proposition 1.1. Let C be a locally small abelian category with exact directlimits and X 2 ObðCÞ. Then there exists a cardinal number @ such that X is@-generated.

Proof. Since C is locally small, we know that ‘ðX Þ (the lattice of subobjectsof X ) is a set. Let @ ¼ j‘ðX Þj. It is then clear that X is @-generated. u

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Proposition 1.2. Let C be a locally small abelian category with exact directlimits; l the least ordinal number with jlj > @; X an @-generated object of C;and Y 2 ObðCÞ such that Y ¼

Pa<lYa for fYa; a < lg a continuous chain of

subobjects of Y . Then for any morphism f : X ! Y there exists b < l such thatf ðX Þ � Yb.

Proof. If X is @-generated then f ðX Þ is also @-generated, and we havef ðX Þ �

Pa<lYa. Therefore f ðX Þ ¼ ð

Pa<lYaÞ \ f ðX Þ ¼ ð

Pa<lYa \ f ðX ÞÞ

since direct limits are exact. We then know there exists a subset S � l,jSj � @ such that f ðX Þ ¼

Pa2SðYa \ f ðX ÞÞ ¼ ð

Pa2SYaÞ \ f ðX Þ; so f ðX Þ �P

a2SYa.Since S � l, we have that jaj � @ for all a 2 S, and if b ¼ Sup S, we see

that jbj � @ since jSj � @, so we get b < l. Therefore,P

a2SYa � Yb and thenFðX Þ � Yb. u

2. THE MAIN THEOREM

Definition 2.1. A ð preÞsheaf F on a topological space X is said to be flabbyprovided that the restriction map FU ;X : FðX Þ ! FðUÞ is surjective for allU 2 O.

The next construction was suggested by [10, Lemma 6.6].For each open set U of X we construct the presheaf of modules SU on

X given by

SUðVÞ ¼R if V � U0 if V U

nand the restriction maps rW ;V : SU ðV Þ ! SU ðW Þ, whenever W � V are opensubsets of X , are defined as

rW;V ¼idR if V � U0 if V U

nWe also define the presheaf TðUÞ ¼ R for all open sets U of X , and

TðW Þ !idRTðV Þ for all open sets W � V of X .

With the help of these two presheaves, as well as their sheafifications,we characterize flabby sheaves in the next result. For this purpose we notethat SU is a subpresheaf of T for any U , and that, if �SU , �T and T=SU are thesheafifications of SU , T and T=SU respectively, the exact sequence in thecategory of presheaves

0! SU !kUT!pU T=SU ! 0

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induces an exact sequence in the category of sheaves

0! �SU !k0U �T!

p0UT=SU ! 0;

in such a way that the diagram of presheaves

0 ! SU !kU T !pU T=SU ! 0

iSU iT iU0 ! �SU !

k0U T !

p0U T=SU ! 0

is commutative.

Proposition 2.2. Let F be any sheaf. Then; ExtðT=SU ;FÞ ¼ 0 for any opensubset U of X if and only if F is flabby.

Proof. It is immediate to see that for any presheaf F, HomðSU ;FÞ ffi FðUÞfor any U 2 O, and HomðT ;FÞ ffi FðX Þ. Thus, a presheaf F is flabby if andonly if the sequence

HomðT;FÞ ! HomðSU;FÞ ! 0

is exact in the category of presheaves for any open set U of X .Let then F be a sheaf and suppose ExtðT=SU ;FÞ ¼ 0 for any open set

U of X . Of course, if we prove that F is flabby as a presheaf, then F will be aflabby sheaf. Thus, we will show that the sequence

HomðT;FÞ ! HomðSU;FÞ ! 0

in the category of presheaves is exact for any open set U of X .Let f : SU ! F be any morphisms of presheaves. Since F is a sheaf, we

get a unique morphism g : �SU ! F such that f ¼ g � iSU. Then, since

ExtðT=SU ;FÞ ¼ 0, there exists a morphism of sheaves h : �T ! F such thath � k 0U ¼ g. Hence, we have a diagram

in which the upper square and the lower triangle are commutative, so thediagram is commutative. This means that h � iT � kU ¼ g � iSU

¼ f , so thesequence

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HomðT;FÞ �!ðkUÞ HomðSU;FÞ ! 0

is exact and F is a flabby presheaf (i.e. actually a flabby sheaf).Conversely, consider for any U 2 O the diagram

T

pT=SU

iU0 ! F !a G !b T=SU ! 0

Since F is flabby, we know that bX : GðX Þ ! T=SU ðX Þ is an epi-morphism (see for example [3, Chapter II, Theorem 5.4]), so there existst 2 GðX Þ such that bX ðtÞ ¼ ððiU ÞX � pX Þð1Þ (note that TðX Þ ¼ R). Considerthen the morphism of presheaves j : T ! G induced by t (recall thatHomðT;GÞ ffi GðX Þ). This is such that b j ¼ iU p.

By the definition of T=SU we see that, if GW ;V : GðV Þ ! GðW Þ are therestriction maps of G, ðbU GU ;X ÞðtÞ ¼ 0 2 ðT=SU ÞðUÞ, that is, GU ;X ðtÞ ¼aU ðyÞ for some y 2 FðUÞ. But F is a flabby sheaf, so there exists u 2 FðX Þsuch that FU ;X ðuÞ ¼ y. If we consider the element t � aX ðuÞ 2 GðX Þ we seethat GU ;X ðt � aX ðuÞÞ ¼ 0 and that bX ðt � aX ðuÞÞ ¼ bX ðtÞ. This means thatwe can consider our original t 2 GðX Þ such that GU ;X ðtÞ ¼ 0, and in thatcase, jV ¼ 0 for all V 2 O, V � U , that is, jjSU

¼ 0. Thus, there exists amorphism of presheaves f : T=SU ! G with f p ¼ j, and so with b f ¼ iUsince p is an epimorphism.

Now, since G is a sheaf, we get a unique morphism of sheavesg : T=SU ! G such that g � iU ¼ f, and then we have bx gx ðiU Þx ¼ðb g iuÞx ¼ ðb fÞx ¼ ðiU Þx for all x 2 X . But ðiU Þx is an isomorphism for allx 2 X , so it follows that bx gx ¼ idðT=SU Þx

8x 2 X and then that b g ¼ idT=SU

,that is, the sequence

0! F!a G!b T=SU ! 0

splits, and ExtðT=SU ;FÞ ¼ 0. u

We recall that a topological space is said to be a Zariski topologicalspace if it satisfies the ascending chain condition on open sets, or equiva-lently, if every open subset is quasi-compact.

The next result is well known (see [7, Remark 3.6 (b) on pg. 113]). Forcompleteness we include a short proof.

Proposition 2.3. Let X be a Zariski topological space and fFa; a < lg anychain of sheaves on X. Then; the direct union F ¼ [a<lFa is a sheaf.

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Proof. It is clear that F is a presheaf, so we only need to argue that ifU ¼ [i2I Ui where Ui 2 O for all i 2 I , and si is an element of FðUiÞ for alli 2 I , such that FUi\Uj ;Ui

ðsiÞ ¼ FUi\Uj ;UjðsjÞ for all i; j 2 I , then there exists a

unique s 2 FðUÞ with FU ;UiðsÞ ¼ si 8i 2 I .

We know that U is quasi-compact since X is a Zariski topologicalspace, so U ¼ [i2J Ui for some finite subset J of I . Then, let us call G theleast element of the chain fFa; a < lg which contains the si for i 2 J , i.e.such that si 2 GðUiÞ 8i 2 J . Since G is a sheaf, there exists an elements 2 GðUÞ such that GUj ;U

ðsÞ ¼ sj 8j 2 J . Therefore we have the desireds 2 FðUÞ. It is clearly unique. u

Corollary 2.4. Any direct sum in the category of presheaves of a family ofsheaves on a Zariski topological space is indeed a sheaf.

Proof. Note that this is the case when the direct sum is finite and thenapply Proposition 2.3. u

The proof of the next result partially follows the argument given byEklof and Trlifaj in [5, Theorem 2]. In their proof they used the fact that thecategory of modules is generated by the class of all projectives, but it isknown that this is not the case for the category of sheaves. However, we willsee that the argument carries through without having to use this fact.

Theorem 2.5. Any sheaf of modules on a Zariski topological space has aspecial flabby preenvelope.

Proof. An immediate consequence of Proposition 2.2 is that the pairð?F ;FÞ is cogenerated by a set, namely, the set fT=SU ; U 2 Og. Then, it isclear that ð?F ;FÞ is cogenerated by the sheaf A ¼ �U2OT=SU (note that A isa sheaf by Corollary 2.4), which clearly belongs to ?F . ButA ffi ð�U2O TÞ=ð�U2O SU Þ, so we have an exact sequence of sheaves

0!MU2O

SU !k MU2O

T!f A! 0 :

Let C be any sheaf on X and j : ð�U2OSU ÞðHomð�SU ;CÞÞ �!C the unique

morphism of sheaves induced by using every f 2 Homð�SU ;CÞ. If we

construct the pushout of the morphisms j and k 0 ¼ kðHomð�SU ;CÞÞ, say C1, wesee that every morphism g0 : �U2O SU ! C can be extended to a morphismg1 : �U2O T ! C1. Moreover, C1=C ffi AðHomð�SU ;CÞÞ which clearly belongs to?F since it is a sheaf, so C1=C 2? F .

Following the same argument, we can find C2 such that C2=C1 2?Fand that every morphism �U2O SU !

g1C1 extends to a morphism

�U2O T ! C2. Thus, by transfinite induction, we find for any ordinal

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number m a continuous chain of objects fCa; a < mg such thatCaþ1=Ca 2?F 8aþ 1 < m and every morphism fa : �U2O SU ! Ca extendsto some faþ1 : �U2O SU ! Caþ1.

Now, by Proposition 1.1 we know that �U2OSU is @-generated for@ ¼ j‘ð�U2OSU Þj, so let l be the least ordinal number such that jlj > @ andconsider the continuous chain fCa; a < lg (in which C0 ¼ C). If we letD ¼ [a<lCa, we get by Proposition 1.2 that every morphism �U2O SU ! D

factors through �U2OSU ! Ca ! D for some a < l, and the morphism�U2O SU ! Ca extends to �U2O T ! Caþ1 by the construction of the chain.Therefore, we get an extension �U2O T ! D of �U2OSU ! D, which meansthat the sequence of sheaves

HomMU2O

T;D

!! Hom

MU2O

SU;D

!! 0

is exact, and so that

HomðT;DÞ ! HomðSU;DÞ ! 0

is exact for all U 2 O. But the proof of Proposition 2.2 shows that this isequivalent to the fact that

HomðT;DÞ ! HomðSU;DÞ ! 0:

is exact in the category of presheaves for all U 2 O, that is, to the fact thatD 2 F .

Let us consider then the sequence of sheaves

0! C! D! D=C! 0: :

The sheaf D=C is the direct union of the continuous chain fCa=C; a < lg,where C0=C ¼ 0 and Caþ1=Ca 2?F 8a < l. Thus, by [5, Lemma 1] we getthat D=C 2?F .

It is now immediate to verify that 0! C ! D is a flabby preenvelope,and in fact a special flabby preenvelope since D=C 2?F . u

Corollary 2.6. If X is a Zariski topological space; then the pair of classesð?F ;FÞ is a cotorsion theory cogenerated by a set; which has enough injec-tives.

Proof. The fact that ð?F ;FÞ is cogenerated by a set follows fromProposition 2.2, and that it has enough injectives is known by Theorem 2.5.

To prove that ð?F ;FÞ is a cotorsion theory we just have to observethat for any sheaf S 2 ð?FÞ? there exists an splitting exact sequence ofsheaves

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0! S! F! C! 0

with F 2 F and C 2 ð?FÞ?. Thus, S is a direct summand of F and so S 2 F(it is immediate that F is closed under direct summands). u

3. REMARKS AND EXAMPLES

Let X be the Sierpinski space consisting of two points with exactly oneof the points open. Then the category of sheaves on X is equivalent to thecategory of representations of the quiver � ! � (see [2, Chapter 4] for thisterminology). The representation M !f N is flabby if and only if f is sur-jective. To get a flabby preenvelope of M ! N , let P! N=f ðMÞ be surjec-tive where P is projective. If P! N is a lifting of P ! N=f ðMÞ then it is easyto check that the inclusion

M ! N

idNM� P ! N

is a flabby preenvelope and that in fact it is an envelope if and only ifP! N=f ðMÞ is a projective cover.

We also note that the Godement embedding of M ! N in a flabbysheaf (see [3, pg. 36]) is

M ! N

idNM�N ! N

From the above we can conclude that the Godement embedding israrely a flabby preenvelope.

REFERENCES

1. Aldrich, S.T.; Enochs, E.; Garcıa Rozas, J.R.; Oyonarte, L. Coversand Envelopes in Grothendieck Categories. Flat Covers of Complexeswith Applications. Preprint.

2. Benson, D.J. Representations and Cohomology I. Cambridge Studiesin Advanced Mathematics No. 30; Cambridge University Press:Cambridge, 1995.

3. Bredon, G.E. Sheaf theory; Second edition, Springer-Verlag, 1997.4. Eklof, P.C. Homological Algebra and Set Theory. Trans. Amer. Math.

Soc. 1977, 227, 2077225.

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5. Eklof, P.; Trlifaj, J. How to Make Ext Vanish. To appear in Bull.London Math. Soc.

6. Enochs, E. Injective and Flat Covers, Envelopes and Resolvents. IsraelJ. Math. 1981, 39, 1897209.

7. Milne, J.S. Etale Cohomology; Princeton University Press: Princeton,New Jersey, 1980.

8. Osofsky, B.L. Projective Dimension of ‘‘Nice’’ Directed Unions.J. Pure Appl. Algebra 1978, 13, 1797219.

9. Salce, L. Cotorsion Theories for Abelian Groups. Symposia Mathema-tica; Academic Press: London-New York, 1979; Vol. XXIII, 11732.

10. Swan, R.S. Cup Products in Sheaf Cohomology, Pure Injectives, and aSubstitute for Projective Resolutions. J. Pure Appl. Algebra 1999, 144,1697211.

11. Xu, J. Flat covers of modules. Lecture Notes in Mathematics No. 1634;Springer-Verlag, 1996.

Received February 2000Revised July 2000

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