Failure of flat descent of model structures on module categories

0.05 MB·English

by  Andrew Salch

FAILURE OF FLAT DESCENT OF MODEL STRUCTURES ON MODULE
CATEGORIES.
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Abstract. Weprove that the collection of modelstructures on(quasicoherent) module
1
categoriesdoesnotobeyflatdescent. Inparticular, itfailstobeaseparatedpresheaf,in
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thefppftopology,onArtinstacks.
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n
a
J
Contents
6

1. Introduction. 1
]
T 2. Modelasapresheaf. 2
A 3. Modelisnotanfppfsheaf. 2
References 4
.
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1. Introduction.
[

1 RecentlyH.Bacardaskedusaquestionwehadourselvesalreadybeenwonderingabout:
v doesthere exist a “local-to-globalprinciple” for modelstructures on modulecategories?
3 In other words, if one has an open cover of a scheme or a stack, and the structure of a
9 modelcategoryonthe(quasicoherent)modulecategoryofeachopeninthecover,canone
0
“paste” the modelstructures into a model structure on (quasicoherent)modulesover the
1
wholeschemeorstack?Inthisnoteweshowthattheanswertothisquestionisno,atleast
0
. intheflattopology.
1
We nowprovidemoredetails. For X aschemeoranalgebraicstack, let Model(X)be
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5 thecollectionofcofibrantlygeneratedclosedmodelstructuresonthecategoryofquasico-
1 herentO -modules. (Ingeneralthiscollectionisnotknowntoformaset,orevenaclass;
X
: itisjustacollection. FormanypracticalchoicesofX,however,itformsasetandevena
v
i finiteset. See[5]fordetails.)
X
Onewantstoknowif Modelisasheafinvarioustopologies. Moreprecisely: letX be
r analgebraicstackinatopologyτandchooseanτ-coverY →X. Onewantstoknowif
a
Model(X) // Model(Y) //// Model(Y×XY)
isanequalizersequence. In otherwords, givena modelstructureonthecategoryofO -
Y
moduleswhichrestrictscompatiblyonoverlapsbetweencomponentsofthecoverY,does
thereexistauniquemodelstructureonthecategoryofO -moduleswhichrestrictstothe
X
chosenoneonO -modules?Thisistheproblemofdescentofmodelstructuresonmodule
Y
categories.
In thisshortnote we givea negativeanswer to this questionforthe fppftopology,by
producing an explicit counterexample in Thm. 3.2. We do not know an answer to the
questionforcoarsertopologiesbutwe(withH.Bacard,M.Frankland,D.Schaeppi)have
exploredthe same question for the Zariski topology, where it seems more plausible that
thisquestionofdescenthasapositiveanswer,andperhapsinterestingresultscanbefound
Date:March2014.
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2 FAILUREOFFLATDESCENTOFMODELSTRUCTURESONMODULECATEGORIES.
in thatdirection. We are gratefulto J. F. Jardinefor hostingusduringa visit to Western
University,wherethesequestionscameupinconversation.
2. Modelasapresheaf.
Definition2.1. Letτbeoneofthefollowingtopologies:Zariski,e´tale,orfppf.LetXbean
algebraicstackinthetopologyτandletAlgStacks (X)bethecategoryofalgebraicstacks
τ
Y intheτ-topologyequippedwithmapsY → X whicharecomponentsofsomecovering
familyinτ.
ForeveryobjectY ofAlgStacks (X),letModel(Y)bethecollectionofcofibrantlygen-
τ
eratedclosedmodelstructuresonthecategoryofquasicoherentO -modules.
Y
Foreverymap f : Y′ → Y inAlgStacks (X),let Model(f) : Model(Y) → Model(Y′)
τ
bethepartially-definedfunctionsendingamodelstructureonquasicoherentO -modules
Y
to the transfermodelstructure, if it exists, onquasicoherentOY′-modules, i.e., themodel
structureinwhichamapg: M → N ofquasicoherentOY′-modulesisaweakequivalence
(respectively,fibration)ifandonlyif f∗g: f∗M → f∗Nisaweakequivalence(respectively,
fibration)inthegivenmodelstructureonquasicoherentO -modules.
Y
In the literature there is a small point of difference between variousdefinitions of al-
gebraic stack: the issue is what condition one requiresof the diagonalmap. The reader
canchoosewhateverdiagonalconditionstheylikebest,forthepurposesofDef.2.1. Our
counterexample(in Thm.3.2) to fppfdescentof modelstructures, however,worksin all
thedefinitionsoffppfalgebraicstackwhichareincommoncirculation,sinceinourcoun-
terexamplealldiagonalmapsareaffine,hencealsoquasicompact,quasiseparated,etc.
Thetransfermodelstructureexistsifandonlyif,foreverymorphismgofquasicoherent
OY′-moduleswhichisatransfinitecompositeofpushoutsofcoproductsofmorphisms f∗h
with h an acyclic cofibrationin the given modelstructure on O -modules, the morphism
Y
f∗g is a weak equivalencein the givenmodelstructure on OY-modules. This is a simple
applicationofCrans’stheoremonexistenceoftransfermodelstructures,from[3].
The assumption that our model structures be cofibrantly generated is an unnecessary
assumptioninmanysituationsofpracticalinterest.In[5]weshowthat,ifacategoryadmits
a“saturatingbasis,” theneverymodelstructureonthatcategoryis cofibrantlygenerated.
It then follows from the Cohen-Kaplansky theorem (from [2]) that, if R is an Artinian
principalidealring,theneverymodelstructureonthecategoryofR-modulesiscofibrantly
generated. We givemoreinformationandexplicitcomputationsin[5]. Itisprobablythe
casethatverymanymodulecategorieshavethepropertythatalltheirmodelstructuresare
cofibrantly generated. In any case, leaving out the adjectives “cofibrantly generated” or
“closed,”orboth,fromDef.2.1doesnotdeterThm.3.2fromprovidingacounterexample
toModelbeingasheaf.
3. Modelisnotanfppfsheaf.
Definition3.1. Letk beafieldofcharacteristic2andletα bethealgebraicgroupover
2
kco-representedbytheHopfalgebrak[x]/x2with xprimitive. Thatis,α isthealgebraic
2
group given on affines by letting α (SpecR) be the group of square-zero elements of R,
2
underaddition.
Let Bα be the (Artin) stack of α -torsors, i.e., the fppf algebraic stack associated to
2 2
the group(oid)affine scheme (Speck,Speck[x]/x2). We will write Speck → Bα for the
2
covernaturallyassociatedtothepresentationofBα givenbythegroupoidaffinescheme
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FAILUREOFFLATDESCENTOFMODELSTRUCTURESONMODULECATEGORIES. 3
(Speck,Speck[x]/x2). Finally,let j , j denotethetwo(partially-defined)maps
1 2
//
Model(Speck) // Model(Speck×Bα2 Speck)
andletidenotethe(partially-defined)map
Model(Bα )→ Model(Speck).
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Theorem3.2. Thereexisttwodistinctelementsa,binModel(Bα )suchthat:
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• i(a)andi(b)arewell-defined,
• i(a)=i(b),
• and j (i(a))and j (i(a))arewell-defined(andconsequently j (i(a))= j (i(a)).
1 2 1 2
Consequently, even if Model is a well-definedpresheaf, it is not a separatedpresheaf in
thefppftopology,hencenotasheafinthefppftopology.
Proof. We first define the model structures a,b on quasicoherentO -modules. By the
Bα2
usualdescentargument,quasicoherentO -modulesareequivalenttok[x]/x2-comodules.
Bα2
First,adefinition:supposeMisarightk[x]/x2-comodulewithstructuremapψ: M →
M⊗ k[x]/x2. WesaythatMisx-trivialifthecompositemap
k
M (idM−⊗→kη)−ψ M⊗ k[x]/x2 id−M→⊗kǫ M⊗ k
k k
iszero,whereηistheunitmapη:k →k[x]/x2. Inotherwords, Mis x-trivialifandonly
if,forallm∈ M,ψ(m)hasnononzerotermsinvolvingx.
Notethateveryrightk[x]/x2-comoduleMhasamaximalx-trivialsubcomoduletriv(M),
andthattheinclusiontriv(M)→ Mexpressesthex-trivialcomodulesasacoreflectivesub-
categoryofthecategoryofk[x]/x2-comodules.
Letmodelstructureaonthe categoryofk[x]/x2-comodulesbethe modelstructurein
which:
• thecofibrationsarethepushoutsofmapsbetweenx-trivialcomodules,
• the weak equivalences are the maps inducing an isomorphism on the maximal
x-trivialsubcomodules,
• andallmapsarefibrations.
In [1] (also, more directly, in our [4]) the necessary results on factorization systems are
provento ensure that, if C is an categorywith finite limits and colimits and A is a core-
flective replete subcategory of C, then there exists a model structure on C in which the
cofibrationsarethepushoutsofmapsbetweenobjectsinA,theweakequivalencesarethe
mapsinducinganisomorphismonapplyingthecoreflectorfunctor,andallmapsarefibra-
tions. Our model structure a is the special case of this theorem where A consists of the
x-trivialcomodules.
Letmodelstructurebonthecategoryofk[x]/x2-comodulesbethediscretemodelstruc-
ture,i.e.,themodelstructureinwhich:
• allmapsarecofibrations,
• theweakequivalencesaretheisomorphisms,
• andallmapsarefibrations.
We claim thati(a)and i(b)bothexist andin fact are eachequalto the discrete model
structureonthecategoryofk-modules. Indeed,undertheidentificationofquasicoherent
O -modules with k[x]/x2-comodules, we have the standard identification of the direct
Bα2
f
imageandinverseimagefunctorsintermsofcomodules:themapofstacksSpeck −→ Bα
2
inducesthedirectimagefunctor
f∗ :Mod(k)→Comod(k[x]/x2)
4 FAILUREOFFLATDESCENTOFMODELSTRUCTURESONMODULECATEGORIES.
sending a k-module M to the extended comodule M ⊗ k[x]/x2, and the inverse image
k
functor
f∗ :Comod(k[x]/x2)→Mod(k)
sendingak[x]/x2-comoduletoitsunderlyingk-module,i.e.,forgettingthecoactionmap.
So in the transfermodelstructurei(a) onk-modules, a morphismg is a weak equiva-
lence if and only if g⊗ k[x]/x2 is a weak equivalencein a. But this implies g must be
k
an isomorphism, since the map induced by g⊗ k[x]/x2 on the maximal x-trivialsubco-
k
modulesisgitself. Hencetheweakequivalencesini(a)arethesameasthediscreteweak
equivalences.Thesameistriviallytrueforfibrations.
Sincebisthediscretemodelstructureonk[x]/x2-comodules,itistriviallytruethatthe
weak equivalencesand fibrationsin i(b) agree with those of the discrete modelstructure
onk-modules.
Consequentlyi(a) andi(b) both existand are equalto the discrete modelstructureon
k-modules.
Nowallthatremainsistoshowthat j (i(a))and j (i(a))bothexist. Thattheyareequal
1 2
if they exist is trivial (for example, because j = j in this setting, since k[x]/x2 is a
1 2
HopfalgebraandnotonlyaHopfalgebroid!). Themodelstructure j (i(a))isthediscrete
1
model structure on k-modules transferred to quasicoherent OSpeck×Bα2Speck (cid:27) OSpeck[x]/x2-
modules, i.e., k[x]/x2-modules, along the direct image map, i.e., restriction of scalars.
So a morphism g of k[x]/x2-modules is a weak equivalence in j (i(a)) if and only if its
1
underlying map of k-modules is a weak equivalence in i(a), i.e., if and only if g is an
isomorphism. Similarly,amorphismgofk[x]/x2-modulesisafibrationin j (i(a))ifand
1
only if its underlying map of k-modules is a fibration in i(a), i.e., all morphisms g of
k[x]/x2-modulesarefibrations. Hencetheweakequivalencesandthefibrationsin j (i(a))
1
eachagreewiththoseinthediscretemodelstructureonk[x]/x2-modules.Hence j (i(a))=
1
j (i(a))= j (i(b))= j (i(b))existsandisthediscretemodelstructureonk[x]/x2-modules.
2 1 2
(cid:3)
References
[1] C.Cassidy,M.He´bert,andG.M.Kelly.Reflectivesubcategories,localizationsandfactorizationsystems.J.
Austral.Math.Soc.Ser.A,38(3):287–329,1985.
[2] I.S.CohenandI.Kaplansky.Ringsforwhicheverymoduleisadirectsumofcyclicmodules.Math.Z.,
54:97–101,1951.
[3] SjoerdE.Crans.Quillenclosedmodelstructuresforsheaves.J.PureAppl.Algebra,101(1):35–57,1995.
[4] AndrewSalch.Theclosedunitballinthespaceofmodelstructuresonacategory.availableonarXiv.
[5] AndrewSalch.Computingallclosedmodelstructuresonagivencategory.inpreparation.