Minimal semi-flat-cotorsion replacements and cosupport

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Journal of Algebra

www.elsevier.com/locate/jalgebra

Minimal semi-flat-cotorsion replacements and cosupport

Tsutomu Nakamura^a, Peder Thompson^b,∗

aGraduateSchoolofMathematics,NagoyaUniversity,Furocho,Chikusaku, Nagoya464-8602,Japan

bInstituttformatematiskefag,NorwegianUniversityofScienceandTechnology, N-7491Trondheim,Norway

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received23July2019 Availableonline15July2020 CommunicatedbyChangchangXi

MSC:

primary13D02 secondary13C13

Keywords:

Minimalcomplex Flatcotorsionmodule Cosupport

Finitisticdimension

Over a commutative noetherian ring R of finite Krull dimension,weshow thatevery complexofflat cotorsion R- modulesdecomposesas a directsum ofa minimal complex andacontractible complex. Moreover,wedefinethenotion of a semi-flat-cotorsion complex as a special type of semi- flat complex, and provide functorial ways to construct a quasi-isomorphism from a semi-flat complexto a semi-flat- cotorsion complex. Consequently, every R-complex can be replaced by a minimal semi-flat-cotorsion complex in the derivedcategoryoverR.Furthermore,wedescribestructureof semi-flat-cotorsionreplacements, bywhichwerecoverclassic theorems for finitistic dimensions. In addition,we improve some results on cosupport and give a cautionary example.

Wealsoexplainthatsemi-flat-cotorsionreplacementsalways existandcanbeusedto describethederivedcategory over anyassociativering.

©2020TheAuthors.PublishedbyElsevierInc.Thisisan openaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).

* Correspondingauthor.

E-mailaddresses:[email protected](T. Nakamura),[email protected] (P. Thompson).

https://doi.org/10.1016/j.jalgebra.2020.07.001

0021-8693/©2020TheAuthors.PublishedbyElsevierInc.ThisisanopenaccessarticleundertheCC BYlicense(http://creativecommons.org/licenses/by/4.0/).

Introduction

Theexistenceofinjectiveenvelopesformodulesoveranyringyieldsminimalinjective resolutions; dually, insettings where projectivecoversexist—such as for finitelygener- ated modules over a semi-perfect noetherian ring—one can build minimal projective resolutions. These classic forms ofminimality areencompassed bythe following defini- tion: a complex is minimal if every self homotopy equivalence is an isomorphism; see Avramov and Martsinkovsky [3]. Infact, Avramov, Foxby, and Halperin show [2] that everycomplexofinjectivemodulesdecomposesasadirectsumofaminimalcomplexand acontractiblecomplex,see alsoKrause[28],thusshowingeverycomplexhasaminimal semi-injectiveresolution.Adualstatement,consideredinitiallybyEilenberg[12],holds insettingswhere projectivecoversexist.

A naturalquestion iswhetheracomplex offlat modulesexhibits similar behaviour.

Although flatcovers do exist for modules over any ring, due to Bican, El Bashir, and Enochs[6],itturnsoutthatminimalityispoorlybehavedforcomplexesofflatmodules ingeneral:indeed,there existquasi-isomorphismsbetweenminimalsemi-flatcomplexes thatarenotisomorphismsofcomplexes (unlikethecaseforminimalsemi-projective or semi-injectivecomplexes),seeforexampleChristensenandThompson[10].Wethusre- strictourfocustocomplexesofaspecialtypeofflatmodules:theflatcotorsionmodules.

Let R be a commutative noetherian ring. Enochs shows [13] that flat cotorsion R- modules—i.e., those flatmodules that are also right Ext-orthogonal to flat modules—

haveauniquedecomposition, whose structureis akinto thatofinjectivemodulesover anoetherianringas shownbyMatlis[30].Further,minimalitycriteriaforcomplexesof flatcotorsionR-moduleswasgivenbyThompson[48].Onegoalofthispaperistoshow thatwhenRhasfiniteKrulldimension,suchcomplexescanbedecomposedanalogously to complexesofinjectivemodules:

Theorem (See 1.9 and 2.4).Assume dimR < ∞. If Y is a complex of flat cotorsion R-modules,thenY =Y⊕Y,whereY isminimalandY iscontractible.

InSection3,wegivetwofunctorialapproachstoconstructacomplexofflatcotorsion R-modules;oneofthembuildsonworkofNakamuraandYoshinoin[37],andtheotheris inspiredbyit.Wealsoturntoconsideringsemi-flat-cotorsioncomplexes,thatis,semi-flat complexesofflatcotorsionR-modules,aswellasreplacementsbysuchcomplexesinthe derivedcategoryoverR;seeAppendixA.IfF isasemi-flatcomplex,thenConstructions 3.1 and 3.3yield functorialways to buildasemi-flat-cotorsion complexY and aquasi- isomorphismF →Y.Inparticular, weobtain:

Theorem (See 3.4). Assume dimR < ∞. Every R-complex has a minimal semi-flat- cotorsionreplacement in thederived category overR.

Although it is immediate from [48, Theorem 5.2] that every R-module hasa minimal semi-flat-cotorsion replacement without the assumption of finite Krull dimension, the

assumptionhereisnaturalinconsideringunboundedcomplexes.Onemotivationforour approach is that not every R-module admits a surjection from, or injection to, aflat cotorsion R-module—see Example 3.11—and so our method differs from the one for complexesofinjectivemodulesgivenin[28, AppendixB].

InSection 4, we employthe functorial construction inConstruction 3.3, along with the Auslander–Buchsbaumformula, to describe the structure of semi-flat-cotorsion re- placements; see Lemma 4.1 and Theorem 4.6. In particular, this extends structure of theminimalpure-injectiveresolutionofaflatmoduledescribedbyEnochs[14],andalso recovers—seeCorollary 4.7—the factthat the finitistic flat dimension of R is at most dimR.Inaddition,thisstructuregivesanewproofofaclassicresultofGrusonandRay- naud[41] andJensen[25]:anR-moduleoffiniteflatdimensionhasprojectivedimension atmostdimR,seeTheorem4.9;inparticular, thefinitisticprojectivedimensionofRis atmostdimRandflatR-moduleshaveprojectivedimensionatmostdimR.

In Section 5, we apply the other functorial construction, Construction 3.1, in the contextofcosupport.ThecosupportofanR-complexX isthesetofprimeidealspsuch thatRHomR(κ(p),X) is nontrivial inthe derived category over R. As an analogue to workofChenandIyengar[8],wegiveinExample5.11anunboundedminimalcomplexY offlatcotorsionR-modulessuchthatcosupp_RY isstrictlycontainedin

i∈Zcosupp_RYⁱ. Thisgivesacounterexampleto[47,Theorem2.7],unfortunately,andweproceedtogive acorrection—and improvement—forthisresult;seeTheorem5.4.

In the appendix, we define the notion of semi-flat-cotorsion replacements for any associativeringA,andpointtohowthesecomplexescanbeusedtodescribethederived category over A. In particular, we note that—due to a result of Gillespie [20]—every A-complexcanbereplacedbyasemi-flat-cotorsioncomplexinthederivedcategoryover A,althoughminimalityremainsopen;seeQuestion A.10.

∗ ∗ ∗

Throughout,let R be acommutative noetherianring. We use standardcohomological notation for R-complexes (that is, complexes of R-modules), and use H(−) to denote thecohomologyfunctor.DenotebyModRthecategoryofR-modules,C(R) thecategory of R-complexes, K(R) the homotopy category of R-complexes, and D(R) the derived category over R. A morphism α : X → Y inC(R) or K(R) is aquasi-isomorphism if H(α) is anisomorphism;an R-complex X isacyclic if H(X)= 0, and iscontractible if X isisomorphicto thezerocomplexinK(R).

1. Decomposingcomplexesofflat cotorsionmodules

Foracomplex P of finitely generated freemodulesover alocal ring (R,m,k),there exists adecompositionP =P⊕P suchthatk⊗RP haszerodifferential and P is contractible;thiswasshownin[2].Althoughsuchaphenomenondoesnotextendtoall complexesofinfinitelygenerated projectivemodules(seeExample1.6),theredoesexist

asimilar decompositionifwetakecomplexesof m-adiccompletionsoffreemodules. In this section,weexplainthis factandextenditto thecaseofcomplexesofflatcotorsion modules.

Westartwiththefollowingelementarylemma,inwhichRisnotrequiredtobelocal.

Lemma 1.1.Let a be an ideal of R, let T and T be a-adic completions of projective R-modules,andletϕ:R/a⊗RT →R/a⊗RT beahomomorphism.

(1) Thereexists ahomomorphismϕ:T →T suchthat R/a⊗Rϕ=ϕ.

(2) Any suchliftingϕ:T →T isanisomorphismifR/a⊗Rϕ=ϕisanisomorphism.

Remark1.2.WriteT = lim_{←−−n≥1}(P/aⁿP) foraprojectiveR-moduleP.Fortheproofofthe lemma, we recallthatthere is anaturalisomorphismT /aⁿT ∼=P/aⁿP foreachn≥1.

This iswell-knownforspecialists;whenP isfinitely generated,[32,§8] issufficient,but even when P is infinitely generated, it is within classic commutative algebra, see [45, Corollary 2.1.10 and Proposition2.2.3] or[37, Lemma2.3].Indeed,itis furtherknown thattheisomorphismholdstrueforanyR-module,see[43,Theorem1.1] or[45,Theorem 2.2.5].

Proof of Lemma 1.1. Setφ₁ =ϕ.For n≥1,wehaveT /aⁿ⁺¹T isaprojectiveR/aⁿ⁺¹- module by Remark 1.2, hence a map φn : T /aⁿT → T/aⁿT lifts to a map φn+1 : T /aⁿ⁺¹T → T/aⁿ⁺¹T. Induction yields maps φn for every n ≥ 1, thus setting ϕ = lim←−−n≥1φ_n yields(1).

For(2),letϕ:T →T beanyliftingofϕsuchthatR/a⊗Rϕ=ϕisanisomorphism.

Define φ_n : T /aⁿT → T/aⁿT as the map induced byϕ for n ≥1, where φ₁ =ϕ. It is enough to show that each φn is bijective since ϕ= lim_{←−−n≥1}φn. Weremark thatany R/aⁿ-module M with(a/aⁿ)M =M iszero sincethe ideala/aⁿ of R/aⁿ is nilpotent.

HencesurjectivityofR/a⊗Rφ_n =ϕimpliesφ_n issurjective, andinfactsplitsurjective since T/aⁿT is projectiveover R/aⁿ. Thusinjectivity of R/a⊗Rφn =ϕ alsoimplies kerφ_n= 0,thatis, φ_n isinjective.

Remark1.3.WhenRisalocalringwithmaximalidealmandTisthem-adiccompletion of afree R-module, the canonical map T → T /mT is aflat cover by [49, Proposition 4.1.6], whichcaninsteadbeused toverifyLemma1.1inthiscase.

TheargumentintheproofofLemma1.1isinspiredbytheproofof[41,II,Proposition 2.4.3.1],whichshowsthatthem-adiccompletionofaflatR-moduleisisomorphictothe m-adiccompletionofafreeR-module;see also[16,Lemma6.7.4].

ForanindexsetAandanR-moduleM,wedenotebyM^(A)or

AM thedirectsum ofA-copiesofM.If(R,m,k) islocal,thenwewrite Mforthem-adiccompletionofM.

Lemma 1.4. Assume (R,m,k) is a local ring. Let A and A be some index sets, and let ∂ :R^(A) →R^(A) be ahomomorphism of R-modules. Thereexist disjoint partitions A=B C andA =B C andacommutative diagram ofR-modules

R^{(A) ∂} R^(A)

R^(B)⊕R^(C)

∼=

1 0 0 ∂

R^(B)⊕R^(C)

∼=

wherek⊗R∂ = 0.

Proof. There are isomorphisms k⊗RR^(A) ∼= k^(A) and k⊗R R^(A) ∼= k^(A), hence we mayviewk⊗R∂ as alineartransformation of k-vector spaces. Since ker(k⊗R∂) and im(k⊗R∂) aresubspaces(andhencedirectsummands),wemayfinddisjoint partitions A=B C andA=B C suchthatthefollowing diagramcommutes:

k^{(A) k⊗R∂} k^(A)

k^(B)⊕k^(C)

∼= α

1 0 0 0

k^(B)⊕k^(C)

∼= β

The maps αand β lift, by Lemma1.1, to isomorphismsα: R^(B)⊕R^(C) → R^(A) and β:R^(B)⊕R^(C)→R^(A).Wethusobtainacommutativediagram:

R^{(A) ∂} R^(A)

R^(B)⊕R^(C)

∼= α

i f g h

R^(B)⊕R^(C)

∼= β

where k⊗Ri= 1 andk⊗Rf =k⊗Rg =k⊗Rh= 0.Thus Lemma1.1implies thati isanisomorphism;theconditionsonf, g, andhallowfor anelementarytranslationof thediagraminto thedesiredone.

Weaimto apply Lemma1.4to acomplexY of m-adiccompletionsof freemodules.

Towards this end, note that application of the lemma to ∂ = d⁰_Y replaces the 4-term complexY⁻¹→Y⁰→Y¹→Y² withthefollowingone:

R^(D)

0

a R^(B)⊕R^(C)

1 0 0 ∂

R^(B)⊕R^{(C) [ 0 b]} R^(D), wherek⊗R∂= 0.HencewecanextractadirectsummandY(0) ofY correspondingto acontractiblecomplex0→R^{(B) =}−→R^(B)→0.ByLemma1.4,wecanfurtherfindsuch contractible direct summands Y(−1) and Y(1) of Y from d⁻¹_Y and d¹_Y respectively.

Then it is clear from the above matrices that the canonical map Y(−1)⊕Y(0)⊕ Y(1)→Y isasplitmonomorphism.Thisobservationcanbeusedtoshowthefollowing lemma.

Lemma1.5. Assume(R,m,k)isalocal ring.IfY isacomplex ofm-adic completionsof free R-modules,then Y =Y⊕Y,such thatthe complex k⊗RY has zero differential and Y iscontractible.

Proof. ApplyingLemma1.4to dⁿ_Y :Yⁿ →Yⁿ⁺¹ foreach n∈Z,extractacontractible direct summand Y(n) of Y such thatthe differential of Y /Y(n) in degree n is zero upon application of k⊗R−. Then Y has a contractible direct summand of the form Y =

n∈ZY(n) =

n∈ZY(n), and the differential of Y = Y /Y is zero upon applicationofk⊗R−.

The next example exhibits the necessityof taking completions to obtain a suitable decomposition.

Example1.6.Let(R,m,k) bealocalringwithdimR≥1.Letx∈mbeanelementthat isnotnilpotent.ThelocalizationRxisthereforenonzeroandhasaprojectiveresolution of the form P = (0 →

NR →

NR → 0); indeed, R_x ∼= R[Y]/(1−xY) for an indeterminateY, hencetheexactsequence

0 R[Y] ^1−xY R[Y] R_x 0

providessucharesolutionP.Sincek⊗RRx= 0 andRxisaflatR-module,thecomplex k⊗RP = (0→

Nk−→^∼=

Nk→0) isexact,thusP hasnononzerodirectsummand P such that k⊗RP haszero differential. However, P is notcontractible sinceRx is nonzero.

ThegoalofthissectionistoextendLemma1.5abovetothecaseofcomplexesofflat cotorsion modules, and so webegin with somebasicfactsabout these.Here we return to thesettingofanycommutativenoetherianringR.

AnR-moduleM isflatcotorsionifitisbothflatandcotorsion,thatis,Misflatand Ext¹_R(F,M)= 0 for everyflatR-moduleF.Enochsshowsin[13] thatan R-moduleM is flat cotorsion ifand only ifM ∼=

p∈SpecRTp, where Tp is thep-adiccompletionof

afreeRp-module.For anideal aofR, letΛ^a = lim_{←−−n≥1}(−⊗RR/aⁿ) denotethe a-adic completionfunctor;foranR-moduleM,also writeΛ^aM =M_a^∧.

AmotivationforstudyingcomplexesofflatcotorsionR-modulesistheirrelationship tocosupport.ThenotionofcosupportwasdefinedbyBenson,Iyengar, andKrause[5], whoseworkwasinspiredbyNeeman’s[40].ForanR-complexX,thecosupportofX is:

cosupp_RX={p∈SpecR|H(RHomR(κ(p), X))= 0},

whereκ(p) standsfortheresiduefieldRp/pRp.Seealsotheequivalentcharacterizations in(2.1).ThisisdualtothenotionofsupportdefinedbyFoxby[18];thesupport ofX is:

supp_RX={p∈SpecR|H(κ(p)⊗^LRX)= 0}. For an index set A, we have supp

AE(R/p) ⊆ {p}, where E(R/p) stands for the injectivehullofR/poverR.Further,[16,Theorem 3.4.1(7)] yieldsanisomorphism

(

ARp)^∧_p ∼= HomR(E(R/p),

AE(R/p)). (1.7) Fromthisandtensor-homadjunction,weseethatcosupp(

ARp)^∧_p ⊆ {p}.Consequently itfollows thata flatcotorsion R-module M hascosupportcontained inasubset W of SpecR if and only if M ∼=

p∈WT_p, where T_p is the p-adic completionof afree R_p- module.WecanthereforetranslateLemma1.5 to:

Lemma 1.8. Let p ∈ SpecR. If Y is a complex of flat cotorsion R-modules with cosupp_RYⁱ⊆ {p}foreveryi∈Z,thenY =Y⊕Y,suchthatthecomplex κ(p)⊗RY has zerodifferential andY iscontractible.

Proof. Reducetoalocalring(R,m,k);this isjustarestatementofLemma1.5.

ForasubsetW ofSpecR,we definedimW asthesupremum ofthelengthsof strict chainsof prime idealsin W. As is standard, dim(SpecR) is denoted bydimR; this is the Krull dimension of R. The next theorem is the main result of this section. In its proof, weuseseveral basicfactsaboutcomplexes offlatcotorsion R-modules; theyare summarizedat theend ofthissection.

Theorem1.9. LetW ⊆SpecR with dimW <∞.If Y isacomplex of flat cotorsionR- moduleswithcosupp_RYⁱ⊆W foreveryi∈Z,thenY =Y⊕Y,suchthatthecomplex κ(p)⊗RHom_R(R_p,Y)has zero differential foreveryp∈W andY iscontractible.

Proof. We proceed by induction on dimW. First suppose dimW = 0. In this case, Y ∼=

q∈WΛ^qY by (1.18), and Λ^qY consists of flat cotorsion R-modules having co- support contained in {q} by (1.13). For each q ∈ W, we apply Lemma 1.8 to obtain

a decomposition Λ^qY = Y(q)⊕Y(q), where κ(q)⊗RY(q) haszero differential and Y(q) iscontractible. Takingaproductover q∈W, weobtainadecomposition

q∈WΛ^qY =

q∈W(Y(q)⊕Y(q))∼=

q∈WY(q)

⊕

q∈WY(q)

. (1.10) A product ofcontractible complexesis contractible,hence

q∈W Y(q) iscontractible;

moreover,(1.14) impliesthatforeveryp∈W thereisanisomorphism κ(p)⊗RHom_R(R_p,

q∈WY(q))∼=κ(p)⊗RY(p), and thelatterhaszerodifferential.

Nextsuppose dimW =n>0.SetZ =

q∈maxW Λ^qY.By(1.17),there isadegree- wisesplitexactsequenceofcomplexes offlatcotorsion R-modules:

0 X Y Z 0.

The complexes X and Z are complexes of flat cotorsion R-moduleswith cosupport in W \maxW and maxW, respectively. As dim(W \maxW) < n and dim(maxW) = 0< n, wemayapply theinductivehypothesis toobtaindecompositions X =X⊕X and Z =Z⊕Z, where κ(p)⊗RHom_R(R_p,X) and κ(p)⊗RHom_R(R_p,Z) have zero differential for every p∈ W and X and Z are contractible; see also (1.15). Letting π : X → X be the canonical projection, there exists a complex P of flat cotorsion R-modulesmakingthefollowingpush-outdiagramcommute:

0 X

π

Y

f

Z 0

0 X P Z 0

The snake lemma yields an exact sequence of complexes of flat cotorsion R-modules 0→X→Y −→^f P →0;evidently,thissequence isdegreewisesplit,and itfollowsfrom the proof of [3, Lemma 1.6] (seealso [10, Propositions 2.5 and 2.6])that thesequence splits inC(R) andf isahomotopyequivalence.

Ontheotherhand,lettingι:Z→Z bethecanonicalinclusion,weobtainacomplex QofflatcotorsionR-modulesmakingthepull-backdiagramcommute:

0 X P Z 0

0 X Q

g

Z

ι

0

(1.11)

The snake lemma yields a degreewise split exact sequence 0→ Q−→^g P → Z → 0 of flat cotorsionR-modules.As Z iscontractible, this sequencesplits inC(R) andg is a

homotopyequivalencebythedualargumentoftheproofof[3,Lemma1.6] (seealso[10, Propositions2.5and 2.6]); letg :P →Qbe asplitting ofg inC(R), andnote thatg isalsoahomotopy equivalence.Thuswehaveasplitexactsequence

0 ker(gf) Y ^g

f

Q 0,

whereker(gf) isacontractible complexofflatcotorsionmodules.

It remains to show that for every p ∈ W, the complex κ(p)⊗R HomR(Rp,Q) has zero differential. To do so, we use the degreewise split exact sequence in the bottom row of (1.11). The modulesin X have cosupport contained inW \maxW as X is a directsummand ofX;similarlythemodulesinZ havecosupportcontainedinmaxW. Ifp∈maxW,thenκ(p)⊗RHomR(Rp,X)= 0 by(1.15).This impliesthat

κ(p)⊗RHomR(Rp, Q) =κ(p)⊗RHomR(Rp, Z)

andthatthelatterhaszerodifferentialbyconstruction.Ifp∈W\maxW,thenwehave HomR(Rp,Z)= 0 by (1.14) andhence

κ(p)⊗RHom_R(Rp, X) =κ(p)⊗RHom_R(Rp, Q), wheretheformerhaszerodifferentialbyconstruction.

Remark 1.12.It is known thata complex of objects in an abelian category admitting injective envelopes can be decomposed as a direct sum of a minimal complex and a contractiblecomplex;see[28,PropositionB.2].Inoursituation,however,itisnotclear howtheargumentsof[28,PropositionB.2] canbeemployed;indeed,flatenvelopesmay not exist, and although flat covers do exist over any ring [6], there exists a minimal complex of flat cotorsion modules thatis not built from flat covers, see Example 2.7.

This is one motivation for modellingthe arguments hereon that of finitely generated freemodulesover alocal ring.

Ontheotherhand,minimalityofacomplexofflatcotorsionmoduleswithcosupport in{p}canbe characterizedbyflatcovers;seeTheorem2.3.

In the remainder of this section, we summarize several basic facts concerning flat cotorsionR-moduleswhichare oftenusedinthis paper.LetF =

q∈SpecRTq be aflat cotorsionR-module,whereTq istheq-adiccompletionofafreeRq-module.Thederived functors LΛ^p and RHom_R(R_p,−) with p ∈ SpecR are useful for working with such a module;inparticular, thefollowinghold:

LΛ^pF∼= Λ^pF ∼=

q⊇pTq; (1.13)

RHom_R(Rp, F)∼= HomR(Rp, F)∼=

q⊆pTq. (1.14)

See[29,§4,p.69] and[48, Lemma2.2].Notsurprisingly, theaboveformulasextendto boundedcomplexesofflatcotorsionR-modules,seealso(5.2).Ifforeachq∈SpecRwe write Tq= (

BqRq)^∧_q forsomeindex setBq,then

κ(p)⊗^LRRHomR(Rp, F)∼=κ(p)⊗RHomR(Rp, F)∼=

Bpκ(p), (1.15) see alsoRemark1.2.

Forafinitelygenerated R-moduleM,there isalsoacanonicalisomorphism

M⊗RHomR(Rp, F)∼= HomR(Rp, M⊗RF). (1.16) Thismapisgivenbythetensorevaluationmap,andweonlyneedtocheckrightexactness of HomR(Rp,−⊗RF). This verificationcanbe reduced to checking right exactness of Hom_R(Rp,−⊗RTq) foreachprimeq, asM isfinitelypresented, whichcanbechecked byusing (1.7).

WenextexplainausefulreductiontechniqueforcomplexesofflatcotorsionR-modules thatisusedanumberoftimes.LetW beasubsetofSpecRandY beacomplexofflat cotorsion R-moduleswithcosuppYⁱ⊆W.Wemaythenwrite

Y = ( · · ·

q∈WT_qⁱ

q∈WT_qⁱ⁺¹ · · · ), where Yⁱ =

q∈W T_qⁱ and each T_qⁱ is the q-adic completion of a free R_q-module. We denote by maxW the subsetof W consisting of primeidealswhich aremaximal inW with respecttoinclusion.Ifp∈maxW,then

Λ^pY = ( · · · T_pⁱ T_pⁱ⁺¹ · · · ) by (1.13). Thus, the chain map Y →

p∈maxWΛ^pY induced by the canonical chain maps Y →Λ^pY yieldsadegreewise splitexactsequence:

0 X Y

p∈maxWΛ^pY 0, (1.17)

where Xⁱ =

p∈W\maxWT_pⁱ. In particular, if dimW = 0, then maxW = W and we have

Y ∼=

p∈WΛ^pY . (1.18)

2. Minimalitycriteriaforcomplexesofflatcotorsionmodules

Wenowaimtorefineandrecover[48,Theorem3.5],whichgivesminimalitycriteriafor complexes offlatcotorsionmodules;ourapproachusestoolsfrom theprevioussection.

LetY beanR-complex.There areimportantbi-implicationsaboutcosupport:

p∈cosupp_RY ⇐⇒ H(LΛ^pRHomR(Rp, Y))= 0 (2.1)

⇐⇒ H(κ(p)⊗^LRRHomR(Rp, Y))= 0.

Thesecharacterizationswereessentiallyshownin[5];see also[42,Proposition4.4].

Thenextlemmaisaversionof[48,Lemma3.1];theproofgivenhereinsteadusesthe notionofcosupport.

Lemma 2.2.Let f be ahomomorphism of flat cotorsion R-modules. The followingcon- ditionsare equivalent:

(1) f isan isomorphism.

(2) Λ^pHomR(Rp,f)isanisomorphism foreveryp∈SpecR.

(3) κ(p)⊗RHomR(Rp,f)isan isomorphismforeveryp∈SpecR.

Proof. AcomplexX satisfiescosupp_RX =∅ifandonlyifXisacyclic,¹seeforexample [5, Theorem 4.5]. By definition, f is an isomorphism if and only if cone(f) is acyclic.

Hencecosupp_R(cone(f))=∅ ifand only if (1) holds. Finally, (2.1) alongwith (1.13), (1.14),and (1.15) yieldthisisalsoequivalent to(2)or (3).

AnR-complexX isminimalifeveryhomotopyequivalenceX →Xisanisomorphism inC(R);see [3].Compare thenext resultto [48,Theorems 3.5 and4.1];conditions (2) and(5)herearenew.LetussimplydenotebyRp thep-adiccompletionofRp.

Theorem2.3. LetY be acomplexof flat cotorsionR-modules.Thefollowingconditions areequivalent:

(1) Thecomplex Y isminimal.

(2) IfY =Y⊕Y andY iscontractible, thenY= 0.

(3) Foranyp∈SpecR,thecomplexκ(p)⊗RHomR(Rp,Y)has zerodifferential.

(4) For any p ∈ SpecR, the complex Λ^pHom_R(Rp,Y) has no direct summand of the form 0→R_p−→⁼ R_p →0.

(5) Foranyp∈SpecR andi∈Z,thecanonicalmapTⁱ⁺¹→coker(dⁱ_T)isaflat cover, where T = Λ^pHomR(Rp,Y).

Proof. (1)⇒(2):Thisfollowsby[3,Proposition1.7(3)].

(2)⇒(3):Fixp∈SpecRandsetX = Hom_R(R_p,Y).AsX isacomplexofflatcotor- sionRp-modules,see(1.14),wemayapplyTheorem1.9toX toobtainadecomposition

1 ThisisadirectconsequenceofNeeman’s[38,Theorem2.8],whichsaysthatD(R) isgeneratedbythe set{κ(p)|p∈SpecR}.

X =X⊕X such thatκ(p)⊗RX has zerodifferentialand X iscontractible. From thecanonicalprojectionπ:X→X, formapush-outdiagram:

0 X

π

Y Y /X 0

0 X P Y /X 0

As intheproofofTheorem 1.9,thesnakelemmayieldsasplitexactsequence

0 X Y P 0.

Theassumption (2)nowimpliesX= 0,thusX =X.Henceitholdsthat κ(p)⊗RHomR(Rp, Y) =κ(p)⊗RX=κ(p)⊗RX, whichhaszerodifferential;(3)follows.

(3)⇒(1): Letf :Y →Y be ahomotopy equivalence.Thusκ(p)⊗RHom_R(R_p,f) is also ahomotopyequivalence foreveryp∈SpecR.However,sincethecomplexκ(p)⊗R

HomR(Rp,Y) haszerodifferential,itfollowsthatκ(p)⊗RHomR(Rp,f) isanisomorphism foreveryp∈SpecR.Lemma2.2nowyields thatf isanisomorphism.

(3) ⇔ (4): Fix p ∈ SpecR and set T = Λ^pHomR(Rp,Y). The forward implication follows by replacing Y by T inthe implication (3) ⇒ (2) already proven above. Con- versely, condition(4) forces dⁱ_T :Tⁱ →Tⁱ⁺¹ per Lemma1.4 to havethe property that κ(p)⊗Rdⁱ_T = 0 foreveryi∈Z.

(3) ⇔ (5): Fix p ∈ SpecR and set T = Λ^pHomR(Rp,Y). For each i ∈ Z, apply Lemma2.5belowtotheexactsequence

T^{i d}

i

T Tⁱ⁺¹ coker(dⁱ_T) 0

to showthatTⁱ⁺¹→coker(dⁱ_T) isaflatcoverifandonlyifκ(p)⊗Rdⁱ_T = 0.

Corollary 2.4. AssumedimR <∞.If Y isacomplex of flat cotorsion R-modules,then Y =Y⊕Y where Y isminimalandY is contractible.

Proof. ApplyTheorem1.9and theequivalence (1)⇔(3) ofTheorem 2.3.

Thenextlemmaisneededfortheequivalence(3)⇔(5) inTheorem2.3above;notice that itsproof shows anR_p-module M having apresentationbyflat cotorsion modules with cosupportin{p}infacthasaresolutionbysuchmodules.

Lemma2.5.Letp∈SpecR andletT⁰ andT¹ bep-adiccompletionsof freeRp-modules.

SupposeT⁰ −→^f T¹ −→^g M →0is an exact sequence of R_p-modules. The mapg is aflat coverof M over Rif andonly ifκ(p)⊗Rf = 0.

Proof. If κ(p)⊗Rf = 0, then Lemma1.4 implies thecomplex T⁰ −→^f T¹ has adirect summandRp

−→= Rp.TheexactsequenceT⁰−→^f T¹−→^g M →0 thusgivesadecomposition g= [ 0 h] :Rp⊕T→M,whereRp⊕T =T¹.Theendomorphism0⊕id_T :Rp⊕T → R_p⊕T isnotanisomorphism,yetitsatisfies g·(0⊕id_T)= 0⊕h=g;henceg isnot aflatcover.

Conversely,suppose thatκ(p)⊗Rf = 0;thisisequivalenttosayingthatκ(p)⊗Rg is anisomorphism.Supposethatthereisacommutativediagram:

T¹

g

T¹

g M

By assumption, all maps in this diagram become isomorphisms upon application of κ(p)⊗R−, and so Lemma 1.1(2) implies that the map T¹ → T¹ is an isomorphism.

Henceit remains to show thatg is aflatprecover, or equivalently, ker(g) iscotorsion.

Toshowthis,we willprovethatthereisanexactsequence

· · · T⁻² T⁻¹ T^{0 f} T^{1 g} M 0, (2.6) where all Tⁱ are p-adic completions of free Rp-modules. Then the truncated complex

· · · →T⁻² → T⁻¹ → T⁰ → 0 is a resolution of ker(g), and we can easily verify that ker(g) iscotorsion,byusingRemark3.2 and(A.1).

SetK = ker(f). Thep-adiccompletionfunctor induces anisomorphismonboth T⁰ andT¹,henceweobtainthefollowingcommutativediagram:

0 K T^{0 f}

=

T¹

=

K_p^∧ T^{0 f} T¹

Byasimple diagram chase,the image of themap K_p^∧ → T⁰ is precisely K,hence the second row is exact. Choose asurjection from afree Rp-module F → K; this induces asurjection F_p^∧ =T⁻¹ → K_p^∧ bya standardargument (seethe proof of[32, Theorem 8.1]),henceweobtainasurjectionT⁻¹→K.Repeating thisprocess, wecanconstruct anexactsequenceas in(2.6).

The existenceof theexactsequence (2.6) is alsoaconsequence ofaresultof Dwyer and Greenlees[11,Proposition5.2],whichimpliesthatM ∼= LΛ^pM inthissetting.

Weendthesectionwithanexampleshowingthatstatement(5)inTheorem2.3may be thebestpossibleintermsofflatcovers:

Example2.7. LetkbeanuncountablefieldandR=k[x,y].Theminimal pure-injective resolutionofRisaminimalcomplexofflatcotorsionR-modulesoftheform0→P^{0 d}

−→0

P¹ −→^d1 P² →0; see [35, Remark3.3 andTheorem 4.8]. Although P² = coker(d⁰), the map d¹:P¹→P² isnotaflatcover.

A similar example can be constructed for the ring k[x,y]_(x,y), using Gruson’s [23, Proposition3.2].

3. Functorialconstructionsofsemi-flat-cotorsionreplacements

Inthissection,wegivetwofunctorialways toconstructachainmapfrom acomplex offlatmodulestoacomplexofflatcotorsionmodulessuchthatitsmappingconeispure acyclic; recallthatacomplex P ispure acyclic if M⊗RP isacyclic forany R-module M.Inparticular,thisapproachyieldsareplacementofasemi-flatcomplexthatisboth semi-flatand semi-cotorsion(definedbelow).

Althoughthesettingofthisfirstconstructionisabitrestricted,theconstructionitself is notcomplicated;moreover,itplaysakeyroleinExample 5.11below.

Construction 3.1. Assume dimR ≤ 1 or R is countable. Let P be an R-complex of projective modules. Let W be the set of maximal ideals of R. The canonical map Pⁱ →

m∈WΛ^mPⁱ isapure-injectiveenvelopefor eachi∈Z, see[16, Remark6.7.12].

Moreover,itfollowsfrom[16,Theorem8.4.12,Corollary8.5.10],[41,II,Corollary3.3.2], and [26, Theorem 5.8] that the pure-injective dimension of Pⁱ is at most 1 (see also Remark3.7)andso thereisashortexactsequenceofcomplexes

0 P

m∈W Λ^mP (

m∈WΛ^mP)/P 0 whereeverytermof(

m∈WΛ^mP)/P isaflatcotorsionmodule,see[16,§8.5].Regarding the abovesequenceas adouble complex, denoteits totalcomplexby XP. Therowsof this doublecomplexarepureexact,thatis,theyareexactuponapplicationofM⊗R− foranyR-moduleM.Abasicargument[27,Theorem12.5.4] ofdoublecomplexesshows thatX_P ispure acyclic.Ontheother hand,there isacommutativediagram

P 0

m∈WΛ^mP (

m∈WΛ^mP)/P

whereallarrowsexpressthecanonicalchainmaps.Regardingbothrowsasdoublecom- plexes,thismorphismbetweendoublecomplexesnaturallyinducesachainmapP →YP, whereYP denotesthetotalcomplexofthesecondrow.Themappingconeofthecanoni- calmapP →Y_P canbe identifiedwithX_P,henceP →Y_P isaquasi-isomorphismwith pureacyclicmappingcone. Moreover,Y_P isacomplexofflatcotorsionR-modules.

AcomplexP issemi-projectiveifHomR(P,−) preservesacyclicityandPⁱisprojective forevery i∈Z; acomplex F is semi-flat if −⊗RF preserves acyclicity and Fⁱ is flat forevery i∈Z. Semi-projective complexes and pure acycliccomplexes of flatmodules arebothsemi-flat.ItfollowsthatifP issemi-projective inConstruction3.1,thenYP is semi-flat.

A complex C is semi-cotorsion if Hom_R(−,C) preserves acyclicity of pure acyclic complexesofflatmodulesandCⁱ iscotorsionforeveryi∈Z(seeAppendixA).Bythe construction, YP consists offlat cotorsion R-modules. Thenextremark shows thatYP

isalsosemi-cotorsion.

Remark 3.2. Let p ∈ SpecR. As R is noetherian, Λ^p is left adjoint to the inclusion of p-adically complete modules into ModR (this follows from [43, Theorem 1.1] which impliesthatΛ^p isidempotent, seealso [45,Theorem 2.2.5] and[27, §4.1]);inaddition, the functor −⊗R R_p is left adjoint to the inclusion of p-local modules into ModR.

Hence,ifM isanyR-moduleandT_p isthep-adiccompletionofafreeR_p-module,then HomR(M,Tp)∼= HomR(Λ^p(Mp),Tp).

LetT be acomplexof p-adiccompletions of free Rp-modules. For any pure acyclic complex X of flat R-modules, we have Hom_R(X,T) ∼= HomR(Λ^p(X_p),T), and so Hom_R(X,T) is acyclic because Λ^p(X_p) is contractible. To see this, we only need to notice that all cycle modules of X are flat, and Λ^p(−⊗RRp) sends a short exact se- quenceofflatR-modulestoasplitshortexactsequenceofflatcotorsionR-modules,see [29,§4,p.69] andthesecondparagraphofRemark1.3.ThereforeT issemi-cotorsion.

LetW beasubsetofSpecRwithdimW <∞andletY beacomplexofflatcotorsion R-moduleswithcosuppYⁱ⊆W foreveryi∈Z.ThenwecaneasilyshowthatY issemi- cotorsionbyaninductiveargument ondimW, usingtheabovefact,(1.17),and(1.18).

Inparticular,itfollowsthatallcomplexesofflatcotorsionR-modulesaresemi-cotorsion whendimR <∞.

Ontheotherhand,whenRiscountablebutofinfiniteKrulldimension,wecaninstead recover finiteness of projective dimension of flat modules, see Remark 3.7 and (3.10).

From this, a standard argument shows that an acyclic complex of cotorsion modules has cotorsion cycle modules. Consequently any complex of cotorsion modules is semi- cotorsion,asitssemi-injectiveresolution(seeAppendixA)yieldsamappingconewhich issemi-cotorsion.

We define a complex Y to be semi-flat-cotorsion if it is both semi-flat and semi- cotorsion. The above remark shows that any semi-flat complex of flat cotorsion R-

modulesissemi-flat-cotorsionaslongasRisoffiniteKrulldimensionorcountable.² In particular, ifP isassumedtobesemi-projective (orsemi-flat)inConstruction3.1,then thecomplexYP constructedthereinissemi-flat-cotorsion.

AssumedimR <∞.Wenowaimtogivetheconstructionofafunctorfrom[37],which in particular sends semi-flatR-complexes to semi-flat-cotorsion ones. If W is asubset of SpecR withdimW = 0, thenwrite ¯λ^W =

p∈W Λ^p(−⊗RRp).Thereis acanonical morphism id_C(R) → λ¯^W; see [37, Notation 7.1].For a non-empty subset W of SpecR, a family of subsets W = {Wi}0≤i≤n is a system of slices of W if W =

0≤i≤nWi, the intersections Wi∩Wj are empty for i = j, dimWi = 0 for 0≤ i ≤ n, and Wi is specialization-closed inW; see[37,Definition7.6].

Construction 3.3.Assume dimR = d < ∞. Let W be a non-empty subset of SpecR orderedbyinclusion.Denote byW0 thesetofmaximalelementsinW.IfW\W0 isnot empty, then define W1 to be the maximal elements of W \W0. Iterating this process, we obtainasystemofslices W ={Wi |0≤i≤n}ofW.Thenaturaltransformations id_C(R)→¯λ^Wi yield(see [37,Remark7.3]) aČechcomplexoffunctors:

L^W =

⎛

⎜⎝

0≤i≤n

λ¯^Wi

0≤i<j≤n

λ¯^Wjλ¯^Wi · · · λ¯^Wn· · ·¯λ^W0

⎞

⎟⎠.

ForanR-complexX,wenaturallygetadoublecomplexL^WX,andthecanonicalchain maps X → λ¯^WiX induce a morphism X → L^WX of double complexes. Totalization yields anaturalchainmap X→totL^WX.

Set A^W = totL^W, as in [34]; we see thatA^W is a functor on C(R) and there is a naturaltransformationa^W : id_C(R)→A^W (thiswaswrittenas^W in[37]).

IfMisanR-module,thenA^WM=L^WM.IfF isaflatR-module,thentheR-module λ¯^WiF =

p∈WiΛ^p(Fp) isflatcotorsion,seethesecondparagraphofRemark1.3;thusif X isacomplexofflatR-modules,thenA^WX isacomplexofflatcotorsionR-modules.

Assumenow thatW = SpecR, sod=n.ForeachflatR-moduleXⁱ,itfollows from [37,Corollary7.12] thata^WXⁱ:Xⁱ →A^WXⁱ isa(pure)quasi-isomorphism;wegivea moreelementaryproofofthisinFact3.6below.Moreover,cone(a^WX) isthetotalization of thedouble complex

0 X

0≤i≤d

λ¯^WiX

0≤i<j≤d

λ¯^Wj¯λ^WiX · · · λ¯^Wd· · ·λ¯^W0X 0,

whose rowsare pure exact, and so thetotalization cone(a^WX) ispure acyclic; see for example [27, Theorem 12.5.4]. It then follows that A^W sends any semi-flat complex

2 Infact,itfollowsfrom[46] or[4] thateverycomplexofflatcotorsionmodulesissemi-cotorsionwithout anyadditionalassumptionsonthering,andsoeverysemi-flatcomplexofflatcotorsionmodulesissemi-flat- cotorsion;seeLemmaA.8.Weprovidethemoreelementaryobservationaboveforthereader’sconvenience.

to a semi-flat complex of flat cotorsion R-modules (cf. [37, Remark 7.13]), that is, a semi-flat-cotorsioncomplexperRemark3.2.

A semi-flat-cotorsion replacement of an R-complex X is an isomorphism in D(R) betweenX and asemi-flat-cotorsionR-complex;seeDefinitionA.5.

Theorem3.4. Assume dimR <∞. EveryR-complex has a minimalsemi-flat-cotorsion replacement inD(R).

Proof. Let X be an R-complex with semi-flat resolution F → X. Construction 3.3 yieldsasemi-flat-cotorsionreplacementF −→A^WF.ByCorollary2.4,thecomplexA^WF decomposes as A^WF = Y ⊕Y where Y is a minimal complex of flat cotorsion R- modulesandY iscontractible.AsA^WF issemi-flat,soisY.Wethenhaveadiagram of quasi-isomorphisms Y ←− F −→ X, where Y is a minimal semi-flat-cotorsion R- complex.

Remark 3.5. Over a commutative noetherian ring, the notion of minimal semi-flat- cotorsionreplacementsisacommongeneralizationofminimalpure-injectiveresolutions offlatmodulesandminimalflatresolutionsofcotorsionmodules. Indeed,ifM isaflat R-module, then its minimal pure-injective resolution P (built from pure-injective en- velopes)consistsofflatcotorsionmodules[16,§8.5],issemi-flatas themappingconeof M→P ispureacyclic,andisminimal[48,Theorem4.1].SeealsoTheorem2.3and[16, Proposition8.5.26].Similarly,ifMiscotorsion,thenitsminimalflatresolutionF (built fromflatcovers)consists offlatcotorsion modules[16,Corollary 5.3.26] andisminimal [48,Theorem4.1];seealso[49,§5.2].Finally,aminimalsemi-flat-cotorsionreplacement (ifitexists) isuniqueuptoisomorphisminC(R);seeLemmaA.4.

In the precedent work [37], the Čech complex L^W naturally appeared as a conse- quenceofthe(generalized)Mayer–Vietoristriangles[37,Theorem3.15].Forthereader’s convenience, we provide an alternative proof of the following fact from [37], which we usedinConstruction3.3.

Fact3.6. Assume dimR <∞ and let W be asystem of slices of SpecR. If F is aflat R-module, then the map a^WF : F →A^WF is aquasi-isomorphism. In particular, the mappingcone of a^WF isapureacyclic complex offlat R-modules.

Proof. SetC= cone(a^WF),whichbydefinitionisofthefollowingform:

0 F

0≤i≤n

λ¯^WiF

0≤i<j≤n

λ¯^Wj¯λ^WiF · · · ¯λ^Wn· · ·λ¯^W0F 0.