Modules $M$ such that $Ext_R^1(M,-)$ commutes with direct limits

MODULES M SUCH THA T Ext 1 R ( M , − ) COMMUTES WITH DIRECT LIMITS SIMION BREAZ Abstract. W e will use W atts’s theorem toget her with Lenzing’s c haracte riza- tion of finitely presente d mo dules via commu ting properties of the induced ten- sor functor in order t o study comm uting properties of co v arian t Ext-functo rs. 1. Intr oduction It is well known that co mm uting prop erties of so me canonical functors (as Hom or tensor functor s induced by a r igh t module) provide imp ortant infor m ation (ab out that mo dule) or some imp ortan t to ols in the study o f some sub categories for the mo dule categor y . F o r insta nce , H. Le nzing prov ed in [19, Sa t z 3] that a right R - mo dule M is finitely presented if and only if the functor Hom R ( M , − ) preser v es direct limits (i.e. filtered colimits) or the tensor pro duct − ⊗ R M commutes with direct pro ducts, [19, Satz 2]. These theorems had a grea t influence in mo dern alge- bra: the first res ult is used to define finitely pr e sen ted ob jects in v arious catego ries, e.g. [1], while the second is an impor t ant ingredient in Chase’s c haracteriza tion of right coher e nt rings [11, Theorem 2.1]. The prop erty that the cov arian t Ho m- functor co mm utes with direct sums pro - vides us with the notion o f small (co mpact) mo dule, [5, p.54]. This notion is useful in many topic s in mo dule theory as generaliza tions o f Morita equiv alences [12], al- most free mo dules [24] or the internal s tructure of the ring, [27]. It is well known that every finitely generated mo dule (these are the mo dules such that the induced cov ariant Hom-functor commutes with direct unions , [29, 24 .10]) is sma ll. More- ov er, for some imp orta nt classes o f r ings (as right no e ther ian or right p erfect b y [20], [13]) these tw o conditions ar e equiv alent, but there are also imp orta nt t ype s of rings for which there are non-finitely g enerated small mo dules (e.g. no n-artinian regular simple ring s), [14]. It is prov ed in [2 6] that every small mo dule is finitely pr esented if a nd only if the r ing is no etherian. A simila r pr oblem can b e prop osed for the cov ariant functor E xt 1 R : Identify classes of rings R such that a functor E xt 1 R ( M , − ) c ommu tes with dir e ct un ions (or dir e ct limits) if and only if it c ommu tes with dir e ct sums. Cor o llary 2.5 pr ovides an a nswer to this pro ble m. Moreov er, in Example 2.6 it is shown that these two co nditions ar e not equiv alen t in general. W e mention here that R is right coher ent exactly if for every r ight R -mo dule M the functor Ext 1 R ( M , − ) commut es with dir ect unions if and only if it c ommut es with direct limits, [7, Co rollar y 7]. Date : Nov em ber 26, 2024. 2000 M athematics Subj e ct Classific ation. 16E30, 16E60, 18G15. Key wor ds and phr ases. Ext-functor, direct limit, hereditary r ing. Researc h supp orted by the CN C S- UEFISCDI gr an t PN- II- RU-TE-2011-3-0065. 1 2 SIMION BREAZ Concerning commuting pro p e r ties o f the derived functors Ext ∗ and T or ∗ , it w as prov ed b y Brown in [10] that Lenz ing ’s results ca n b e e x tended to pro duce more finiteness conditions. In or der to s tate this re s ults, let us recall from [17, p. 10 3 ] that a right R mo dule M is F P n , for a fixed integer n ≥ 0, if M has a pro jective resolution which is finitely g e nerated in every dimension ≤ n . Theorem 1.1. [10, Theore m 1 ] , [22, Theorem A] The fol lowing ar e e quivale nt for a right R mo dule M and a non-ne gative inte ger n : (1) M is F P n +1 ; (2) E x t k R ( M , − ) c ommutes with dir e ct limits for al l 0 ≤ k ≤ n , (3) T or R k ( M , − ) c ommutes with dir e ct pr o ducts for al l 0 ≤ k ≤ n . The pr o of of this theorem is based on Lenzing’s c haracter izations of finitely presented mo dules via commuting prop er ties of cov aria nt Hom-functors with direct limits, r esp ectively comm uting prop erties of tensor functors with direct pro ducts, and these are applied to obtain indepe ndently (1) ⇔ (2) and (1) ⇔ (3). F or the case of Ext-functors , this theor em was r efined and completed by Streb el in [22]. W e can as k if there is a cyclic pro of. A connectio n is sugge s ted in [2 2, Corolla ry 1], where it is prov ed that a rig ht R - mo dule M of pr o jective dimension at most n has the prop er ty that E xt n R ( M , − ) co mmutes with direct limits if and o nly if M has a pro jective r esolution which is finitely generated in dimension n . In the present pap er we give such a pro of in Theor e m 2.4, where mo dules of pro jective dimens io n at mo st 1 such that the induced cov ariant Ext 1 -functor com- m utes with direct limits are characterized via Lenzing’s c haracteriz a tion of finitely presented mo dules by commut ing prop er ties of the tensor pr o duct. This theorem is used to obtain s ome r esults from Streb el’s pap er, [2 2]. Similar techniques were used by K rause in [1 8] in o rder to character iz e gener al coherent functors. In the end o f this intro duction let us o bs erve that there are commuting prop erties which are no n-trivial for the Hom-functors but they are tr ivial for Ext-functors . F or instance, it is well known that the cov a riant Ho m-functors commute with inv erse limits. How ever, a s in [17, Exa mple 3.1 .8], we can use [6 , Theorem 2] to write every mo dule as an in verse limit of injective mo dules. T her efore Prop ositi on 1. 2. L et M b e a mo du le. The fun ctor Ext 1 R ( M , − ) c ommutes with inverse limits if and only if M is pr oje ctive. Dualizing the definition of small mo dules w e obtain the notion o f slender mo d- ule. The structure of these mo dules ca n be very complica ted (see [15] or [17]). T r ansferring this approach to the contrav ariant Ex t-functor, w e say tha t the co- trav ariant functor Ext n R ( − , M ) inv erts pro ducts if for every countable family F = ( M i ) i<ω of right R -mo dules the canonica l homomo rphism L i<ω Ext n R ( M i , M ) → Ext n R ( Q i<ω M i , M ) is an iso mo rphism. How ever, in the cas e when M has the in- jective dimension at most n this prop erty is very r estrictive. Prop ositi on 1.3. L et M b e a right R -mo dule of inje ctive dimension at most n ≥ 1 . The functor Ext n R ( − , M ) inverts pr o ducts with c ountable m any factors if and only if M is of inje ctive dimension at most n − 1 . Pr o of. By the dimension shifting formula, it is eno ug h to assume n = 1. MODULES M SUCH THA T Ext 1 R ( M , − ) COMMUTES WITH DIRECT LIMITS 3 In this hypothesis, we s ta rt with a module N and with the ca nonical monomor- phism 0 → N ( ω ) → N ω . Applying the contrav ariant E xt-functor and [4, Prop o s i- tion 2.4] we obtain the natural homomorphism Ext 1 R ( N , M ) ( ω ) ∼ = Ext 1 R ( N ω , M ) → Ext 1 R ( N ( ω ) , M ) ∼ = Ext 1 R ( N , M ) ω , which coincides with the cano nical homomor phism Ext 1 R ( N , M ) ( ω ) → Ext 1 R ( N , M ) ω , and it is an epimorphism. This is poss ible only if Ext 1 R ( N , M ) = 0.  W e do not know wha t happ ens in P rop osition 1 .3 in case w e do no t assume an y bo und on the injectiv e dimension. It is also an op en question when the contra v a riant Ext 1 -functor preser ves pro ducts. In [16] the authors provide an a nswer for ab elia n groups, and we refer to [8] for the case o f contrav ariant Hom-functors. Other commuting prop erties of these functors, for the case of ab elia n groups, ar e studied in [3] a nd [21]. 2. W hen Ext com m u tes with direct limits Let R b e a unital ring and M a right R -mo dule. If F = ( M i , υ ij ) i,j ∈ I is a direct system of mo dules and υ i : M i → lim − → M i are the canonical ho momorphisms, then for every non-negative integer n there is a canonica l homomorphis m Φ n,M F : lim − → Ext n R ( M , M i ) → Ext n R ( M , lim − → M i ) , the natura l homomorphism induced b y the family Ext n R ( M , υ ij ), i, j ∈ I . F o r the case o f direc t sums, if we co nsider F = ( M i ) i ∈ I a fa mily of mo dules and w e denote by u i : M i → L i ∈ I M i the c anonical homomorphisms then w e will obtain a natur a l homomor phis m Φ n,M F : M i ∈ I Ext n R ( M , M i ) → Ext n R ( M , M i ∈ I M i ) , induced by the family E xt n R ( M , u i ), i ∈ I . W e will us e these homomorphis ms for the cases n ∈ { 0 , 1 } , and we will denote Φ 0 ,M F := Ψ M F , Φ 1 ,M F = Φ M F , resp ectively Φ 0 ,M F := Ψ M F , Φ 1 ,M F = Φ M F . W e say that Ext n R ( M , − ) commutes with direct limits ( direc t s ums ) if the ho - momorphisms Φ n,M F (resp ectively Φ n,M F ) are iso mo rphisms for all dir ected systems F (resp ectively all families F ). Lenzing’s theor em says us that M is finitely presented if and o nly if Ψ M F are isomorphisms for all F . Moreover, using Theorem 1.1 for n = 1 , we o bserve that M is an F P 2 mo dule if and only if Ψ M F and Φ M F are iso mo rphisms for a ll families F . W e will start with a slight improvemen t of this res ult, replacing the co ndition “Ψ M F is an isomorphism” (i.e. M is finitely pr esented) by the condition “M is finitely generated”. Therefore in the ca se o f finitely gener ated mo dules the hypo thesis o f [17, Lemma 3 .1.6] is sha rp. F o llowing the terminolo gy used in [7 ], we will call a mo dule M an fp- Ω 1 -mo dule ( fg- Ω 1 -mo dule , resp ectively small- Ω 1 -mo dule ) if there is a pro jective r esolution ( P ) : · · · → P 2 → P 1 α 1 → P 0 → M → 0 such that the first syzygy Ω 1 ( P ) = Im( α 1 ) is finitely presented (finitely generated, resp ectively sma ll). 4 SIMION BREAZ Lemma 2.1. If M is fp- Ω 1 (fg- Ω 1 , r esp e ctivel y smal l- Ω 1 ) right R -mo dules then ther e is a pr oje ctive mo dule L su ch t hat M ⊕ L is a dir e ct su m of an F P 2 -mo dule (finitely pr esente d mo dule, r esp e ct ively fi n itely gener ate d mo dule) and a pr oje ctive mo dule. Pr o of. Let 0 → K α → P β → M → 0 b e an exact s equence such that P is pro jective and K is finitely presented (finitely gener ated, r esp ectively small). If C is a pro- jective mo dule such that P ⊕ C ∼ = R ( I ) is free, then we cons ider the induced exact sequence 0 → K α → P ⊕ C β ⊕ 1 C → M ⊕ C → 0. Since K is small (r ecall that every finitely generated mo dule is sma ll) a s a right R -mo dule ther e is a finite subse t J of I s uch that Im( α ) ⊆ R ( J ) . Then M ⊕ C ∼ = R ( J ) / Im( α ) ⊕ R ( I \ J ) = H ⊕ L , where H is an F P 2 -mo dule (finitely presented, r esp ectively finitely g enerated) and L is pro jective.  The clos ure under direct summands of the clas s of fg-Ω 1 -mo dules can b e c harac- terized by the co mmut ing prop er t y of the cov ariant Ext 1 -functor with dir ect unions (i.e. direct system of monomorphis ms ). W e state this result for sake of complete- ness. Theorem 2.2 . [7, Theorem 5] The fol lowing ar e e quiva lent for a right R -mo dule M : (1) M is a dir e ct summ and of an fg- Ω 1 -mo dule; (2) The functor Ext 1 R ( M , − ) c ommutes with dir e ct systems of monomorphisms. F o r the cases of fp-Ω 1 -mo dules and small-Ω 1 -mo dules we are able to prove a similar res ult only in the ca se o f finitely genera ted mo dules. In fact, it is easy to observe, us ing [1 7, Lemma 3.1 .6] that the cov a riant Ext 1 -functor induced by an fp- Ω 1 -mo dules commutes with direct limits. The conv erse is true for finitely generated mo dules, [7, Lemma 1]. A similar result is v a lid for s mall-Ω 1 -mo dules. Theorem 2. 3. L et M b e a r ight R -mo dule M . (1) If M is an fp- Ω 1 -mo dule then Ext 1 R ( M , − ) c ommutes with dir e ct limits. (2) If M is a smal l- Ω 1 -mo dule then Ext 1 R ( M , − ) c ommutes with dir e ct sums. The c onverses of these statemen ts ar e tru e if M is fin itely gener ate d. Pr o of. W e will prov e (2). The pro o f for (1) follows the same steps. Using Lemma 2.1, we can supp ose that M is finitely gener ated. If 0 → K → P → M → 0 is an e x act sequence with P a pro jectiv e mo dule then fo r every family F = ( M i ) i ∈ I of right R -mo dules, we have the following useful commutativ e diag ram ( ♯ ) 0 − − − − → ⊕ i ∈ I Hom R ( M , M i ) − − − − → ⊕ i ∈ I Hom R ( P, M i ) − − − − →   y Ψ M F   y Ψ P F 0 − − − − → Hom R ( M , ⊕ i ∈ I M i ) − − − − → ⊕ i ∈ I Hom R ( P, M i ) − − − − → − − − − → ⊕ i ∈ I Hom R ( K, M i ) − − − − → ⊕ i ∈ I Ext 1 R ( M , M i ) − − − − → 0   y Ψ K F   y Φ M F − − − − → Ho m R ( K, ⊕ i ∈ I M i ) − − − − → E xt 1 R ( M , ⊕ i ∈ I M i ) − − − − → 0 whose rows are exact. MODULES M SUCH THA T Ext 1 R ( M , − ) COMMUTES WITH DIRECT LIMITS 5 Since M and P ar e finitely generated, the arr ows Ψ M F and Ψ P F are iso mo rphisms by [29, 24.10]. Using Five Lemma we o bserve that Ψ K F is an iso morphism if a nd only if Φ M F is an is omorphism.  W e a re rea dy to state the main r esult o f this pap er. It states that fp-Ω 1 -mo dules of pro jective dimension a t most 1 can b e characterized by commuting pr op erties of the induced co v a r iant E xt 1 -functor. Theorem 2.4. The fol lowing ar e e quivale nt for a right R - mo dule M of pr oje ctive dimension 1: (1) M is an fp- Ω 1 -mo dule. (2) Ther e is a pr oje ctive mo dule L su ch that M ⊕ L is a dir e ct sum of an F P 2 -mo dule ( F P 1 -mo dule) and a pr oje ctive mo dule. (3) E x t 1 R ( M , − ) c ommutes with dir e ct limits. (4) E x t 1 R ( M , − ) c ommutes with dir e ct sums of c opies of R . Pr o of. W e only need to pr ov e (4) ⇒ (1 ). Let M be a mo dule of pr o jective dimension at most 1 such that Ext 1 R ( M , − ) commutes with direct sums of copies of R . Therefor e E xt 1 R ( M , − ) is a rig ht exact functor which co mmut es with dir ect sums o f copies of R . B y W atts’s theore m [2 8, Theorem 1] (and its pro of ) we conclude that the functor Ext 1 R ( M , − ) is naturally equiv a lent to the functor − ⊗ R Ext 1 R ( M , R ). It follows that the tensor pro duct func- tor − ⊗ R Ext 1 R ( M , R ) preser ves the pro ducts , hence that Ext 1 R ( M , R ) is a finitely presented le ft R -mo dule. Let n b e a p o sitive int eger such that the left R - mo dule Ext 1 R ( M , R ) is genera ted by n elements. Using [23, Lemma 6 .9] w e conclude that there is a n exact sequence of right modules 0 → R n → C → M → 0 such that Ext 1 R ( C, R ) = 0. Starting with this exact sequence we o bta in for every cardinal κ a commutativ e diagram Hom( R n , R ) ( κ ) − − − − → Ext 1 R ( M , R ) ( κ ) − − − − → Ext 1 R ( C, R ) ( κ ) − − − − → 0   y   y   y Hom( R n , R ( κ ) ) − − − − → Ext 1 R ( M , R ( κ ) ) − − − − → Ext 1 R ( C, R ( κ ) ) − − − − → 0 , where the vertical ar rows are the canonical homomorphisms induced b y the uni- versal pr op erty of direct sums. Mor eov er the first v ertical is an isomo rphism since every finitely genera ted mo dule is sma ll, while the seco nd is also a n iso mo rphism by our hypothesis . Therefore the third vertical a rrow is also a n isomorphism. It fol- lows that Ext 1 R ( C, R ( κ ) ) = 0 for all c a rdinals κ . Since C is o f pro jective dimension at most 1, it is not hard to see that C is pro jective.  Corollary 2.5. L et R b e an right her e ditary ring and M a right R -mo dule. The fol lowing ar e e qu ivalent: (1) E x t 1 R ( M , − ) c ommutes with dir e ct limits; (2) E x t 1 R ( M , − ) c ommutes with dir e ct sums; (3) E x t 1 R ( M , − ) c ommutes with dir e ct sums of c opies of R ; (4) M = N ⊕ P , wher e N is finit ely pr esente d and P is pr oje ctive. Pr o of. Since every right he r editary r ing is right coherent, every finitely pr e sented mo dule is a n F P 2 -mo dule. Therefore only (3 ) ⇒ (4) r e quires a pr o of. 6 SIMION BREAZ Suppo se that Ext 1 R ( M , − ) commutes with direct sums o f copies o f R . Using Theorem 2.4, we observe that there is a pro jective mo dule L such that M ⊕ L = F ⊕ U with F finitely presented and U pro jective. The conclusion follows using the same techniques as in [2]. If π U : F ⊕ U → U is the cano nical pro jection, then π U ( M ) is pro jective. Then F ∩ M = Ker( π U | M ) is a dir ect summand o f M . Therefore N = F ∩ M is a direct summand of F ⊕ U = M ⊕ L . Using this, it is not hard to see that N = F ∩ M is a direct summand of F . Hence N is finitely presented a nd M = N ⊕ P , where P ∼ = π U ( M ) is pro jective.  Example 2.6. The equiv ale nc e (3) ⇔ (4) from Theore m 2.4 is not v alid for general finitely gener ated mo dules . T o see this, it is eno ugh to consider a ring R which has a non-finitely genera ted small r ight ideal I . F or instance, it is proved in [2 5] tha t every direct pr o duct of infinitely many rings has such an ide a l. Since R /I is a finitely generated right R - mo dule, we can apply Theorem 2.3 to see that Ext 1 R ( R/I , − ) commutes with direct sums, but it does not commute with direct limits. Remark 2.7. Corolla ry 2.5 was prov ed for ab elian groups in [9] without the con- dition that the isomorphisms L i ∈ I Ext 1 R ( M , M i ) ∼ = Ext 1 R ( M , L i ∈ I M i ) are the natural ones. Corollary 2 .8. [2 2, Coro llary 1] L et M b e right R -mo dule of finite pr oje ctive di- mension ≤ n . Then M admits a pr oje ctive r esolution which is finitely gener ate d in dimension n if and only if E xt n R ( M , − ) pr eserves dir e ct sums of c opies of R . If one of the e quiva lent c onditions is fulfil le d, then the functors Ext n R ( M , − ) and − ⊗ R Ext n R ( M , R ) ar e n atur al ly isomorphic . Pr o of. Let ( P ) b e a pro jectiv e resolution o f M . Then the ( n − 1)-th syz y gy Ω n − 1 ( P ) is of pr o jective dimens ion at most 1. Using the dimensio n s hifting formula we observe that E xt n R ( M , − ) is na turally isomorphic to Ex t 1 R (Ω n − 1 ( P ) , − ), and the conclusion follows from Theorem 2 .4 and W atts’s theo rem.  Lemma 2.9. L et L b e a right R -mo dule of pr oje ctive dimension at most 1 and K a fin itely pr esente d submo dule of L . The mo dule M = L/K has the pr op erty that Ext 1 R ( M , − ) pr eserves dir e ct sum s of c opies of R if and only if M is an fp- Ω 1 - mo dule. Pr o of. Let I be a set. It induces a commutativ e diagra m Hom R ( K, R ) ( I ) − − − − → Ext 1 R ( M , R ) ( I ) − − − − → Ext 1 R ( L, R ) ( I ) − − − − →   y Ψ K   y Φ M   y Φ L Hom R ( K, R ( I ) ) − − − − → E xt 1 R ( M , R ( I ) ) − − − − → Ext 1 R ( L, R ( I ) ) − − − − → − − − − → Ext 1 R ( K, R ) ( I ) − − − − → Ext 2 R ( M , R ) ( I ) − − − − → 0   y Φ K   y − − − − → Ext 1 R ( K, R ( I ) ) − − − − → Ext 2 R ( M , R ( I ) ) − − − − → 0 where a ll arrows repr esent the ca no nical homomorphisms. Since K is finitely pre- sented, the ar rows Ψ K and Φ K are isomorphisms (for Φ K we use Theorem 2.3). Moreov er, Φ M is an isomorphism from our hypothesis and it is obvious that the last vertical arrow is also a monomo rphism by [22, Lemma 2.2]. Ther efore Ext 1 R ( L, − ) MODULES M SUCH THA T Ext 1 R ( M , − ) COMMUTES WITH DIRECT LIMITS 7 commutes with direct sums of co pies of R . By Theorem 2.4, it follows that there is a pro jective re s olution 0 → R n → P → L → 0. Using a ll these data we construct a pullback diagra m 0 0   y   y 0 − − − − → R n − − − − → U − − − − → K − − − − → 0      y   y 0 − − − − → R n − − − − → P − − − − → L − − − − → 0   y   y M M   y   y 0 0 . Since U is finitely presented, the pro o f is co mplete.  W e also obtain the principal ingr e dient for the pro of of [22, Theorem B]. Corollary 2.10. [22, Lemma 2.6 ] Le t M b e a right R -mo dule such that Ext 1 R ( M , − ) c ommu tes with dir e ct sums of c opies of R . S upp ose that ther e is a pr oje ctive r eso- lution ( P ) P 2 → P 1 α 1 → P 0 α 0 → M → 0 such that Ω 2 ( P ) is finitely gener ate d. Then ther e is a pr oje ctive r esol ution ( P ′ ) P 2 → P ′ 1 → P ′ 0 → M → 0 with P ′ 1 finitely gener ate d such that Ω 2 ( P ) = Ω 2 ( P ′ ) . Pr o of. W e can supp os e that there is a set I such that P 1 ∼ = R ( I ) . If J is a finite subset of I such that Ω 2 ( P ) ⊆ R ( J ) and K = R ( J ) / Ω 2 ( P ) then we have a n exact sequence 0 → K ⊕ R ( I \ J ) α → P 0 → M → 0 , where α is induced by α 1 . If M ′ = P 0 /α ( R ( I \ J ) ) then we hav e an exact sequence 0 → K → M ′ → M → 0. Obs erve that M ′ is of pro jectiv e dimens ion at mos t 1. By Lemma 2.9, the mo dule M is an fp-Ω 1 -mo dule. T her efore there is an exact sequence 0 → U → Q → M → 0 with Q pro jective and U finitely pres ented, and 8 SIMION BREAZ we can c onstruct a co mmu tative dia gram with exa ct rows 0 0 0   y   y   y 0 − − − − → Ω 2 ( P ) − − − − → V − − − − → U − − − − → 0   y   y   y 0 − − − − → P 1 − − − − → W − − − − → Q − − − − → 0   y   y   y 0 − − − − → Ω 1 ( P ) − − − − → P 0 − − − − → M − − − − → 0   y   y   y 0 0 0 Since the middle vertical row is splitting, V is a finitely gener ated pro jectiv e mo dule. 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