Some Remarks on Global Dimensions for Cotorsion Pairs.
EDGARE. ENOCHS(*) - HAE-SIKKIM(**)
ABSTRACT- We introduce several global dimensions associated with a cotorsion pair of classes of leftR-modules and consider the relations between them.
Cotorsion theories were introduced by Salce in [10]. Recently several authors use the terminology cotorsion pair for the same notion(e.g. Hovey in [8]).
DEFINITION1. Let R bea ring and let F and C be classes of (left) R-modules. We say that (F;C) is a cotorsion pairif for any R-module F, F2 F if and only if Ext1R(F;C)0 for all C2 C and similarly for any R-module C, C2 C if and only if Ext1R(F;C)0 for all F2 F.
Wenotethat if (F;C) is a cotorsion pair, then bothF andCareclosed under extensions and summands. Also F contains all projective modules andCall injective modules. In fact a result of Eklof says thatF is closed under transfinite extensions (see Eklof [2] for the result and Hovey [8, Section 6] for the definition of transfinite extensions).
Examples of cotorsion pairs are (P;M) and (M;E), wherePis theclass of projective modules,Eis that of injective modules, andMis theclass of all modules. A less trivial example is (F;C), where F is theclass of flat modules and C is the class of cotorsion modules (see Xu [11], Definition 3.1.1 and Lemma 3.4.1).
PROPOSITION2. Let(F;C)be a cotorsion pair of (left)R-modules and let nsupfproj.dimFjF 2 F g(so,0n 1). If0n <1, thennis the least s with 0s <1 such that if 0!N !E0 !E1! !Es 1! (*) Indirizzo dell'A.: Department of Mathematics University of Kentucky Lexington, KY 40506-0027, USA; e-mail: enochs@ms.uky.edu
(**) Indirizzo dell'A.: Department of Mathematics Kyungpook National Uni- versity Taegu 702-701, Korea; e-mail: hkim@dreamwiz.com
!C!0is a partial injective resolution of anyR-moduleN thenC2 C.
Andn 1precisely when there is no suchs.
PROOF. Wefirst notethat weareusing theconvention that 0!N!
!E0!E1! !Es 1!C!0 means an exact sequence 0!N!
!C!0 whe ns0. Now assumethat 0n<1. Supposethats with 0s<1issuchthatforanyR-moduleNandanypartialinjective resolution 0!N!E0!E1! !Es 1!C!0 ofN, wehaveC2 C. Notethat Exts1R (F;N)Ext1R(F;C) for all F2 M. The n if F2 F, wehave Exts1R (F;N)Ext1R(F;C)0 for all N. Hence proj.dim Fs for all F2 F. This givesns.
But also if 0!N!E0!E1 ! !En 1!C!0 is any partial injective resolution of any R-module N, then Extn1R (F;N)0 for all F2 F sinceproj.dim Fn. So Ext1R(F;C)Extn1R (F;N)0 for all F2 F. HenceC2 C. This gives that ournis theleast suchsas desired.
The same type argument gives thatn 1precisely when there is no
suchswith 0s<1. p
A dual argument gives us;
PROPOSITION3. Let(F;C)be a cotorsion pair of (left)R-modules and letpsupf inj.dimCjC2 C g. If0p <1, thenp is the leastt with 0t <1 such that if 0!F !Pt 1 !Pt 2! !P0!M !0 is any partial projective resolution of any R-module M we have F 2 F.
Furthermore,p 1if and only if there is no sucht.
Now weletmbe the left global dimension ofR (withm 1a possi- bility).
PROPOSITION4. If(F;C)is a cotorsion pair ofR-modules and ifm; n andpare as above, thenmnp.
PROOF. If either n 1or p 1, this inequality trivially holds. So assumethat n;p<1. Le t M bean R-moduleand let 0!F!
!Pp 1!Pp 2! !P0!M!0 be a partial projective resolution of M. By Proposition 3 wehaveF2 F. Also by thedefinition ofnwehave proj.dim Fn. So proj.dim Mnp for all R-module M. He nce
mnp. p
REMARK5. There is a special case of Proposition 3 given in Xu ([11, Theorem 3.3.2]). So we thought that it is of interest to consider these
dimensions in the framework of a cotorsion pair and then to deduce Pro- position 4.
For a given cotorsion pair (F;C), it is of interest to findnandp. Also it is of interest to know whether all possibilities subject ton;pmnp occur for somecotorsion pair (F;C). Webriefly indicatehow all possibi- lities form;nsubject tonmdo occur.
LEMMA6. Given such m;n with nm, let R be any ring with left global dimension equal to m. Let L consist of all R-modules L with proj.dim Ln and letHbe the class of all R-module H such that Ext1R(L;H)0for all L2 L. Then(L;H)is a cotorsion pair.
PROOF. Ifn 1, then we getL MandH EwithMall modules andEthe injective modules. So we just get the cotorsion pair (L;H). Now supposethatn<1. ThenLis a Kaplansky class(see [5] for the definition and [1, Proposition 4.1] for theproof). A basic result of Auslander says that L is closed under transfinite extensions. Since L contains all projective modules and is closed under summands, we get (L;H) is a complete co-
torsion pair. p
This fact is a consequence of [6, Theorem 10] and of the proof of that theorem. Also note that in the statement of that theorem the cotorsion pair is said to have enough injectives and projectives. To say that the pair is complete means precisely this.
IfRis commutative Noetherian and (F;C) is thecotorsion pair withF the class of flat modules, a result of Jensen says that proj.dimFK-dimR (theKrull dimension ofR) for everyF2 F( [9] or see [3, Proposition 3.1]
for an alternate proof). Hence nK-dimR in this situation. There are easy examples with K-dimR<1and global dimensionm 1. So in this situation wehavep 1.
For our next result we need the notion of a hereditary cotorsion pair.
DEFINITION7. ([4]) A cotorsion pair (F;C) of R-modules is said to be hereditary if it satisfies one (so all) of the three equivalent con- ditions.
(1) If 0!F0!F!F00!0 is exact withF;F002 F, thenF02 F.
(2) If 0!C0!C!C00!0 is exact withC0;C2 C, thenC002 C.
(3) ExtiR(F;C)0 for allF2 F,C2 Candi1.
To get (1))(3) letF2 Fand let 0!F0!P!F!0 beexact withP projective. SinceP2 F, wegetF02 F. Consider the long exact sequence !Ext1R(F;C)!Ext1R(P;C)!Ext1R(F0;C)!
!Ext2R(F;C)!Ext2R(P;C)!Ext2R(F0;C)! withC2 Cand useinduction oni1.
To see (3))(1) note that for the given exact sequence 0!F0!F!
!F00!0, the long exact sequence !Ext1R(F;C)!Ext1R(F0;C)!
!Ext2R(F00;C)! withC2 Cgives usExt1R(F0;C)0. ThenF02 F. The equivalence (2),(3) can be proved in a similar manner.
Wenow giveanother characterization of then;pdefined earlier.
PROPOSITION8. Let (F;C) be a hereditary cotorsion pair ofR-mod- ules. Letnandpbe as previously defined. Ifp <1, thenpis the smallestt with 0t <1 such that if 0!Ft!Ft 1 ! !F0!M !0 is an exact sequence with F0; F1; ; Ft 12 F thenFt2 F. Also p 1if and only if there is no suchtwith0t <1.
Dually, if n<1, then n is the smallest s with0s<1such that if 0!N!C0!C1! !Cs 1!Cs !0 is an exact sequence with C0;C1; ;Cs 12 C then Cs 2 C. And n 1precisely when there is no such s with0s<1.
PROOF. Weprovethefirst claim. Theproof of thesecond is dual to this proof. Giventwith 0t<1let 0!Ft!Ft 1! !F0!M!0 be any exact sequence withF0;F1; ;Ft 12 F andManyR-moduleand let 0!F!Pt 1!Pt 2! !P0!M!0 be any partial projective re- solution ofM. Then there is a commutative diagram (with exact rows) ;
Wethink of thetwo rows as complexes and form themapping cone (see[7, pg. 154]). So weget thecomplex 0!F!FtPt 1! Ft 1Pt 2! !F1P0!F0M!M!0. This comple x is e xact since the two original complexes are exact. This complex has the exact sequence 0!M!M!0 with theidentity map M!M as a sub- complex. So taking the quotient complex we get the exact complex 0!F!FtPt 1!Ft 1Pt 2! !F1P0!F0!0. Thus we
have an exact sequence 0!F!FtPt 1 !G!0, where G ker(Ft 1Pt 2!Ft 2Pt 3). Since(F;C) is hereditary, G2 F. So F2 F if and only ifFt 2 F. With this observation we see that the result
follows. p
REMARK 9. IfF is theclass of flat modules, thefirst claim of this proposition is a familiar result. The usual argument in this case involves using theTor functors.
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Manoscritto pervenuto in redazione il 14 luglio 2005
