Rings without a Gorenstein analogue of the Govorov-Lazard Theorem

RINGS WITHOUT A GORENSTEIN ANALOGUE OF THE GO V ORO V-LAZARD THEORE M HENRIK HOLM AND PETER JØRGENSEN Abstract. It was pro ved b y Be lig iannis and K rause that ov er certain Artin a lgebras, there are Gorenstein flat mo dules which are not direct limits of finitely gener ated Gore ns tein pro jective modules. That is, these algebra s ha ve no Gor enstein analogue of the Go vorov-Lazar d Theorem. W e show that, in fact, there is a large cla ss o f r ings without such an analogue. Namely , let R b e a commutativ e lo cal noetherian ring. Then the analog ue fails for R if it ha s a dualizing complex, is henselian, not Gorenstein, a nd has a finitely generated Gorenstein pro jective module which is no t free. The pro of is based o n a theor y of Gor e ns tein pr o jectiv e (pre)en- velopes. W e show, a mo ng other things, that the finitely g e ne r ated Gorenstein pro jectiv e mo dules form an env elo ping class in mod R if and only if R is Gorenstein o r has the prop erty tha t each finitely generated Gorenstein pro jectiv e mo dule is free. This is analogous to a recent result on co vers by Christensen, Piepmeyer, Striuli, and T ak ahashi, and their methods are an im- po rtant input to our work. 0. Intr oduction Gor enstein h o molo gic al algebr a w a s founded b y Auslande r and Bridger in [1]. Some of its main concepts are the so-called Go r enstein pro jectiv e and Gorenstein fla t mo dules, see [8] and [1 0 ]. These mo dules inhabit a theory par a llel to classical ho mo lo gical algebra. F or instance, just as pro jectiv e mo dules can b e used to define pro jectiv e dimension, so Gorenstein pro jectiv e mo dules can b e used to define Gorenstein pr o - jectiv e dimension. A commutativ e lo cal no etherian ring is Gorenstein if and only if all its mo dules ha v e finite G orenstein pro jectiv e dimen- sion. A go o d introduction is giv en in [4]; in particular, the definitions 2000 Mathematics Subje ct Classific ation. 13 H10, 18G25 . Key wor ds and phr ases. Algebra ic duality , closur e under direct limits, covers, en- velopes, Gore ns tein flat modules , Gorenstein pr o jective modules, pr ecov ers, preen- velopes, sp ecial precovers, special preenv elop es. 1 2 HENRIK HOLM AND PETER JØR GENSEN of Gorenstein pro jectiv e and Gorenstein flat mo dules can b e found in [4, (4.2.1) and (5 .1 .1)]. The Govor ov-L azar d The or em say s that the closure under direct limits of the class of finitely generated pro jectiv e mo dules is equal to t he class of flat mo dules ; see [13] and [14, thm. 1.2]. It is natural to ask if this has a Gorenstein ana lo gue. Namely , if G denotes t he class of finitely generated Gor enstein pro jectiv e modules, is lim − → G equal to the class of Gorenstein flat mo dules? In some cases the answ er is yes , for instance o v er a ring whic h is Gorenstein in a suitable sense; this w a s es tablished b y Eno c hs and Je nda in [9, thm. 10.3.8]. Ho w ev er, Beligiannis a nd Krause prov ed in [2, 4.2 and 4.3] that for certain Artin alg ebras, the answ er is no. W e sho w for a considerably larger class of rings that there is no Go ren- stein analogue of the Gov oro v-Lazard Theorem. Namely , let R b e a comm utativ e lo cal no etherian ring and let F b e the class of finitely generated free mo dules. The follo wing is our Theorem 2 .7. Theorem A . I f R has a dualizing c omplex, is hen selian, not Gor en- stein, and has G 6 = F , then lim − → G is strictly c ontaine d in the class of Gor enstein flat mo dules. The pro of is based o n a theory of G -preen v elop es, the dev elopmen t of whic h tak es up most of the pap er. The bac kground is that the existence of G -preco v ers has b een considered at length. That is, if M is a finitely generated mo dule, do es there exist a homomorphism G γ → M with G in G suc h that an y other homomorphism G ′ → M with G ′ in G factors through γ ? A breakthrough w as achie ved recen tly in [6] b y Christensen, Piepmey er, Striuli, and T ak ahashi who pro ved, among other things, that if R is henselian, then G -precov ers exist for all finitely generated modules in precisely tw o cases: If R is Gorenstein, or if G = F . W e will consider the dual question: Existence of G -preenv elopes. Th a t is, if M is a finitely generated mo dule, do es there exist a homomorphism M µ → G with G in G suc h tha t a n y other homomorphism M → G ′ with G ′ in G factors through µ ? W e giv e criteria for the existence of v arious t yp es of G -preen v elop es in Theorem 2.5. One a sp ect is the f ollo wing precise analogue of the precov ering case. Theorem B. If R is henselian then al l finitely gener ate d R -mo dules have G -pr e env e lop es if and on ly if R is Gor en stein or G = F . Note that the metho ds a nd results of [6] a re an imp orta nt input to o ur pro of. RINGS WITHOU T GORENSTEIN GOVOR OV-LAZARD 3 The pap er is organized as follows : Section 1 prepares the g round b y examining the connections b et w een G -preco ve rs a nd G -preenv elop es whic h are induced by the algebraic duality functor ( − ) ∗ = Hom R ( − , R ). Section 2 pro v es Theorems A a nd B, among other things. Section 3 sho ws a metho d f or constructing a Gorenstein flat mo dule outside lim − → G . 1. Algebraic duals of precovers and preenvelop es This section pro v es Theorems 1.5 and 1.6 b y whic h algebraic duals of v arious t yp es of G - preco v ers give the corresp onding types of G - preen v elop es, and vice v ersa. Setup 1.1. Throughout the paper, R is a commutativ e no etherian ring. By mo d R is denoted the category of finitely generated R -mo dules. Recall that F is the class of finitely g enerated f r ee R -mo dules and G is the class of finitely generated Gorenstein pro jectiv e R -mo dules. Remark 1.2. The follo wing prop erties of G will b e used b elow. (i) Ext > 1 R ( G , R ) = 0. (ii) R is in G . (iii) The class G is closed under the algebraic duality functor ( − ) ∗ = Hom R ( − , R ). (iv) The bidualit y homomorphism G δ G − → G ∗∗ , as defined in [4 , (1.1.1)], is an isomorphis m for eac h G in G . (v) Eac h mo dule in G is isomorphic to a mo dule G ∗ where G is in G . Here (i) and (iv) are part of the definition of G , see [4, def. ( 1 .1.2)]. (ii) is by [4, rmk. (1.1.3)] and (iii) is b y [4, obs. (1.1 .7 )]. (v) is immediate from (iii) and (iv). Lemma 1.3. If C is an R -mo dule satisfying Ext 1 R ( C , R ) = 0 , then Ext 1 R ( G, C ∗ ) ∼ = Ext 1 R ( C , G ∗ ) for e ach G in G . Pr o of. W e ha v e H < 0 RHom( C , R ) = 0 , (1.a) so RHom( C , R ) can b e represen ted in the deriv ed category D ( R ) by a complex conce ntrated in non-negativ e cohomological degrees. Hence there is a canonical morphism in D ( R ) from t he zeroth coho mology 4 HENRIK HOLM AND PETER JØR GENSEN H 0 RHom( C , R ) ∼ = C ∗ to RHom( C , R ). Complete it to a distinguished triangle, C ∗ χ → RHom( C , R ) → M → , (1.b) and consider the long exact cohomology sequence which consists of pieces H i ( C ∗ ) H i χ − → H i RHom( C , R ) − → H i M . Since C ∗ is a mo dule, H i ( C ∗ ) = 0 for i 6 = 0. Com bined with equation (1.a), the long exact se quence hence implies H 6 − 2 M = 0. Moreo v er, H 0 χ is an isomorphism b y the construction of χ , and b y assumption, H 1 RHom( C , R ) = Ext 1 ( C , R ) = 0. So in f act, the long exact sequence also implies H − 1 M = H 0 M = H 1 M = 0. Consequen tly , the complex M admits an injec tive resolution of the form I = · · · → 0 → I 2 → I 3 → · · · , and in particular, H 6 1 RHom( G, M ) ∼ = H 6 1 Hom( G, I ) = 0 (1.c) for eac h R -mo dule G . No w let G b e in G . It follows fr om Remark 1 .2(i) that there is an isomorphism RHom( G, R ) ∼ = G ∗ in D ( R ), and henc e b y “sw a p” , [4, (A.4.22)], w e get RHom( G, RHom( C , R )) ∼ = RHom( C , RHom( G, R )) ∼ = RHom( C , G ∗ ) . Th us, b y applying RHom( G, − ) to the distinguished triangle (1.b) w e obtain RHom( G, C ∗ ) → RHom( C , G ∗ ) → RHom( G, M ) → . Com bining the long exact cohomolog y sequence of this with equation (1.c) pro v es the lemma.  Lemma 1.4. L et C b e an R -mo dule. (i) If Ext 1 R ( C , G ) = 0 then Ext 1 R ( G , C ∗ ) = 0 . (ii) If Ext 1 R ( C , R ) = 0 and Ext 1 R ( G , C ∗ ) = 0 , then Ext 1 R ( C , G ) = 0 . Pr o of. Com bine Lemma 1.3 with Remark 1.2, parts (ii) and ( iii) , re- sp ectiv ely , part (v).  Let G γ → N b e a G -preco v er. F or the following theorems, recall that γ is called a sp ecial G -precov er if Ext 1 R ( G , Ker γ ) = 0 , and that γ is called a cov er if eac h en domo r phism G ϕ → G with γ ϕ = γ is an automorphism. Special G -preen ve lop es and G -en v elop es are defined dually . RINGS WITHOU T GORENSTEIN GOVOR OV-LAZARD 5 Theorem 1.5. L et M b e in mo d R , let G b e in G , and let G γ → M ∗ b e a h o momorphism. C onsider the c omp osition M δ M − → M ∗∗ γ ∗ − → G ∗ wher e δ denotes the biduality homomorphism a gain. Th en (i) If γ is a G -pr e c over then γ ∗ δ M is a G -pr e envelop e. (ii) If γ is a sp e cial G -pr e c over then γ ∗ δ M is a sp e cial G -p r eenve- lop e. (iii) If γ is a G -c ove r then γ ∗ δ M is a G -envelop e. Pr o of. There is a comm utative diagram G ∼ = δ G   γ / / M ∗  _ δ M ∗   G ∗∗ γ ∗∗ / / M ∗∗∗ ( δ M ) ∗ O O O O (1) δ M ∗ γ = γ ∗∗ δ G , (2) γ = ( δ M ) ∗ γ ∗∗ δ G . Here (1) just says that the biduality homomorphism is nat ura l. By the pro of of [4 , prop. (1.1.9)] w e hav e ( δ M ) ∗ δ M ∗ = 1 M ∗ , so δ M ∗ is (split) in- jectiv e, ( δ M ) ∗ (split) surjectiv e. No w ( 2 ) follo ws fro m δ M ∗ ( δ M ) ∗ γ ∗∗ δ G = δ M ∗ ( δ M ) ∗ δ M ∗ γ = δ M ∗ γ since δ M ∗ is injectiv e. (i). Supp o se that γ is a G -precov er and let e G be in G . Remark 1.2(iv) and “sw ap” in the form [3, I I. Ex er. 4] give the follo wing natural equi- v a lences of functors, Hom( − , e G ) ≃ Hom( − , e G ∗∗ ) ≃ Hom( e G ∗ , ( − ) ∗ ) . This giv es the (top) tw o squares of the comm utat iv e diagram b elo w, where w e ha v e a bbreviated Hom( − , − ) to ( − , − ). The (b ottom) com- m utativ e triangle comes from applying Hom( e G ∗ , − ) to part (2) from the b eginning of the proo f. ( G ∗ , e G ) ∼ =   ( γ ∗ , e G ) / / ( M ∗∗ , e G ) ∼ =   ( δ M , e G ) / / ( M , e G ) ∼ =   ( e G ∗ , G ∗∗ ) ( e G ∗ ,γ ∗∗ ) / / ( e G ∗ , M ∗∗∗ ) ( e G ∗ , ( δ M ) ∗ ) / / ( e G ∗ , M ∗ ) ( e G ∗ , G ) ∼ = ( e G ∗ ,δ G ) h h R R R R R R R R R R R R R R ( e G ∗ ,γ ) 6 6 6 6 l l l l l l l l l l l l l l 6 HENRIK HOLM AND PETER JØR GENSEN Since e G ∗ is in G b y Remark 1.2(iii), the map Hom( e G ∗ , γ ) is surjec- tiv e, and the diagra m implies that so is Hom( δ M , e G ) ◦ Hom( γ ∗ , e G ) = Hom( γ ∗ δ M , e G ). Hence γ ∗ δ M is a G -preenv elope. (ii). Supp ose that γ is a special G -preco v er; in particular w e ha ve Ext 1 ( G , Ker γ ) = 0. Part (i) sa ys that γ ∗ δ M is a G - preenv elop e, and it remains t o sho w Ex t 1 ( C , G ) = 0 where C = Cok er ( γ ∗ δ M ). T o pro v e this w e use Lemma 1.4(ii). Th us we need to sho w that Ext 1 ( G , C ∗ ) = 0 and Ext 1 ( C , R ) = 0. Applying ( − ) ∗ to the exact s equence M γ ∗ δ M / / G ∗ π / / C / / 0 giv es the second exact row in G ∼ = δ G   γ / / M ∗ 0 / / C ∗ π ∗ / / G ∗∗ ( δ M ) ∗ γ ∗∗ / / M ∗ where t he square is commutativ e by part (2) at the b eginning of the pro of. It follows that C ∗ ∼ = Ker γ , and hence Ext 1 ( G , C ∗ ) = 0. T o prov e Ext 1 ( C , R ) = 0, w e will arg ue that eac h short exact sequence 0 → R → E → C → 0 splits . Consider the diagram with ex act ro ws, M µ        γ ∗ δ M / / G ∗ ϕ ~ ~ | | | | | | | ν        π / / C χ ~ ~ } } } } } } } / / 0 0 / / R ρ / / E ε / / C / / 0 . By Remark 1.2, (i) and (iii), w e ha v e Ext 1 ( G ∗ , R ) = 0, so the functor Hom( G ∗ , − ) preserv es the exactness of the b o ttom row . In part icular, there exists G ∗ ν → E with εν = π . By the univ ersal prop ert y of the k ernel of ε , there exists a (unique) M µ → R with ρµ = ν γ ∗ δ M . Since γ ∗ δ M is a G -preen ve lop e and since R is in G b y Remark 1 .2(ii), there exists G ∗ ϕ → R satisfying ϕγ ∗ δ M = µ . It follo ws that ( ν − ρϕ ) γ ∗ δ M = ν γ ∗ δ M − ρϕγ ∗ δ M = ν γ ∗ δ M − ρµ = 0 , so by t he univ ersal property of the cok ernel o f γ ∗ δ M , there exists a (unique) C χ → E with χπ = ν − ρϕ . Consequen tly , εχπ = ε ( ν − ρϕ ) = ε ν − ε ρϕ = π − 0 = id C π , RINGS WITHOU T GORENSTEIN GOVOR OV-LAZARD 7 and since π is surjectiv e w e g et εχ = id C . This prov es that ε is a split epimorphism as desired. (iii). Supp ose that γ is a G -co v er. Part (i) says that γ ∗ δ M is a G - preen v elop e, and it remains to sho w that each endomorphism G ∗ ϕ → G ∗ with ϕγ ∗ δ M = γ ∗ δ M (1.d) is an automorphism. Ho w ev er, suc h an endomorphism has γ δ − 1 G ϕ ∗ = ( δ M ) ∗ γ ∗∗ ϕ ∗ = ( δ M ) ∗ γ ∗∗ = γ δ − 1 G where the first a nd third = are by part (2) at the b eginning of the pro of while the second = is ( − ) ∗ of equation (1.d). Hence γ ( δ − 1 G ϕ ∗ δ G ) = γ , and since γ is a G -cov er and δ − 1 G ϕ ∗ δ G is an endomorphism of G , it follo ws that δ − 1 G ϕ ∗ δ G is an automorphism. Therefore ϕ ∗ , and hence also ϕ ∗∗ , is an automorphism. Applying Re- mark 1.2, (iii) a nd (iv), and natura lit y of the biduality homomorphism giv es ϕ = δ − 1 G ∗ ϕ ∗∗ δ G ∗ whence ϕ is an automorphism as desired.  Theorem 1.6. L et M b e in mo d R , let G b e in G , and let M µ → G b e a h o momorphism. C onsider the algebr aic dual G ∗ µ ∗ → M ∗ . Then (i) If µ is a G -pr e envelop e then µ ∗ is a G -pr e c over. (ii) If µ is a sp e cial G -pr e envelop e then µ ∗ is a s p e cial G -pr e c over. (iii) If µ is a G -envelop e then µ ∗ is a G -c over. Pr o of. (i). W e ha v e Hom( G, µ ∗ ) ∼ = Hom( µ, G ∗ ) b y “ sw ap”, [3, I I. Exer. 4], and com bined with Remark 1.2(iii) this implies the claim. (ii). Supp o se that µ is a sp ecial G -preenv elop e; in particular w e ha v e Ext 1 (Cok er µ, G ) = 0 . P art (i) sa ys that µ ∗ is a G -preco v er, and it remains to sho w Ext 1 ( G , Ker( µ ∗ )) = 0. But this follow s from Lemma 1.4(i) b ecause Ker( µ ∗ ) ∼ = (Cok er µ ) ∗ . (iii). Supp ose that µ is a G -en v elop e. P art (i) sa ys that µ ∗ is a G -pre- co v er, and it remains to show that eac h G ∗ ϕ → G ∗ with µ ∗ ϕ = µ ∗ is an automorphism. The bidualit y homomorphism is natural so δ G µ = µ ∗∗ δ M , and sinc e δ G is an isomorphism b y Remark 1.2(iv), it follow s that µ = δ − 1 G µ ∗∗ δ M . Applying ( − ) ∗ to µ ∗ ϕ = µ ∗ giv es ϕ ∗ µ ∗∗ = µ ∗∗ . Com bining these giv es ( δ − 1 G ϕ ∗ δ G ) µ = ( δ − 1 G ϕ ∗ δ G )( δ − 1 G µ ∗∗ δ M ) = δ − 1 G ϕ ∗ µ ∗∗ δ M = δ − 1 G µ ∗∗ δ M = µ . Since µ is a G -en v elop e and δ − 1 G ϕ ∗ δ G is a n endomorphism of G , it follo ws that δ − 1 G ϕ ∗ δ G is an automorphism. 8 HENRIK HOLM AND PETER JØR GENSEN The argumen t used at the end of the pro of of Theorem 1.5 no w shows that ϕ is an automor phism a s desired.  2. Existence of pree nvelopes and the Gov or ov-Lazard Theorem This section prov es Theorems A a nd B of the introduction; see Theo- rems 2.7 and 2.5. Setup 2.1. In this section, the comm utativ e no etherian ring R is as- sumed to b e lo cal with residue class field k . W e write d = depth R . In the follo wing lemma, the case d = 0 is trivial, d = 1 is closely inspired b y a pro of of T a k aha shi, and d > 2 is classical. Recall that Ω d ( k ) denotes the d th syzygy in a minimal free resolution of k o ver R . Lemma 2.2. T h er e ex i s ts an M in mo d R such that Ω d ( k ) is isomor- phic to a dir e ct summand of M ∗ . Pr o of. d = 0. W e can use M = k since Ω d ( k ) = Ω 0 ( k ) = k and since M ∗ = Ho m( k , R ) ∼ = k e with e 6 = 0 b ecaus e d = 0. d = 1. W e will show that M = Ω d ( k ) ∗ w orks here; in fact, we will sho w that the bidualit y homomorphism fo r Ω d ( k ) is an isomorphism so Ω d ( k ) ∼ = Ω d ( k ) ∗∗ = M ∗ . There is a short exact sequence 0 → m µ → R → k → 0 (2.a) where m is the maximal ideal o f R and µ is the inclusion, so Ω d ( k ) = Ω 1 ( k ) = m . If R is regular then k has pro jectiv e dime nsion 1 by the Auslander- Buc hsbaum formula, so (2.a) sho ws that m is pro jectiv e whence the bidualit y homomorphism δ m is an isomorphism as desired . Assume that R is not regular. F o r r easons o f clarit y , w e start b y re- pro ducing, in our notation, part of T ak a hashi’s pro of of [1 8 , thm. 2.8]. Applying ( − ) ∗ and its deriv ed functors to the short exact seque nce (2.a) giv es a long exact sequence con taining 0 → R ∗ µ ∗ → m ∗ → k e → 0 (2.b) where w e ha v e written k e instead of Ex t 1 ( k , R ), and where e 6 = 0 since d = 1. Applying ( − ) ∗ again g ives a left exact sequence 0 → ( k e ) ∗ → m ∗∗ µ ∗∗ → R ∗∗ ; here ( k e ) ∗ = 0 b ecause d = 1, so µ ∗∗ is injectiv e. RINGS WITHOU T GORENSTEIN GOVOR OV-LAZARD 9 Consider the comm utat iv e square m   µ / /  _ δ m   R ∼ = δ R   m ∗∗   µ ∗∗ / / R ∗∗ where δ m is injectiv e because δ R µ is injectiv e. The re are inclusions Im( µ ∗∗ δ m ) ⊆ Im( µ ∗∗ ) ⊆ R ∗∗ . (2.c) W e hav e R ∗∗ / Im( µ ∗∗ δ m ) = R ∗∗ / Im( δ R µ ) ∼ = R/ Im( µ ) ∼ = k where the first ∼ = is because δ R is an isomorphism. This quotien t is simple so one of the inclusions (2.c) m ust b e a n equality ; this means that either µ ∗∗ or δ m is an isomorphism . Supp ose that µ ∗∗ is an isomorphism ; w e will pro v e a con tradiction whence δ m is an isomorphism as desired . T o g et the con tradiction, we now depart from T ak ahashi’s pro of. Since µ ∗∗ is a n isomorphism, so is R ∗∗∗ µ ∗∗∗ − → m ∗∗∗ , and so m ∗∗∗ ∼ = R . But ( δ m ) ∗ δ m ∗ = id m ∗ b y the pro of o f [4, prop. (1 .1.9)], so m ∗ δ m ∗ − → m ∗∗∗ is a split monomorphism. It follo ws that m ∗ is a direct summand of R , so m ∗ is pro jectiv e. Hence the exact sequ ence (2.b) g iv es a pro jectiv e re- solution of k e , and since e 6 = 0 it follows that gldim R 6 1 con tra dicting that R is no t regular. d > 2. Here w e hav e Ω d ( k ) = Ω 2 (Ω d − 2 ( k )), so it is enough to sho w t ha t a sec o nd syz ygy o f a finitely generated mo dule is a direct summand of some M ∗ . In fact, suc h a second syzygy Ω 2 is isomorphic to a n M ∗ . Namely , Ω 2 sits in a short exact sequence 0 → Ω 2 → P π → Q whe re P and Q are finitely generated pro jectiv e mo dules. Consider the right- exact sequence Q ∗ π ∗ → P ∗ → M → 0 and apply ( − ) ∗ to get a left- exact sequence 0 → M ∗ → P ∗∗ π ∗∗ − → Q ∗∗ . Since π ∗∗ is isomorphic to π , w e get Ω 2 ∼ = M ∗ .  The follow ing lemma is implicitly in [6], but it is handy to ma ke it explicit for reference. Recall fro m [6, defs. (2.1)] that if B is a full sub category o f mo d R , then a B -appro ximation of an M in mo d R is a short exact sequence 0 → K → B → M → 0 where B is in B and Ext > 1 R ( B , K ) = 0. Lemma 2.3. Consider a sp e cia l G -pr e c over and c omplete it with its kernel. The r esulting short exact se quenc e 0 → K → G → M → 0 is a G -appr oximation of M . 10 HENRIK HOLM AND PETER JØR GENSEN Pr o of. W e kno w Ext 1 ( G , K ) = 0. By [4, cor. (4.3.5 ) (a)] eac h G in G sits in a short exact sequence 0 → G ′ → P → G → 0 where P is a finitely generated pro jectiv e mo dule and G ′ is in G , and it follows by an easy induction that Ext > 1 ( G , K ) = 0 as desired.  Remark 2.4. L et us giv e a brief summary of a part o f [6]. Recall from [6, (1.1)] that if B is a full subcategory o f mo d R , then h B i denotes the closure under direct summands and extensions. The class of finitely generated Gorenstein pro jectiv e mo dules G is a so-called reflexiv e sub category of mo d R b y [6, def. (2.6)]. It follow s from [6, prop. (2.10)] that h b R ⊗ R G i is a reflex ive sub category of mo d b R . No w supp ose that there is a n h b R ⊗ R G i -cov er of Ω d b R ( k ). The co ve r is an h b R ⊗ R G i -a ppro ximation by [6, (2.2)(b)]. But when suc h an appro ximation exists, the pro of of [6, thm. (3 .4)] giv es that either, b R is Gorenstein, or h b R ⊗ R G i consists of fr ee b R -mo dules. An imp ortan t input to the pro of of the next theorem are the metho ds and results dev elop ed by Christensen, Piepme yer, Striuli, a nd T ak a - hashi in [6]. Theorem 2.5. The fol lowing thr e e c onditions ar e e q uivalent. (i) Each mo dule in mo d R has a G -envelop e. (ii) Each mo dule in mo d R has a sp e cial G -pr e envelop e. (iii) R is Gor e n stein or G = F . They i m ply the fol lowing c ond i tion . (iv) Each mo dule in mo d R has a G -pr e env e lop e. Mor e over, if R is henseli a n then (iv) implies (i) , (ii) , and (iii) . Pr o of. (i) ⇒ (ii). Holds by W ak amatsu’s Lemma, [20, lem. 2.1.2]. (ii) ⇒ (iii). By Lemma 2.2 the mo dule Ω d R ( k ) is a direct summand in a mo dule of the form M ∗ where M is in mo d R . If (ii) holds then M has a sp ecial G -preen v elop e, and b y Theorem 1.6( ii) it follows that M ∗ has a sp ecial G - preco v er. Completing with the k ernel g iv es a short exact sequence 0 → K → G → M ∗ → 0 whic h is a G - appro ximation of M ∗ b y Lemma 2.3 . T ensoring the sequence with b R giv es an h b R ⊗ R G i -a ppro ximation of b R ⊗ R M ∗ b y [6, prop. 2.4 ]. In pa rticular, there is a n h b R ⊗ R G i -precov er of RINGS WITHOU T GORENSTEIN GOVOR OV-LAZARD 11 b R ⊗ R M ∗ , and the same mus t hold for its direct summand b R ⊗ R Ω d R ( k ) ∼ = Ω d b R ( k ). Hence there is an h b R ⊗ R G i -cov er of Ω d b R ( k ) b y [17, cor. 2.5]. But no w the results of [6] imply that either, b R is Gorenstein, or h b R ⊗ R G i consists of free b R - mo dules; see Remark 2.4. In t he former case, R is Gorenstein by [16, thm. 18.3]. In the latter case, in particular, b R ⊗ R G is a free b R -mo dule whene ver G is in G . But then G is a free R -mo dule whence G = F ; cf. [16, cor. p. 53, exe r. 7.1, and (3), p. 6 3]. (iii) ⇒ (i). First, supp ose that R is Go r enstein. Then eac h finitely ge- nerated R -mo dule ha s a G - cov er b y unpublished w ork of Auslander; see [11 , thm. 5 .5 ]. Existence of G -en v elop es no w follow s from Theorem 1.5(iii). Secondly , suppose G = F . Then eac h finitely generated R -mo dule has an F - en v elop e by [19, Prop. 2.3(3)], whic h does not need that pap er’s assumption that the ring is henselian. (i) ⇒ (iv). T rivial. No w assume tha t R is henselian. (iv) ⇒ (i). Supp ose that (iv) holds. Then Theorem 1.6(i) implies that eac h R -mo dule of the form M ∗ with M in mo d R has a G - preco v er. Since R is henselian, eac h M ∗ has a G -cov er b y [17, cor. 2.5], and so eac h M has a G -en v elop e b y Theorem 1 .5(iii).  Remark 2.6. As a consequence, the f ollo wing conditions are equiv a - len t. (i) Eac h mo dule in mo d R has a G -cov er. (ii) Eac h mo dule in mo d R has a sp ecial G -precov er. (iii) Eac h mo dule in mo d R has a G -env elop e. (iv) Eac h mo dule in mo d R has a sp ecial G -preen v elop e. (v) R is Gorenstein or G = F . Namely , (i) ⇒ (ii) is b y W ak amatsu’s Lemma, [20, lem. 2.1.1]. (ii) ⇒ (iv) follo ws from Theorem 1.5(ii). Conditions (iii), ( iv), and (v) are equiv a - len t b y Theorem 2.5. And (v) ⇒ (i) follow s from unpublished w ork b y Auslander; see [11, thm. 5.5]. Note t ha t the equiv alence of (i), (ii), and (v) w as first established in [6], and that our proof dep ends on that pap er. 12 HENRIK HOLM AND PETER JØR GENSEN No w assume that R is hense lian. Com bining with a result of Cra wley- Bo ev ey sho ws that the follo wing conditions are also equiv alen t, where lim − → G denotes the closure o f G under direct limits. (i) Eac h mo dule in mo d R has a G -precov er. (ii) Eac h mo dule in mo d R has a G -preen ve lo p e. (iii) R is Gorenstein or G = F . (iv) lim − → G is closed under set indexed direct pro ducts. Namely , (i) ⇒ (iii) holds b y [6, (2.8) and thm. (3.4)]. (iii) ⇒ (i) fol- lo ws from unpu blished work by Auslander as abov e; see [11 , thm. 5.5]. (ii) ⇔ (iii) is b y Theorem 2.5. And (ii) ⇔ ( iv) holds by [7, (4.2)]. Theorem 2.7. If R has a dualizing c omplex, is henselian, not Gor en- stein, and has G 6 = F , then lim − → G is strictly c ontaine d in the class of Gor enstein flat mo dules. Pr o of. Eac h mo dule in G is Gorenstein flat , cf. [4, Thm. (5.1.11)], and the class of Gorenstein flat mo dules is closed under direc t limits by [12], so lim − → G is con tained in the class of Gorenstein flat modules. The class of Gorenstein flat mo dules is closed under set indexed pro- ducts by [5, thm. 5.7]. O n the other hand, b y the last four conditions of Remark 2.6, the assumptions on R imply that lim − → G is not closed under set indexed pro ducts.  Example 2.8. It is easy to find rings of the type required b y Theorem 2.7. F or instance, let us sh ow that the 1-dimensional r ing T = Q [ X , Y , Z, W ℄ / ( X 2 , Y 2 , Z 2 , X Y ) satisfies the conditions of the theorem. First note t ha t since T is complete, it has a dualizing complex and is henselian. Next consider S = Q [ X , Y ℄ / ( X 2 , Y 2 , X Y ) whic h is no t Gorenstein. The ring T is S [ Z , W ℄ / ( Z 2 ); that is, T is t he ring of dual num bers o v er S [ W ℄ . Since S is not Gorenstein, neither is S [ W ℄ or T . Finally , let z be the image of Z in T . Then the complete pro jectiv e resolution · · · → T z · → T z · → T → · · · sho ws that the non-pro jectiv e mo dule T / ( z ) is Gorenstein pro jectiv e, so G 6 = F . RINGS WITHOU T GORENSTEIN GOVOR OV-LAZARD 13 Remark 2.9. Assum e that R is artinian. Then it has a dualizing complex and is henselian (in fact, R is complete). Moreo ver, it is easy to pro ve that eac h G orenstein flat mo dule is Gorenstein pro jective . If R is not Gorenstein a nd has G 6 = F , then Theorem 2 .7 shows that lim − → G is strictly con tained in the class of Gorenstein pro jectiv e modules. Hence [2, 4.2] sho ws that R is not a so-called virtually Gorenstein ring. 3. A spe cial Gorenste in fla t module This short section sho ws a metho d for constructing a G orenstein flat mo dule outside lim − → G . Construction 3.1. Let { G i } i ∈ I b e a set of represen tativ es of the iso- morphism classes of indecomp osable mo dules in G . Let M b e in mo d R . F o r eac h i in I , view H ( i ) = Hom R ( M , G i ) as a set and consider the direct pro duct G H ( i ) i indexed by that set. Define Λ( M ) = Q i ∈ I G H ( i ) i . Prop osition 3.2. Assume that R has a d uali z ing c omplex. L et M b e in mo d R and supp ose that M do es no t have a G -pr e envelop e. Then Λ( M ) is a Gor enstein flat mo dule outside lim − → G . Pr o of. As in the proo f of The orem 2.7, the mo dules in G are Gorenstein flat and the class of Gorenstein flat mo dules is closed under set indexed pro ducts, so Λ( M ) is G o renstein flat. F o r eac h i in I , consider the homomorphism M µ i → G H ( i ) i , m 7→ ( h ( m )) h ∈ H ( i ) . Let Λ( M ) π i → G H ( i ) i b e the i ’th pro jection, and let M µ → Λ( M ) b e the unique homomorphism whic h satisfies π i µ = µ i for eac h i in I . Then eac h homomorphism M η → G with G in G factors through µ , M η   µ / / Λ( M ) . λ { { w w w w w G Namely , w e may assum e G = G i for some i , since each G in G is isomorphic to a finite direct sum of mo dules from the set { G i } i ∈ I . But then η is an elemen t of H ( i ), and w e can let λ equal the comp o sition of the pro jections Λ( M ) π i → G H ( i ) i → G i where the second one is on to the η th copy of G i . 14 HENRIK HOLM AND PETER JØR GENSEN No w, M is finitely presen ted, so if Λ( M ) w ere in lim − → G then [15, prop. 2.1] would giv e that µ could b e factored as M e µ → e G → Λ( M ) with e G in G . Since eac h homomorphism M η → G factors through µ by the ab ov e, it w ould also factor through e µ whic h w ould hence be a G -preenv elope of M . Since there is no suc h G -preenv elope, Λ ( M ) is outside lim − → G .  A c k n ow le dg e ment. W e thank Luc hezar Avramo v, Apo stolos Beli- giannis, and Lars Winther Christe nsen for commen ts to previous v er- sions of the pap er. In particular, LA p ointed out reference [13], and AB informed us that [2] had already f o und Artin algebras without a Gorenstein analogue of t he Go voro v-Lazard Theorem. Reference s [1] M. Auslander a nd M. Bridger, “Stable mo dule theory”, Mem. Amer. Math. So c., no . 94, American Ma thematical So ciety , P rovidence, 1969. [2] A. Beligiannis and H. Krause, Thick sub catego ries and virtually Gorenstein algebras , preprint (2006). math .RA/0 608710 . [3] H. Cartan and S. Eilenberg, “Ho mo logical a lgebra”, Princeton University Press, P r inceton, 1956 . Reprinted in Princeton Landmarks Math., Princeto n Univ er sity Press, Princ e to n, 199 9 . [4] L. W. Christensen, “Go renstein dimensio ns”, Lecture Notes in Math., V ol. 1747, Spr inger, Ber lin, 200 0. [5] L. W. Christens en, A. F rankild, and H. Holm, On Gor enstein pr oje ctive, in- je ctive and flat dimensions — a functorial description with applic ations , J . Algebra 302 (20 06), 231– 279. [6] L. W. Christensen, G. Piepmeyer, J. Str iuli, and R. T ak ahashi, Finite Go r en- stein r epr esen t ation typ e implies simple singularity , Adv. Math. 218 (2 0 08), 1012– 1026. [7] W. 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Dep ar tment of Basic Sciences and Environment, F acul ty of Life Sci- ences, University of Copenhagen, Thor v aldsensvej 40, 6th floor, 1871 Frederiksber g C, Denmark E-mail addr ess : hhol m@life .ku.dk URL : ht tp://w ww.di na.kvl.dk/~hholm/ School of Ma thema tics and S t a tistics, New castle University, New- castle upon Tyne NE 1 7RU, United Kingdo m E-mail addr ess : pete r.jorg ensen@ ncl.ac.uk URL : ht tp://w ww.st aff.ncl.ac.uk/peter.jorgensen