STRONGLY GORENSTEIN PROJECTIVE MODULES OVER UPPER TRIANGULAR MATRIX ARTIN ALGEBRAS. Shanghai , P. R. China. Shanghai , P. R.

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1 STRONGLY GORENSTEIN PROJETIVE MODULES OVER UPPER TRINGULR MTRIX RTIN LGERS NN GO PU ZHNG Department o Mathematics Shanhai University Shanhai 2444 P R hina Department o Mathematics Shanhai Jiao Ton University Shanhai 224 P R hina bstract We determine all the stronly complete projective resolutions and all the stronly Gorenstein projective modules over upper trianular matrix artin alebras n example is iven showin that upper trianular matrix artin alebras really produce new initely enerated stronly Gorenstein projective modules Key words: Gorenstein projective modules; stronly Gorenstein projective modules; upper trianle matrix artin alebras Introduction and preliminaries Since Eilenber and Moore [EM] the relative homoloical alebra especially the Gorenstein homoloical alebra has been developed to an advanced level: the analoues or projective and injective modules are respectively the Gorenstein projective and the Gorenstein injective modules introduced by Enochs and Jenda ([EJ]); and one considers the Gorenstein projective and the Gorenstein injective dimensions o modules and complexes (see e [F] [EJ] [Y] [EJ2] [] [M] [H] [T] [FH] [V] [V]) the existence o proper Gorenstein projective resolutions ([J] [H] [M] [T]) the Gorenstein derived unctors ([H2] [EJ] [M] [EJ2]) the Gorensteinness in trianulated cateories ([S] [] [2]) and the Gorenstein derived cateories ([GZ]) This concept o Gorenstein projective module even oes back to uslander and rider [] where they introduced the G-dimension o initely enerated module M over a twosided Noetherian rin; and then vramov Martisinkovsky and Rieten have proved that M is Goreinstein projective i and only i the G-dimension o M is zero (the remark ollowin Theorem (426) in []) There is also an analooue or ree module namely the stronly Gorenstein projective module ([M]) s observed by ennis and Mahdou a module is Gorenstein projective i and only i it is a direct summand o a stronly Gorenstein projective module ([M] Theorem 27) This also provides a more operatin way to obtain the Gorenstein projective modules i one can construct the stronly Gorenstein projective modules eectively However very ew is known about this (c [M] and also [EJ]) The correspondin author Supported by the National Natural Science Foundation o hina and o Shanhai ity (Grant No 7254 and ZR6449) nanaoshueducn pzhansjtueducn

2 2 NN GO PU ZHNG The aim o this paper is to ive a concrete construction o stronly Gorenstein projective modules via the existed construction o upper trianular matrix artin alebras ([RS] [R]) We determine all the stronly complete projective resolutions and all the stronly Gorenstein projective modules over upper trianular matrix artin alebras (Theorem 22) The eature o this construction is illustrated by Example 23 showin that it really produces new initely enerated stronly Gorenstein projective modules However onepoint extensions do not ive new stronly Gorenstein projective modules (orollary 33) 2 Let R be a rin ll R-modules considered are let and unital y R-Mod and R-mod we denote the cateory o R-modules and the cateory o initely enerated R-modules respectively Followin Enochs and Jenda ([EJ]) a R-module M is said to be Gorenstein projective (or G-projective or short) in R-Mod (resp in R-mod) i there is an exact sequence o projective modules in R-Mod (resp in R-mod) P P P d P P 2 with Hom Λ(P ) exact or any projective module in R-Mod (resp in R-mod) such that M kerd Such a P is called a complete projective resolution o R-Mod (resp o R-mod) Followin ennis and Mahdou ([M]) i P in R-Mod (resp in R-mod) is o the orm P P P then M is said to be a stronly Gorenstein projective module (or SG-projective or short) in R-Mod (resp in R-mod) Such a complete projective resolution P o R-Mod (resp o R-mod) is said to be stron There hold the ollowin (see [M]) and {projectives in R-Mod} {SG-projectives in R-Mod} {G-projectives in R-Mod} {projectives in R-mod} {SG-projectives in R-mod} {G-projectives in R-mod} Denote by R-SGProj (resp R-SGproj) the ull subcateory o SG-projective modules in R-Mod (resp in R-mod) Note that R-SGProj (resp R-SGproj) is closed under takin arbitrary direct sums (resp inite direct sums) The ollowin act is useul Proposition For any rin R there holds (R-SGProj) R-mod R-SGproj Proo Let X R-SGproj Then by deinition we have an exact sequence P : P P P such that P is initely enerated projective and that Hom R(P P ) is exact or any initely enerated projective R-module P For any projective R-module there exists a projective R-module such that R (I) or some index set I (may be ininite) Since P is initely enerated it ollows that Hom R(P ) Hom R(P ) Hom R(P R (I) ) Hom R(P R) (I)

3 STRONGLY GORENSTEIN PROJETIVE MODULES 3 Since Hom R(P R) is exact so is Hom R(P ) ie X R-SGProj R-mod Let X R-SGProj R-mod Then we have a stronly complete projective resolution P : P P P with X Ker y the exact sequence X σ P X and X R-mod we know that P is initely enerated In the view o this act in this paper all modules considered are initely enerated althouh the most part o Section 2 also works in eneral 3 From now on we consider an artin alebra Λ ie Λ is an alebra over a commutative artin rin R with R in the center o Λ via the canonical embeddin and Λ is initely enerated as an R-module ([RS] p26) I R is a ield k then Λ is exactly a initedimensional k-alebra For the representation theory o Λ we reer to uslander Reiten and Smalφ [RS] and Rinel [R] In particular Λ-mod is a Krull-Schmidt cateory Thus any initely enerated Λ-module has a unique direct decomposition into indecomposable ([RS] p33; [R] p52) 4 Let T and U be rins and M a T-U-bimodule onsider the upper trianular T M matrix rin Λ with multiplication iven by the one o matrices We U assume that Λ is an artin alebra throuhout this paper: this is exactly the case when there is a commutative artin rin R such that T and U are artin R-alebras and M is initely enerated over R which acts centrally on M ([RS] p72) X Recall rom [RS] the cateory M Λ: an object is a pair ( φ) where X is a T- Y module Y is a U-module and φ : M U Y X is a T-morphism; and a morphism o M Λ X X is a pair : ( φ) ( φ ) where : X X is a T-morphism Y Y : Y Y is a U-morphism such that φ φ (id M U ) It is an isomorphism i X and only i so are and Note that F(( φ)) X Y induces an equivalence Y r m between M Λ and Λ-mod where the Λ-action on X Y is iven by (x y) : s X X (rx + φ(m U y) sy) For convenience we use the convention : ( φ) Y Y In the ollowin we identiy Λ-mod with M Λ Under this identiication any Λ-module X is written as with X a T-module Y a U-module toether with a T-morphism Y φ : M U Y X Thus the Λ-action is written as r m x rx + φ(m U y) : s y sy P Then the indecomposable projective Λ-modules are exactly either o the orm with P an indecomposable projective T-module or o the orm M U with an

4 4 NN GO PU ZHNG indecomposable projective U-module toether with the identity map; the indecomposable I injective Λ-modules are exactly either o the orm with I an indecomposable injective T-module toether with the canonical map or o the orm with Hom T(M I) J J an indecomposable injective U-module; and the simple Λ-modules are exactly either o S the orm with S a simple T-module or o the orm with S a simple S U-module ompare [RS] p77; and [R] p9 2 SG-projective modules over upper trianular matrix artin alebras The aim o this section is to determine the stronly complete projective resolutions and hence all the SG-projective modules over an upper trianular matrix artin alebra T M Λ U 2 Denote by X : P (M U ) : β id M : X X () with P a projective T-module a projective U-module and β : P M U a T- morphism where the Λ-action on X is iven by : M U P (M U ) id M U P Note that X M U is a projective Λ-module and any projective Λ- module is o this orm It is clear that : X X is a Λ-morphism Here P and could be zero onsider the ollowin conditions (i) (v): (i) is an exact sequence o projective U-modules; (ii) T : P P P is a complex o projective T-modules such that Hom T(T P ) is exact or any indecomposable projective T-module P ; (iii) β + (id M )β ; (iv) I (p) β(p) + (id M )(x) then there exists (p x ) P (M U ) such that p (p ) x β(p ) + (id M )(x ); (v) For an arbitrary indecomposable projective U-module i (st) Hom T(P M U ) Hom U( ) with s + (id M t)β and t then there exists (s t ) Hom T(P M U ) Hom U( ) such that s s + (id M t )β and t t

5 STRONGLY GORENSTEIN PROJETIVE MODULES 5 Lemma 2 With the notations above i the conditions (i) (v) are satisied then P : X X X (2) is a stronly complete Λ-projective resolution; conversely any stronly complete Λ-projective resolution is the orm (2) where X and are iven in () satisyin the conditions (i) (v) Proo First we prove the suiciency It ollows rom (i) (iii) that 2 β id M β id M β + (id M )β id M 2 Note that (iv) implies that Ker Im β id M β id M ; it ollows that Ker Im β id M β id M and hence by (i) the sequence P is exact For any projective T-module P Hom Λ( M U P ) Since by as- sumption Hom T(T P ) is exact it ollows that Hom Λ(P P ) is exact For any projective U-module (we may assume that is indecomposable) we have P (M U ) M U Hom Λ( ) Hom T(P M U ) Hom U( ) Put Y : Hom T(P M U ) Hom U( ) y (i) (iii) we see that the sequence φ Y φ Y φ Y φ is a complex where φ(st) : (s + (id M t)β t) or (st) Hom T(P M U ) Hom U( ); and by (v) it is exact This means that Hom Λ(P M U ) is exact It ollows that P is a stronly complete Λ-projective resolution Secondly we prove the necessity Let P : X X X be an P arbitrary stronly complete Λ-projective resolution Then X is o the orm M U P (M U ) with P a projective T-module and a projective U-module Write as γ P (M U ) P (M U ) β id M :

6 6 NN GO PU ZHNG with T-morphisms β : P M U and γ : M U P Since : X X is a Λ-morphism it ollows that we have the commutative diaram M U id M M U id M id M P (M U ) β P (M U ) γ id M and hence γ ie is iven as in () y Ker Im we have Ker Im and Ker β id M Im β id M These imply that (i) (iii) (iv) are satisied and 2 y the exactness o Hom Λ(P P ) and o Hom Λ(P M U ) we see that Hom T(T P ) is exact and that (v) is satisied This completes the proo 22 Keep the notations T U M Λ Put M : { L T-mod Ext i T(L M) i }; (T-SGproj) M : { P Λ-mod P (T-SGproj) M }; M U (U-SGproj) M U : { Λ-mod U-SGproj } U- SGproj M U where the Λ-module structure o is iven by the identity map M U M U T M Theorem 22 Let Λ be an artin alebra and N a Λ-module Then N U is a SG-projective Λ-module i and only i one o the ollowin holds: (T-SGproj) M ) N ; M U L M U (U-SGproj) 2) N where L Ker with stronly L U-SGproj complete U-projective resolution such that Ker(id M ) M U Ker

7 where 3) N Ker β STRONGLY GORENSTEIN PROJETIVE MODULES 7 id M {(p x) P (M U ) (p) β(p) + (id M )(x) } : Ker P (M U ) P (M U ) P and are respectively arbitrary non-zero projective T-module and U-module and : P P and β : P M U are T-morphisms satisyin the conditions (i) (v) in 2 Remark direct sum o a module in ) and a module in 2) o Theorem 22 is o course aain a SG-projective Λ-module: in act it is in 3) In order to see this just takin β in 3) We stress that 3) really produces new SG-projective Λ-modules See 23 M U (U-SGproj) 2 I M U is lat then by 2) o Theorem 22 each module in U-SGproj is a SG-projective Λ-module However this is not true in eneral See Example 24 Proo o Theorem 22 First we justiy the suiciency I N then N Ker where T : P P P (T-SGproj) M is a stronly complete T- projective resolution Note that N M implies that Hom Λ(T M U U ) Hom T(T M U U) Hom T(T M) U is exact Since Hom Λ(T M U ) Hom T(T M U ) is a direct summand o Hom Λ(T M U U m ) or some m it ollows that Hom Λ(T M U ) is exact or any projective U-module ie T is also a stronly complete Λ-projective resolution and hence N is a SG-projective Λ-module M U L M U (U-SGproj) Let N where L Ker U-SGproj L U-SGproj with stronly complete U-projective resolution such that Ker(id M ) M U Ker onsider the sequence o projective Λ-modules P : M U M U M U id M M U with : : M U y assumption we have Ker(id M ) M U Ker M U Im Ker Im; Ker Ker Im

8 8 NN GO PU ZHNG M U since Hom Λ( P ) and Hom Λ( M U M U ) Hom U( ) it ollows that P is a stronly complete Λ-projective resolution Thus N Ker is a SG-projective Λ-module The case 3) ollows directly rom Lemma 2 Secondly we justiy the necessity I N is a SG-projective Λ-module then N Ker where is the Λ-map occurred in a stronly complete Λ-projective resolution P y Lemma 2 is o the orm (2) where X and are iven in () satisyin the conditions (i) (v) in 2 I in () then β and hence by (ii) and (iv) in 2 we know that T is a stronly complete T-projective resolution; and by takin U in (v) we see that Hom T(T M) is exact which implies N Ker M Thus N is o the orm ) I P in () then β y (i) and (v) we see that is a stronly complete U-projective resolution; and by (iv) in 2 we have Ker(id M ) M U Ker It ollows that N Ker Ker(id M ) Ker M U Ker Ker The remainin case is 3) This completes the proo M U (U-SGProj) U-SGProj T M 23 Note that in the construction o upper trianular matrix artin alebra Λ U T-mod is embedded into Λ-mod via the unctor F T : T-mod Λ-mod iven by F T(X) X ; and U-mod is embedded into Λ-mod via the unctor F U : U-mod Λ-mod M U Y M U Y iven by F U(Y ) where the Λ-module structure o is Y Y iven via id M U Y lso note that U-mod is embedded into Λ-mod via the unctor X G U : U-mod Λ-mod iven by F U(Y ) Λ-module is said to Y Y be newly produced or simply new provided that X Y / F T(T-mod) F U(U-mod) G U(U-mod) In this sense a direct sum o module in ) and a module in 2) o Theorem 22 is not new What we want to stress is that 3) o Theorem 22 really produces new stronly Gorenstein projective Λ-modules as the ollowin example shows This example also shows that a upper trianular matrix artin alebra really produces new Gorenstein projective Λ-modules

9 STRONGLY GORENSTEIN PROJETIVE MODULES 9 Example 23 Let Λ be the k-alebra iven by the quiver 2 y a x with relations x 2 y 2 ay xa (We write the conjunction o paths rom riht to let) Let T k[x]/ x 2 U k[y]/ y 2 and TM U ka kxa ka kay with the natural T-U-actions by the conjunction T M o paths Then Λ with T M TT and M U UU Write P or U P Λe ke kx: it is the unique indecomposable projective T-module; and write or U ke 2 ky: it is the unique indecomposable projective U-module Let : P P be the T-morphism iven by multiplication by x : the U-morphism iven by multiplication by y and β : P M the T-morphism iven by the riht multiplication by a: β(e ) a β(x) xa Note that id M : M U M U is iven by the riht multiplication by y i M U is identiied with M Then β satisy all the conditions (i) (v) in 2 and by Lemma 2 we et a stronly complete Λ-projective resolution P : P (M U ) P (M U ) P (M U ) with : β id M y Theorem 22 3) we obtain a stronly Gorenstein projective Λ-module {(p m) P (M U ) (p) β(p) + (id M )(m) } Ker Ker {(cx ca + day) P M c d k} ky whose Λ-module structure is iven via the T-morphism M U ky {(cx ca + day) P M c d k} : a y ( ay) ay y () Observe that there is a T-isomorphism T {(cx ca + day) P M c d k} : e (x a) x ( ay) From these we see that Ker is a new stronly indecomposable Gorenstein projective Λ-module

10 NN GO PU ZHNG Let P 2 Λe 2 ke 2 ky ka kay ke 2 ky ka kxa lternatively P can be written as β P P 2 β P P 2 β P P 2 β In this expression we have Ker {(cx ca + day + by) P P 2 b c d k} The ollowin example shows that a module in SG-projective Λ-module M U (U-SGproj) U-SGproj may not be a Example 24 Let Λ be the k-alebra iven by the quiver in Example 23 with relations x 2 y 2 ay and TM U ka kxa with the natural T-U-actions iven by the conjunction T M o paths where T and U are same as in Example 23 Then Λ with U TM TT; M U S2 S 2 where S 2 is the unique simple U-module (and also the simple Λ-module correspondin vertex 2) Note that ky S 2 is a SG-projective U-module M U ky and M U ky We claim that S 2 is not a stronly ky ky Gorenstein Λ-module y Theorem 22 it suices to prove that S 2 is neither in 2) nor in 3) o Theorem 22 I S 2 is in 2) o Theorem 22 then we have a stronly complete U-projective resolution such that ky Ker and Ker(id M ) M U Ker Then dim k dim k Ker + dim k Im 2 and U and is iven by multiplication by y Thus we have the desired contradiction Ker(id M ) Ker M U U M M U ky M U Ker I S 2 is in 3) o Theorem 22 then in Theorem 22 3) we have ky Ker and by the same arument above we have U and that is iven by multiplication by y Usin the same notation in Theorem 22 3) we have S 2 Ker {(p m) P (M U ) (p) β(p) + (id M )(m) } Ker {(p m) P M (p) β(p) } ky Write Ker X ky Then X since m M aain a contradiction 3 SG-projective modules over one-point extensions This section is to specialize the result in the last section when U is a ield Thus T M throuhout this section Λ where T is a inite-dimensional k-alebra k is k a ield which is assumed to be alebraically closed and M is a inite-dimensional T-module

11 STRONGLY GORENSTEIN PROJETIVE MODULES 3 Denote by X : P M 2n : β e : X X (3) with P a projective T-module where the Λ-module structure o X is iven by : id M 2n M k P M 2n (here we identiy M k with M 2n ); : and e : M 2n M 2n are iven by the block matrix Note that X is a Λ-morphism P (4) M2n is a projective Λ-module It is clear that : X X onsider the ollowin conditions (i) (iii): (i) P P P is a stronly complete T-projective resolution; (ii) β 2i β 2i β 2i i n where β (iii) For any s : P M with s P i n there exists s : P M such that s s + P i t2iβ2i β β 2 β 2n : P M2n ; t 2iβ 2i or some t 2i k i n Lemma 3 With the notations above i the conditions (i) (iii) are satisied then P : X X X is a stronly complete Λ-projective resolution; conversely any stronly complete Λ-projective resolution is o this orm where X and are iven in (3) and (4) satisyin the conditions (i) (iii) Proo First we prove the suiciency It is clear that Ker Im y (ii) we have β + β e and hence Im β e ker For (p β e m m 2 ) Ker β e m 2n Then (p) β 2i(p) m 2i β 2i (p) i n y Ker

12 2 NN GO PU ZHNG Im we have p P such that p (p ) y (ii) we veriy that m β (p ) m m 2 (p ) m 3 β 3(p ) (p β e m 2n It ollows that Ker β e M 2n Im β e P m 2n β 2n (p ) ie P is exact Since Hom Λ ( ) it ollows rom (i) that Hom Λ(P ) is exact For the exactness o Hom Λ(P M k k m ) we only need the exactness in the k m M k k m M M case o m The reason is k m k k Since Hom Λ( P M ) M 2n M Hom T(P M) and Hom Λ( k k ) Hom k ( k) it ollows that the exactness o Hom Λ(P M k k m ) is same as the exactness o k m ) Hom T(P M) Hom k ( φ k) Hom T(P M) Hom k ( φ k) where φ is deined by φ(s t) : (s + etβ t) or (st) Hom T(P M) Hom k ( k) t (t t 2 t 2n) : k here et (t t 2 t 2n) : M 2n M is iven by m m 2 m 2n P P i timi y (ii) we have Imφ Kerφ; and (iii) implies that Kerφ Imφ This completes the suiciency Secondly we prove the necessity Let P be a stronly complete Λ-projective resolution Then P is o the orm with X P X Y and Y M k k r with M r Since Hom Λ( X Y M k k r k r M r k r X Y ( ) Note that P and r could be We identity P ) it ollows that is o the orm (3)

13 STRONGLY GORENSTEIN PROJETIVE MODULES 3 with Hom T(P P) β Hom T(P M r ) Hom k (k r k r ) and e Hom T(M r M r ) Since k r k r k r is exact it ollows rom the Jordan canonical orm o that r is even say r 2n and that is o orm (4) y Ker β e Im β e Hom Λ(P P we et (ii) and Ker Im (we omit the details) y the exactness o ) we et (i) y the exactness o Hom Λ(P This completes the proo M k k k ) we can et (iii) We omit the details 32 In order to simpliy the stronly complete Λ-projective resolution obtained in Lemma 3 we note that i a complex is isomorphic to a stronly complete projective resolution then this complex itsel is also a stronly complete projective resolution with the same SG-projective module up to an isomorphism onsider the ollowin complex o projective Λ-modules h P M2n h P M2n h P M2n h (5) with h : e (6) where : and e : M 2n M 2n are iven by the block matrix (4); : P P satisies the condition (i) and the ollowin condition (iv) For any s : P M with s there exists s : P M such that s s Observe that (iv) is equivalent to (iii) in 3 Theorem 32 sequence P o Λ-modules is a stronly complete Λ-projective resolution i and only i it is isomorphic as a complex to a sequence o orm (5) with satisyin (i) and (iv) Proo y Lemma 3 any stronly complete Λ-projective resolution is o the orm P : P M2n P M2n P M2n where is iven in (3) with and β satisyin the conditions (i) (iii) in 3

14 4 NN GO PU ZHNG Put β e : β β 3 β 2n : P M 2n Then by condition (ii) in 3 we have eβ + β e e β and hence we have the commutative diaram o Λ-morphisms P M 2n β e P M 2n id P eβ id M 2n id P M 2n h e P M 2n id P eβ id M 2n id Note that id P eβ id M 2n is an isomorphism id onversely the assertion ollows rom Lemma 3 This completes the proo 33 Keep the notations as in 22 The ollowin result shows in particular that one-point extensions do not produce new stronly Gorenstein projective modules T M orollary 33 Let Λ be a inite-dimensional k-alebra with k an alebraically closed ield and N a Λ-module Then N is a SG-projective Λ-module i and only k i N is one o the ollowin orm ) N 2) N (T-SGproj) M ; M n k n ie N is the n copy o the last projective Λ-module any positive inteer n; 3) direct sum o a module in ) and a module in 2) M or k

15 STRONGLY GORENSTEIN PROJETIVE MODULES 5 Proo y Theorem 32 we need to compute Kerh where h : e P M 2n : P M2n P is an arbitrary projective T-module; : and e : M 2n M 2n are iven by the block matrix in (4) (here n could be zero); : P P satisy the condition (i) in 3 and (iv) in 32 Thus Write x m m 2 Kerh m 2n {(p x) P M 2n (p) e (x) } Ker M2n Note that Ker k n : more precisely i we write k k 2 with k k 2 k then Ker k n k k 3 (it is understood to be zero i n ) Then we rewrite Kerh as the ollowin orm Kerh {(p x) P M 2n (p) (x) e } Ker m m 3 {(p P M 2n (p) } Ker M2n m 2n k k 3 With this expression we immediately see that Kerh Ker via the isomorphism m m 3 c m c 3 Kerh (p ) (p m 3 c 2n m 2n m 2n Mn k n c c 3 c 2n ) Ker M2n

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