Triangulated Structures for projective Modules

T riangulated Strutures fo r p rojetive Mo dules Bory ana Dimitro v a No v em b er 15, 2018 Abstrat: W e giv e a  haraterisation of those lo al not neessary omm utativ e rings, for whi h the ategory of pro jetiv e mo dules admits a triangulation with the iden tit y as translation funtor. By "admits a triangulation" w e mean that the ategory an b e giv en the struture of a triangulated ategory that satises the standard set of axioms inluding the o tahedral axiom. 1 Intro dution Among ategories of mo dules that admit triangulated struture, the simplest example is probably that of k -v etor spaes for a eld k . Here the translation funtor is the iden tit y and distinguished triangles are giv en b y sequenes of the form X u / / Y v / / Z w / / X , su h that X u / / Y v / / Z w / / X u / / Y is an exat sequene of k -mo dules. An alternativ e desription is to sa y that the distinguished triangles are preisely the on tratible triangles, as dened b y Neeman in [Nee, 1.3.5℄. It is almost immediately lear that all the axioms are satised thanks to the fat that all k - mo dules are pro jetiv e. This example an b e generalized without m u h eort: w e adopt the analogous denition of distinguished triangles to the ategory of R -mo dules for an y ring R , su h that ev ery mo dule is pro jetiv e, and w e indeed obtain a triangulated struture for this ategory with translation funtor the iden tit y . What kind of rings ha v e this prop ert y? The answ er an b e giv en straigh tforw ard - all mo dules b eing pro jetiv e is equiv alen t to all mo dules b eing injetiv e. In partiular all submo dules of the mo dule R are diret summands, whi h implies that R is semisimple. On the other hand, all mo dules o v er a semisimple ring are pro jetiv e. The Artin-W edderburn theorem giv es us a preise  haraterization of the semisimple rings: They are of the form R = R 1 × R 2 × · · · × R n , where R i is a full matrix ring M n i ( d i ) o v er some sk ew eld d i for ev ery i from 1 to n . Lo oking at this example it seems reasonable to try to giv e a triangulation for a ategory of pro jetiv e mo dules o v er more general rings b y just taking exat sequenes of mo dules. It is in fat true that if su h a ategory admits a triangulation then the distinguished triangles are exat as a sequene of mo dules, but one should b e a w are of the fat that on v ersely , this naiv e onstrution do es not pro due a triangulation in general. F or a oun terexample 2 Preliminaries one an think of the ategory of pro jetiv e Z -mo dules: learly there is no exat sequene of pro jetiv e (th us free) Z -mo dules of the required form Z · 3 / / Z / / ? / / Z · 3 / / Z , starting with the map Z · 3 → Z , so the rst axiom of a triangulated ategory an not b e satised. This is just one of the problems that ma y o ur in the ab o v e onstrution when not all mo dules o v er the giv en ring are pro jetiv e. A tually , there are plen t y of examples of rings for whi h the ategory of pro jetiv e mo dules do es not admit a triangulation at all. Indeed, a neessary ondition for this is that the ring is Quasi-F rob enius. Ho w ev er, it is b y no means suien t, and so the question for a  haraterization of rings whi h allo w a triangulation of their ategories of pro jetiv e mo dules arises. In [HL℄ an answ er is giv en in the ase where the onsidered rings are omm utativ e, and so our aim is to generalize this result to not neessarily omm utativ e rings. W e are though going to restrit our atten tion to lo al rings. Denote b y P and P fg the ategories of pro jetiv e left R -mo dules and nitely generated pro jetiv e left R -mo dules. Our main theorem is: Theorem 3.5 . L et R b e a lo  al ring with maximal ide al m . Then ther e exists a triangulation for P , or P fg r esp e tively, with Σ = Id if and only if m 2 = 0 , m = Rx = xR for al l x ∈ m \{ 0 } and in addition one of the fol lowing  onditions holds: (i) R is a skew eld, i.e., m = { 0 } (ii) m = 2 R (iii) char R = 2 , i.e., 1 + 1 = 0 in R , and for some x ∈ m \ { 0 } ther e is a nontrivial element r x ∈ R / m suh that a) σ x ( r x ) = r x b) σ 3 x ( t ) = r − 1 x tr x for every t ∈ R / m Here σ x is a ertain automorphism of the sk ew eld R/ m that dep ends on the  hosen generator x of m (Lemma 3.3 ). In the pro of of these results w e are going to onstrut an expliit triangulation for the dieren t ases, using an idea from the artile of F. Muro, S. S h w ede, N. Stri kland [ MSS℄. In teresting questions that ome up are whether there are dieren t triangulations in the onsidered ases and - if so - ho w they lo ok lik e. W e will also giv e answ ers to these, for the most part. A  kno wledgemen ts: This artile is a revised v ersion of m y diploma thesis, written at the Univ ersit y of Bonn. I w ould lik e to thank m y advisor Stefan S h w ede for the in teresti topi and the enour- agemen t throughout the w a y , as w ell as the German A ademi Ex hange Servie for the nanial supp ort during the en tire time of m y studies. No w, let us start with some 2 Prelimina ries W e b egin b y xing some notation and reolleting the needed terminology . Let C b e an additiv e ategory and Σ : C → C an equiv alene. A triangle in C (with resp et 2 2 Preliminaries to Σ ) is a diagram in C of the form X u / / Y v / / Z w / / Σ X su h that the omp osites v u , wv and (Σ u ) w are the zero morphisms. A morphism of triangles is then a omm utativ e diagram of the form X u / / f   Y v / / g   Z w / / h   Σ X Σ f   X ′ u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X ′ . In this on text w e also ha v e the notion of a mapping  one whi h is analogous to the denition of the mapping one of a  hain map, namely , w e dene the mapping one of the ab o v e morphism to b e the triangle Y ⊕ X ′ „ − v 0 g u ′ « / / Z ⊕ Y ′ “ − w 0 h v ′ ” / / Σ X ⊕ Z ′ „ − Σ u 0 Σ f w ′ « / / Σ Y ⊕ Σ X ′ . No w, a triangulate d  ate gory T is an additiv e ategory together with a selfequiv alene Σ : T → T and a lass of triangles (with resp et to Σ ), alled distinguished, su h that: A.1 · The lass of distinguished triangles is losed under isomorphisms of triangles. · The triangle X id / / X / / 0 / / Σ X is distinguished for ev ery X ∈ Ob( T ) . · F or an y morphism u : X → Y in T there exists a distinguished triangle of the form X u / / Y / / Z / / Σ X . A.2 · A triangle X u / / Y v / / Z w / / Σ X is distinguished i its translate Y v / / Z w / / Σ X − Σ u / / Σ Y is. A.3 · F or an y omm utativ e diagram with distinguished ro ws of the form X u / / f   Y v / / g   Z w / / Σ X Σ f   X ′ u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X ′ , there is a morphism h : Z → Z ′ ompleting the diagram to a morphism of triangles, in a w a y that the mapping one is distinguished. This set of axioms is equiv alen t to the ommon one inluding the o tahedral axiom as sho wn b y Neeman in [Nee, 1.4.6℄ and [Nee2 , 1.8℄, and it will turn out to b e more on v enien t for our purp ose. F urther, t w o maps of triangles will b e alled homotopi if they dier b y a homotopy X u / / f ′   f   Y Θ ~ ~ } } } } } } } } } } } } } } } } } v / / g ′   g   Z Φ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ w / / h ′   h   Σ X Σ f ′   Σ f   Ψ } } { { { { { { { { { { { { { { { { { { X ′ u ′ / / Y ′ v ′ / / Z ′ w ′ / / Σ X ′ . 3 2 Preliminaries This means there are morphisms Θ , Φ and Ψ as in the diagram su h that g − g ′ = Φ v + u ′ Θ h − h ′ = Ψ w + v ′ Φ Σ( f − f ′ ) = Σ(Θ u ) + w ′ Ψ . As one should exp et homotopi morphisms ha v e isomorphi mapping ones. F or a pro of of this fat see [Nee, 1.3.3℄. Another notion that w e are going to mak e use of are the  ontr atible triangles . W e all a triangle on tratible if the iden tit y map is homotopi to the zero map. A homotop y b et w een b oth maps is alled a nul lhomotopy . Later on w e are going to use the fats that a on tratible triangle in a triangulated ategory is alw a ys distinguished, and that a morphism ha ving a on tratible triangle as soure or target is homotopi to the zero map ([Nee℄ Prop osition 1.3.8 and Lemma 1.3.6). Coming ba k to what w e originally w an ted to do, w e set the follo wing notation : P will denote the ategory of pro jetiv e R -mo dules and P fg the ategory of nitely generated pro jetiv e R -mo dules, where R is a (not neessarily omm utativ e) ring. Later on in this setion R will b e assumed to b e a lo al ring. When w e write R -mo dules w e will mean left R -mo dules and b y an ideal (without adjetiv e) w e will understand a t w o-sided ideal. As usual H om R ( P, Q ) denotes the set of all R -mo dule homomorphisms b et w een the R -mo dules P and Q . As w e men tioned in the in tro dution, Ho v ey and Lo  kridge ga v e a omplete  haraterization of the omm utativ e rings for whi h P and P fg admit a triangulation with Σ = Id . Some of their preliminary results apply also in the general ase and w e are going to use them. Reall that a ring is alled Quasi-F r ob enius (short QF) if ev ery injetiv e R -mo dule is pro je- tiv e or equiv alen tly ev ery pro jetiv e R -mo dule is injetiv e. There are man y other  harater- izations of QF rings, f. [Lam2, 15.1℄ and [F, 24.20℄. In [HL, 3.4℄ it is sho wn that if P admits a triangulation, then R is a QF ring. This is an easy onsequene of the fat that ev ery distinguished triangle in P (or also P fg ) is exat, sine the funtor H om R ( R, − ) on v erts a giv en distinguished triangle in to an exat sequene of R -mo dules isomorphi to the original triangle, and for this reason ev ery injetiv e mo dule an b e em b edded in to a pro jetiv e one, th us, is pro jetiv e itself: P f / / Q     / / T / / Σ P M /  ? ? ~ ~ ~ ~ ~ ~ ~ ~ Here, P f / / Q / / / / M / / 0 is a presen tation of the injetiv e mo dule M b y pro jetiv e mo dules. A v ery similar argumen t [HL, 3.5℄ pro vides that if P fg admits a triangulation then R is a left and righ t IF ring. Reall that R is said to b e a left resp etiv ely righ t IF ring if all injetiv e left resp etiv ely righ t R -mo dules are at. A dding the ondition that R is left or righ t no etherian implies that R is ev en a QF ring ([Co℄). No w, as men tioned in the b eginning w e are going to diret our atten tion to lo al rings. The reason for this is that only in this ase w e ha v e suien t information ab out the struture of QF rings so that it is p ossible to extrat enough information for a p ossible triangulation. In the omm utativ e ase an arbitrary QF ring is a pro dut of lo al QF rings, and so, onsidering only the lo al ase is not really a restrition. W e should sp eify what w e mean b y a lo  al ring. R is dened to b e lo al if there is a unique maximal left ideal. This turns out to b e the same as requiring a unique maximal righ t ideal (that indeed oinsides with the maximal left ideal) or also asking for R/ rad R to b e a sk ew eld (see for example [Lam1 ℄). Note, that 4 3 Main results this is in general not the same as ha ving a maximal t w o-sided ideal. F rom no w on R will denote a lo al ring and in the ligh t of this w e in tro due the follo wing additional notation : m the unique maximal ideal of R d := R/ m the residue sk ew eld of R [ r ] ∈ d the residue lass of an elemen t r of R ˜ t ∈ R a lift of an elemen t t of d R × the units in R Z ( R ) the en ter of R Z ( d ) the en ter of d By a theorem of Kaplansky [Ka , 2℄ w e kno w that the pro jetiv e mo dules o v er a lo al ring are exatly the free mo dules, th us, w e are going to w ork in ategories of free resp etiv ely nitely generated free mo dules. 3 Main results Before b eginning with the pro of of the en tral theorem w e ha v e to do some preparations. Throughout, w e are going to dieren tiate b et w een P and P fg only if neessary . First, w e dedue some restritions on the struture of the rings R that migh t o ur, pro vided P or P fg is triangulated. The follo wing prop osition an also b e found in [HL ℄, but for our further w ork it is reasonable to reall the pro of. Prop osition 3.1. If the  ate gory of pr oje tive mo dules P over a lo  al ring R or the  ate gory of nitely gener ate d pr oje tive mo dules P fg over a lo  al left or right no etherian ring R admits a triangulation with tr anslation funtor Σ = Id , then the fol lowing pr op erties hold for the maximal ide al m of R : (i) m 2 = { 0 } (ii) m = R x = xR for al l x ∈ m \ { 0 } Pr o of. Without loss of generalit y m is not the zero ideal. As w e sa w, under the assumptions of the theorem the ring R is QF, and further it is also assumed to b e lo al. Hene b y [Ni, 3.9℄, so c R R = so c R R (write so c R from here on) is a simple left and righ t mo dule. In other w ords, so c R is an ideal that is b oth simple as a left and as a righ t R -mo dule. Clearly , so c R is generated b y one elemen t, and w e ha v e so c R = R x = xR for an y x ∈ so c R \{ 0 } . W e w an t to pro v e that m equals so c R . Let us x some non trivial, th us generating elemen t x of so c R . Beause of the triangulation there is a distinguished triangle starting with the map · x (righ t m ultipliation b y the elemen t x ). This is of the form R · x / / R f / / P g / / R, where P is some free mo dule. Sine the triangle is exat, w e ha v e a short exat sequene of R -mo dules: 0 / / R/R x / / P / / k er ( · x ) / / 0 Note that k er ( · x ) equals the left annihilator ann l ( xR ) = ann l (so c R ) of the so le of R , and that is preisely m (see for exemple [Ni, 8.11℄). W e reall that sine R is left artinian and left no etherian (b eause of QF), all nitely generated left R -mo dules ha v e nite length, so w e onlude that P also has nite length and w e ha v e: l ( P ) = l ( m ) + l ( R/R x ) = l ( m ) + l ( R ) − 1 ( ⋆ ) Sine m is maximal, l ( R/ m ) is simple and so l ( m ) + 1 = l ( R ) . F urther, the length of P is a m ultiple of the length of R . Coming ba k to ( ⋆ ) the ab o v e onsiderations apply: 5 3 Main results k l ( R ) = l ( R ) − 1 + l ( R ) − 1 = 2 l ( R ) − 2 . Th us, k equals 1 and l ( R ) = 2 . Therefore, 0 ⊂ so c R ⊂ R is already a omp osition series for R and so c R equals m . It is no w an easy alulation that m 2 is trivial. R emark 3.2 . W e observ e that under the assumptions of the prop osition a distinguished triangle for the morphism · x for some non trivial elemen t x of so c R = m is giv en b y R · x / / R · s / / R · t / / R , where s and t are elemen ts of R . Sine x is non trivial, · s and · t an not b e isomorphisms. Beause x is not a unit, they an not b e trivial, either. Hene, s and t are elemen ts of m \{ 0 } . As su h w e an represen t them in the form s = xr 1 and t = r 2 x for some appropriate units r 1 and r 2 . Then w e ha v e an isomorphism of triangles R · x / / R · xr 1 / / R · r − 1 1   · r 2 x / / R R · x / / R · x / / R · ¯ r x / / R, where ¯ r = r 1 r 2 is also a unit. Another onsequene of the prop osition is the follo wing Lemma 3.3. Every x ∈ m \{ 0 } denes an automorphism σ x : d → d of the r esidue skew eld d = R / m , with the pr op erty x e t = ] σ x ( t ) x for al l t ∈ d. Pr o of. W e dene σ x via the prop ert y x ˜ t = ] σ x ( t ) x : Let t b e an elemen t of d . Sine Rx = xR , for a lift ˜ t of t w e ha v e an elemen t s of R with the prop ert y sx = x ˜ t . W e set σ x ( t ) = [ s ] ∈ d . This assignmen t is w ell dened: T w o lifts ˜ t and ¯ t dier b y an elemen t m ∈ m , and sine xm = 0 , w e ha v e x ¯ t = x ( ˜ t + m ) = x ˜ t . If ¯ s also satises x ˜ t = ¯ sx , then ¯ sx = sx , i.e., ( ¯ s − s ) x = 0 . Hene, ( ¯ s − s ) is in m , and therefore [ ¯ s ] equals [ s ] . Next, w e ha v e to sho w that σ x is a sk ew eld homomorphism: F or t 1 and t 2 in d , e t 1 + e t 2 is a lift of t 1 + t 2 . If s 1 and s 2 are elemen ts with s i x = x e t i for i = 1 , 2 , then ( s 1 + s 2 ) x = x ( e t 1 + e t 2 ) . No w [ s 1 + s 2 ] = [ s 1 ] + [ s 2 ] implies w e ha v e a homomorphism of groups with resp et to + . Similarly for m ultipliation, e t 1 e t 2 is a lift of t 1 t 2 . W e ha v e x e t 1 e t 2 = s 1 x e t 2 = s 1 s 2 x , and hene σ x ( t 1 t 2 ) = [ s 1 s 2 ] = [ s 1 ][ s 2 ] = σ x ( t 1 ) σ x ( t 2 ) . It is lear that σ x (1) = 1 . Analogously , w e an onstrut σ x − 1 via the prop ert y ˜ tx = x ^ σ − 1 x ( t ) for ev ery t ∈ d . R emark 3.4 . One an ask what the onnetion b et w een t w o su h automorphisms σ x and σ y is. Sine w e an write y = x ˜ r for an appropriate r ∈ d , w e observ e for an y t ∈ d : ] σ y ( t ) y = y ˜ t ⇔ ] σ y ( t ) x ˜ r = x ˜ r ˜ t ⇔ ] σ y ( t ) ] σ x ( r ) x = x ˜ r ˜ t ⇔ σ y ( t ) σ x ( r ) = σ x ( rt ) Th us, σ y ( t ) is equal to σ x ( r ) σ x ( t )( σ x ( r )) − 1 . In ase σ x ( r ) is in the en ter of d , w e ha v e an equalit y b et w een σ x and σ y . In partiular, if d is omm utativ e all non trivial elemen ts of m dene the same automorphism. No w w e on tin ue with the main result: 6 3 Main results Theorem 3.5. L et R b e a lo  al ring that satises pr op erties (i) and (ii) of Pr op osition 3.1 and m its maximal ide al. Then ther e exists a triangulation for P (r esp. P fg ) with Σ = Id if and only if one of the fol lowing  onditions hold: (i) R is a skew eld, i.e., m = { 0 } (ii) m = 2 R (iii) char R = 2 , i.e., 1 + 1 = 0 in R , and for some x ∈ m \ { 0 } ther e is a nontrivial element r x ∈ R / m suh that a) σ x ( r x ) = r x b) σ 3 x ( t ) = r − 1 x tr x for every t ∈ R / m W e ar e going to  al l these thr e e  ases the "semisimple", the "mixe d-har ateristi" and the "e quihar ateristi"  ase, r esp e tively. Pr o of. Considering the unique map Z → R w e are often going to talk of elemen ts of Z as elemen ts of R . What is mean t should b e lear from the on text. In the rst part of the pro of w e sho w that R should satisfy one of the upp er onditions: If m is trivial, it is lear that R is a sk ew eld. F or m 6 = { 0 } there are t w o p ossibilities: m is generated b y some prime n um b er p ∈ Z , or R on tains a eld. Indeed, if ev ery prime n um b er p is a unit in R , then so is ev ery other elemen t of Z , and therefore Q an b e em b edded in R . If p = 0 in R for some prime p , then there is a ringhomomorphism F p → R that is a monomorphism. If none of the ab o v e ases o urs, then there exists a prime p with p an elemen t of m \{ 0 } . Therefore p is a generator of m . Before on tin uing w e w an t to mak e the follo wing note: a distinguished triangle in a triangu- lated ategory is isomorphi to the triangle that is pro dued b y altering t w o of the maps b y a sign. It is in general not true that  hanging only one of the signs pro dues a distinguished triangle, but in our ase sine the translation funtor is the iden tit y , this is also true as a onsequene of the seond axiom for triangulated ategories. Remark 3.2 allo ws us to onsider the follo wing omm utativ e diagram with distinguished triangles as ro ws: R · r x / / R · x / / R · x / / · t   R R · r x / / R · x / / R ·− x / / R , By Axiom 4 there is a ller · t ompleting the ab o v e to a morphism of triangles. This implies that t is a unit in R represen ting the lass of 1 ∈ d , and th us x = − x . In the ase, that a eld F p an b e em b edded in R , w e onlude from 2 x = 0 that 2 equals 0 or in other w ords F p = F 2 . In the ase that the maximal ideal is generated b y a prime p , 2 p = 0 fores p to b e 2 . In b oth of the ases where R 6 = d , w e also ha v e the llers in the diagrams R · x / / R · x / / R · r x / /   R R · x / / R · r x / / R · x / / R R · x / / · ^ σ x ( s )   R · x / / · ˜ s   R · r x / /   R · ^ σ x ( s )   R · x / / R · x / / R · r x / / R , Here, x still denotes some generating elemen t of m , s is some elemen t of d and ˜ s a lift of s in R . Call the ller in the left diagram · t . W e ha v e xt = r x and tx = rx , th us σ x ([ t ]) = [ r ] 7 3 Main results and [ t ] = [ r ] . In other w ords, σ x ([ r ]) equals [ r ] . Denote the righ t ller b y · ˜ u , for u ∈ d . W e ha v e the follo wing onditions: σ x ( u ) = s and [ r ] σ 2 x ( s ) = u [ r ] . T ogether they fore σ 3 x ( u ) = [ r ] − 1 u [ r ] , where u equals σ − 1 x ( s ) . Sine s w as an arbitrary elemen t of d and σ − 1 x is bijetiv e, w e onlude that σ 3 x ( u ) = [ r ] − 1 u [ r ] for ev ery u ∈ d . Hene, w e are done with the rst part of the pro of. A tually w e sho w ed more than w as stated: w e sho w ed that under the assumption that a triangulation exists, in b oth the mixed- harateristi and the equi harateristi ase the prop erties a ) and b ) are satised for an y x ∈ m \{ 0 } together with some appropriate r x . The seond step is to onstrut a triangulation for ea h of the three ases. The ategory of (nite dimensional) v etor spaes o v er a sk ew eld has a unique triangulated struture analogous to v etor spaes o v er a eld, that w e in tro dued at the b eginning. The other t w o ases are more in teresting. W e are going to treat them together. W e are going to use the onstrution from [MSS ℄, but omplete and mo dify it for our purp ose. When p ossible, w e are going to sti k to their notation. First x an x and an r x with the prop erties a ) and b ) . Note that in ase m = (2) , w e an tak e x = 2 and r x = 1 or in general an y other r x that is in the en ter of d . W e also ha v e x = − x in b oth ases. Dene distinguished triangles to b e triangles isomorphi to a diret sum (nite when w orking in P fg ) of on tratible triangles and triangles △ R of the form: △ R : R · x / / R · x / / R · f r x x / / R F or a free R -mo dule M w e will denote b y △ M the triangle △ M : M · x / / M · x / / M · f r x x / / M , that is a diret sum of the triangles △ R ( hoie of a basis is in v olv ed!). W e ha v e to v erify the three axioms. A.1: Distinguished triangles are losed under isomorphisms b y denition and the triangle P id / / P / / 0 / / P is on tratible for an y ob jet P . No w w e need the follo wing lemma: Lemma 3.6. In the ab ove  ontext, for any morphism f : P → Q in the  ate gory P (r esp. P fg ), ther e is a  ommutative diagr am P f / / ∼ =   Q ∼ =   M ⊕ N ⊕ V „ 1 0 0 0 · x 0 0 0 0 « / / M ⊕ N ⊕ W , for appr opriate fr e e mo dules M , N , V and W . F rom this lemma it is lear that f is a part of a distinguished triangle as required, sine there is a triangle starting with f that is isomorphi to the distinguished triangle M ⊕ N ⊕ V „ 1 0 0 0 · x 0 0 0 0 « / / M ⊕ N ⊕ W „ 0 0 0 0 · x 0 0 0 1 « / / V ⊕ N ⊕ W „ 0 0 0 0 · f r x x 0 1 0 0 « / / M ⊕ N ⊕ V . 8 3 Main results Pr o of of L emma 3.6 . W e an use Zorn's lemma to determine the largest free submo dule of P su h that the restrition of f on it is trivial: Let W denote the set of all free submo dules with this prop ert y . As usual, a partial ordering is giv en b y the inlusion of submo dules and the empt y set is an elemen t of W . W e ha v e to sho w that for ev ery totally ordered subset { W i } i ∈ I ′ , the union S i ∈ I ′ W i is also in W , i.e. is a free R -mo dule. Though a ltered olimit of free mo dules o v er an arbitrary ring need in general not b e a free mo dule, w e are in the sp eial situation where ev ery inlusion of free mo dules splits (free = pro jetiv e = injetiv e), and the omplemen tary mo dule is as a diret summand of a free mo dule a pro jetiv e = free mo dule itself. F or a preise argumen t one needs transnite indution sine w e ha v e a p ossibly unoun table union. One sho ws that there is a ompatible system of bases of the mo dules W i . The suessor ase is exatly the fat disussed ab o v e. A basis of W i an b e ompleted to a basis of W i +1 sine there is a free omplemen t. F or the limit ase, assumed there are ompatible bases for all the mo dules W i for i < α for a giv en limit ordinal α , then a basis for S i<α W i is giv en b y the union of the bases for W i , where elemen ts of dieren t bases are b eing iden tied via the inlusions. No w Zorn's lemma giv es us a maximal elemen t, sa y V , whi h again has a free omplemen t in P . Use the same metho d to split this omplemen t in to t w o mo dules M and N , su h that M is maximal free with the prop ert y that f is injetiv e on it. Using the maximalit y of M and V one easily alulates that f restrited on N is (after an appropriate  hoie of a basis) just m ultipliation b y the elemen t x . The statemen t follo ws. No w ba k to the seond axiom: A.2: A triangle is on tratible i its translate is. The translate of △ R is R · x / / R · f r x x / / R · x / / R , and it is isomorphi to the original one b eause of σ x ( r x ) = r x : R · x / / R · x / / R · f r x x / / · f r x   R R · x / / R · f r x x / / R · x / / R On the other hand, △ R is the translate of the triangle R · f r x x / / R · x / / R · x / / R , whi h is also isomorphi to △ R b y the same argumen t. A.3: W e start b y sho wing that an y diagram of distinguished triangles of the form A f / / α   B i / / β   C q / / A α   A ′ f ′ / / B ′ i ′ / / C ′ q ′ / / A ′ an b e ompleted to a morphism of triangles. Assume rst that one of the ro ws is on- tratible. If this is the upp er ro w, w e an op y this part in the pro of of Theorem 1 of [ MSS℄ without an y  hange. Ho w ev er, there is no p ossibilit y of using the dualit y funtor men tioned there for the ase where the lo w er triangle is on tratible sine w e are also w orking in P , so 9 3 Main results w e giv e an expliit onstrution: Let (Θ , Φ , Ψ) denote a n ullhomotop y for the lo w er triangle: A f / / α   B i / / β   C q / / A α   A ′ 1   0   f ′ / / B ′ 1   0   Θ } } } ~ ~ } } } i ′ / / C ′ 1   0   Φ | | | ~ ~ | | | q ′ / / A ′ 1   0   Ψ } } } ~ ~ } } } A ′ f ′ / / B ′ i ′ / / C ′ q ′ / / A ′ Sine the ro ws are exat w e an fatorize i ′ β through the ok ernel of f (in the ategory of R -mo dules), and then use that C ′ is injetiv e, to onstrut a map γ ′ su h that γ ′ i = i ′ β : B i / / β       ? ? ? ? C γ ′   ok er f /  ?       ? ? ? ? B ′ i ′ / / C ′ No w set γ = γ ′ + Ψ( αq − q ′ γ ′ ) . W e ha v e γ i = γ ′ i + Ψ( αq i − q ′ γ ′ i ) = i ′ β q ′ γ = q ′ γ ′ + q ′ Ψ( αq − q ′ γ ′ ) = q ′ γ ′ + (1 − Θ f ′ )( αq − q ′ γ ′ ) = = q ′ γ ′ + αq − q ′ γ ′ − Θ f ′ αq + Θ f ′ q ′ γ ′ = αq Th us, γ is the desired map. If b oth triangles are isomorphi to diret sums △ R , then, using the isomorphisms, w e are sear hing a ller in P · x / / f   P · x / / g   P · f r x x / /   P f   Q · x / / Q · x / / Q · f r x x / / Q , for appropriate f and g . W e an think of these maps as matries with en tries g ij and f ij resp etiv ely , in R . Beause of the omm utativit y of the rst square w e ha v e the relation f ij x = xg ij . W e dene the desired ller b y setting h ij = ^ σ − 1 x ([ g ij ]) for some lift ( ho ose one) of σ − 1 x ([ g ij ]) . This denition mak es the middle square omm ute, and w e only need to  he k that the righ t square omm utes. Using the relation b et w een g ij and f ij as w ell as the prop erties of x and r x w e alulate e r x xf ij = e r x ^ σ x ([ f ij ]) x = e r x ^ σ 2 x ([ g ij ]) x = e r x ^ σ 3 x ([ h ij ]) x = g [ h ij ] e r x x = h ij e r x x. W e w an t to mak e a few ommen ts: • Note that w e an alter the ller b y adding µ ( · x ) + ( · x ) ν for an y µ, ν : P → Q . • It is also w orth remarking that if f is an isomorphism, so are g and h : Indeed, assumed f is an isomorphism, applying Lemma 3.6 on g and using the equalit y g ( · x ) = ( · x ) f one easily sees that g is an isomorphism. The same argumen t pro vides that h is also one. 10 3 Main results • Giv en the map f , one an alw a ys dene g and h . Th us, in the ase where f is an isomorphism, w e an omplete it to an isomorphism of the distinguished triangles (of ourse this applies only in this sp eial ase where the triangles are isomorphi to diret sums of △ R ). No w w e are ready to pro eed and sho w that the ompletion an b e done su h that the mapping one is distinguished. W e ha v e the follo wing situation: φ =  φ 11 φ 12 φ 21 φ 22  : X ⊕ T → Y ⊕ U , where X ⊕ T and Y ⊕ U are distinguished triangles with T and U on tratible triangles, X and Y isomorphi to diret sums of △ R , and φ = ( α, β , γ ) and ea h of the φ ij = ( α ij , β ij , γ ij ) morphisms of triangles. W e should mo dify γ in an appropriate w a y . Observ e that φ 12 , φ 21 and φ 22 are n ullhomotopi sine in ea h ase the soure or the target is on tratible. Putting the three homotopies together giv es a homotop y from  φ 11 φ 12 φ 21 φ 22  to  φ 11 0 0 0  , whi h means that the orresp onding mapping ones are isomorphi. On the other hand, the one of  φ 11 0 0 0  is itself isomorphi to the diret sum of the mapping ones of φ 11 , U and the translate of T . Sine the last t w o are on tratible, our problem redues to the ase φ 11 : X → Y with X and Y as ab o v e. F or suitable free mo dules M , N , V and W w e an represen t the map α 11 as in Lemma 3.6 : X 1 α 11 / / ∼ =   Y 1 ∼ =   M ⊕ N ⊕ V „ 1 0 0 0 · x 0 0 0 0 « / / M ⊕ N ⊕ W As w e observ ed, the v ertial isomorphisms an b e extended to isomorphisms of exat trian- gles. Th us, w e an restrit our in v estigation to the ase: M ⊕ N ⊕ V · x / / „ 1 0 0 0 · x 0 0 0 0 «   M ⊕ N ⊕ V · x / / ¯ β 11   M ⊕ N ⊕ V · f r x x / / ¯ γ 11   M ⊕ N ⊕ V „ 1 0 0 0 · x 0 0 0 0 «   M ⊕ N ⊕ W · x / / M ⊕ N ⊕ W · x / / M ⊕ N ⊕ W · f r x x / / M ⊕ N ⊕ W Let ¯ α 11 and δ denote the maps from M ⊕ N ⊕ V to M ⊕ N ⊕ W giv en b y ¯ α 11 =  1 0 0 0 · x 0 0 0 0  and δ =  0 0 0 0 1 0 0 0 0  . The morphism ¯ β 11 is of the form ¯ α 11 + Φ( · x ) for some Φ : M ⊕ N ⊕ V → M ⊕ N ⊕ W (f. the relation b et w een f and g in the onstrution of a ller for the ase where the ro ws where diret sums of △ R ). No w, set ¯ γ 11 to b e ¯ β 11 + Φ( · x ) + ( · x ) δ + ( · x )Φ = ¯ α 11 + ( · x ) δ + ( · x )Φ . W e laim that ( δ, Φ , 0 ) is a homotop y from ( ¯ α 11 , ¯ β 11 , ¯ γ 11 ) to the morphism ζ = ( ǫ, ǫ, ǫ ) , where ǫ =  1 0 0 0 0 0 0 0 0  . The three equations for a homotop y hold: ¯ β 11 − ǫ = Φ( · x ) + ( · x ) δ ¯ γ 11 − ǫ = ¯ α 11 + ( · x ) δ + ( · x )Φ − ǫ = ( · x )Φ = 0( · e r x x ) + ( · x )Φ ¯ α 11 − ǫ = δ ( · x ) = δ ( · x ) + ( · e r x x )0 It only remains to observ e that the mapping one of ζ is isomorphi to the diret sum of △ N , △ W , the translate of △ N , the translate of △ V with the mapping one of the iden tit y on △ M , whi h an b e easily  he k ed to b e on tratible. 11 3 Main results F rom the t w o theorems in this setion w e onlude that P is triangulated with translation funtor the iden tit y if and only if R satises prop erties ( i ) and ( ii ) of Prop osition 3.1 as w ell as one of the onditions ( i ) , ( ii ) or ( iii ) of the ab o v e theorem. The analogous statemen t holds for P fg pro vided R is left or righ t no etherian. In the pro of of the theorem w e ha v e giv en an expliit triangulation for the studied at- egories. A natural question is whether there are dieren t triangulated strutures. In the semisimple ase the triangulation is alw a ys unique. This is due to the fat that there ev ery morphism in the ategory P (resp. P fg ) is up to isomorphism of the form U 1 ⊕ U 2 ( 0 1 0 0 ) / / U 2 ⊕ V 1 , and this is part of the triangle U 1 ⊕ U 2 ( 0 1 0 0 ) / / U 2 ⊕ V 1 ( 0 0 0 1 ) / / U 1 ⊕ V 1 ( 0 0 1 0 ) / / U 2 ⊕ U 1 , whi h is on tratible. Th us, this triangle is distinguished in an arbitrary triangulation. Sine ev ery t w o distinguished triangles starting with the same morphism are (non anonially) isomorphi, and the lass of distinguished triangles is losed under isomorphisms, w e see that the triangulation is unique. Let us no w onsider the other t w o ases. W e will denote the triangulation indued b y the generating triangle R · x / / R · x / / R · f r x x / / R b y a r x . In Theorem 3.5 ( iii ) , in order to ha v e a triangulation, w e required the existene of an elemen t r x of d with σ x ( r x ) = r x and σ 3 x ( t ) = r − 1 x tr x for some generator x of m . On the other hand, as w e sa w in the rst part of the pro of of the theorem, in b oth the mixed- harateristi and the equi harateristi ase, a giv en triangulation denes for ev ery generator x ∈ m \ { 0 } an elemen t r x of d that satises the ab o v e onditions. Clearly , the triangulation indued b y x and r x oinides with the original one. All these fats imply that to ompare triangulations it is enough to x one partiular generator of the maximal ideal and observ e those elemen ts of d \{ 0 } , that together with this generator satisfy onditions a ) and b ) of Theorem 3.5 . In the mixed  harateristi ase it is reasonable to tak e x = 2 . W e ha v e a map  triangulations of P ( resp. P fg )  → Z ( d ) i r x 7→ r x where Z ( d ) denotes the en ter of the residue eld d . Indeed, as w e men tioned in the pro of of the theorem, ev ery elemen t r x ∈ d × that denes a triangulation together with x = 2 , should b e in the en ter of d . In the equi harateristi ase w e an also onstrut su h a map. F or this w e x a triangulation a r x and dene  triangulations of P ( resp. P fg )  → Z ( d ) σ x . i r ′ x 7→ r − 1 x r ′ x Here Z ( d ) σ x is the eld of xed p oin ts of Z ( d ) under σ x . There is to  he k that r − 1 x r ′ x lies in Z ( d ) σ x . W e use again onditions (a) and (b) of Theorem 3.5 : σ x ( r − 1 x r ′ x ) = σ x ( r − 1 x ) σ x ( r ′ x ) = r − 1 x r ′ x , t = r x σ 3 x ( t ) r − 1 x = r ′ x σ 3 x ( t )( r ′ x ) − 1 for an y t ∈ d 12 3 Main results The seond equation an b e written as r − 1 x r ′ x σ 3 x ( t ) = σ 3 x ( t ) r − 1 x r ′ x whi h together with the fat that σ x is an automorphism implies that r − 1 x r ′ x is a en tral elemen t of d . W e are no w ready to form ulate the results ab out dieren t triangulations in the mixed-  harateristi and the equi harateristi ase: Corollary 3.7. Pr ovide d P ( r esp. P fg )  an b e triangulate d, the ab ove maps indu e bije tions (i) mixe d-har ateristi  ase:  triangulations of P ( P fg )  1:1 ↔ { nontrivial elements of Z ( d ) } (ii) e quihar ateristi  ase:  triangulations of P ( P fg )  1:1 ↔ { nontrivial elements of Z ( d ) σ x } Pr o of. In the mixed- harateristi ase w e ha v e tak en x = 2 as usual. Th us σ 3 2 is the iden tit y and an elemen t r 2 of d \{ 0 } denes a triangulation via the generating triangle R · 2 / / R · 2 / / R · 2 e r 2 / / R if and only if r 2 is in Z ( d ) \{ 0 } . On the other hand, t w o dieren t elemen ts r 2 and r ′ 2 of Z ( d ) \{ 0 } dene dieren t triangula- tions. Else there should exist the ller in the diagram R · 2 / / R · 2 / / R · 2 e r 2 / /   R R · 2 / / R · 2 / / R · 2 e r ′ 2 / / R, whi h w ould fore r 2 = r ′ 2 . In the equi harateristi ase in order to ha v e more than one triangulation w e again ne- essarily need at least t w o dieren t elemen ts r x , r ′ x of d \{ 0 } that satisfy the onditions a ) and b ) with resp et to the xed generator x of m . The same argumen t as ab o v e sho ws that the indued triangulations are dieren t. Hene w e only ha v e to sho w that the giv en map is surjetiv e. F or this it is enough to see that for ev ery elemen t z of Z ( d ) σ x , r x z denes a triangulation a r x z . Indeed, r x z is a xed p oin t of σ x sine r x and z are, and moreo v er σ 3 x ( t ) = r − 1 x tr x = z − 1 r − 1 x tr x z = ( r x z ) − 1 t ( r x z ) holds b eause z is in the en ter of d . W e kno w no w whether there are dieren t triangulations or not, but the question remains, if, whenev er they exist, these dieren t triangulations are equiv alen t or ev en isomorphi? That is, is there an equiv alene or ev en an isomorphism F of the ategory P (resp. P fg ), that together with a natural transformation η : F Σ → Σ F maps distinguished triangles of one triangulation to distinguished triangles of another triangulation. T o answ er these ques- tions in full generalit y w e p ossibly need a full lassiation of the o urring rings. W e will restrit our atten tion to determine isomorphi triangulations in a v ery strong sense: where the funtor F is the iden tit y . 13 3 Main results Let us rst mak e some observ ations: The en ter of R , Z ( R ) is a subring of R that is itself lo al. Denote b y d ′ the residue eld of Z ( R ) , then learly the follo wing diagram omm utes: Z ( R )       / R     d ′   / Z ( d ) σ x   / Z ( d )   / d W e note that d/ Z ( d ) is a sk ew eld extension, and Z ( d ) /d ′ and Z ( d ) σ x /d ′ are eld extensions. Using the result of the previous orollary , w e w an t to desrib e the isomorphi lasses of triangulations in the sense disussed ab o v e. W e laim that the iden tiation is realized b y the pro jetion maps Z ( d ) × → Z ( d ) × /d ′× ( Z ( d ) σ x ) × → ( Z ( d ) σ x ) × /d ′× In other w ords: Corollary 3.8. F or the e quivalen e lasses of isomorphi triangulations with F = Id as disusse d ab ove, ther e ar e the fol lowing bije tions: (i) mixe d-har ateristi  ase:  e quivalen e lasses of triangulations of P ( P fg )  1:1 ↔  elements of Z ( d ) × /d ′×   a r x  7→ [ r x ] (ii) e quihar ateristi  ase:  e quivalen e lasses of triangulations of P ( P fg )  1:1 ↔  elements of ( Z ( d ) σ x ) × /d ′×  h a r ′ x i 7→ [ r − 1 x r ′ x ] wher e a r x is a xe d triangulation. Pr o of. A t the b eginning w e treat ( i ) and ( ii ) together.W e laim that t w o non trivial elemen ts r x and r ′ x of d dene, together with the generator x of m , equiv alen t triangulations if and only if there is a lift of r − 1 x r ′ x in R that is in Z ( R ) . Assume rst r ∈ Z ( R ) is a lift of r − 1 x r ′ x . T o dene a natural transformation η  ho ose for ev ery P ∈ P an isomorphism f P : P → L I P R for an appropriate set I P . W e insist that in ase P = L I P R , f P is the iden tit y . Set η P : P → P to b e the omp osite P f P / / L I P R · r / / L I P R f − 1 P / / P. Note that η P is an isomorphism. Sine r is an elemen t of the en ter of R , this giv es us a natural transformation η : Id ◦ Σ = Id → Id = Σ ◦ Id . The triangle R · x / / R · x / / R · f r x x / / R is mapp ed under (Id , η ) to the generating triangle R · x / / R · x / / R · f r ′ x x / / R . 14 4 Examples F or on tratible triangles w e ha v e: if (Θ , Φ , Ψ) is a n ullhomotop y for P u / / Q v / / T w / / P , then (Θ , Φ , Ψ η − 1 P ) is one for P u / / Q v / / T η P w / / P . Clearly , diret sums of on tratible triangles with triangles isomorphi to sums of generat- ing triangles are also mapp ed to distinguished triangles under (Id , η ) . Therefore (Id , η ) : ( P , a r x ) → ( P , a r ′ x ) is an isomorphism of triangulated ategories. Con v ersely , if a natural transformation η : Id → Id is giv en, it denes an elemen t r in Z ( R ) via the map η R = · r . If (Id , η ) : ( P , a r x ) → ( P , a r ′ x ) is an isomorphism of triangulated ategories, then the generating triangle of a r x should b e mapp ed to the generating triangle of a r ′ x . Hene, w e see that r x [ r ] equals r ′ x . This fores [ r ] = r − 1 x r ′ x and so r is a lift of r − 1 x r ′ x in the en ter of R . T ranslating what w e sho w ed in to other language: in the ligh t of Corollary 3.7 , the elemen ts of Z ( d ) σ x that are in the same equiv alene lass as r x are preisely those of the form r x z for z ∈ ( d ′ ) × . This yields the result for the equi harateristi ase. T aking x = 2 and using σ 2 = id giv es the one for the mixed- harateristi ase. A diret result from this orollary is that if the ring R is omm utativ e, then all the trian- gulations are isomorphi. 4 Examples In this setion w e w an t to giv e examples for the rings o urring in Theorem 3.5 . In partiular, w e are in terested in the existene of non omm utativ e ones. F or the semisimple ase there are learly plen t y of examples - for ev ery sk ew eld the ategories of v etor spaes and nitely generated v etor spaes are triangulated. W e therefore start with the mixed- harateristi ase. Reall the ring of Witt v etors W ( k ) o v er a p erfet eld k of  harateristi p . It is a omm utativ e disrete v aluation ring with maximal ideal ˆ m = p W ( k ) and residue eld k . The ring of Witt v etors of length 2 then is dened as W 2 ( k ) := W ( k ) / ˆ m 2 , a lo al ring with only one non trivial ideal, namely ¯ m = p W 2 ( k ) . Another desription of W 2 ( k ) is in terms of v etors ( a 0 , a 1 ) ∈ k × k where the addition and the m ultipliation are giv en b y: ( a 0 , a 1 ) + ( b 0 , b 1 ) = ( a 0 + b 0 , a 1 + b 1 + 1 p ( a p 0 + b p 0 − ( a 0 + b 0 ) p ) ( a 0 , a 1 ) · ( b 0 , b 1 ) = ( a 0 · b 0 , b 0 p a 1 + b 1 a 0 p ) F or example, for p = 2 this redues to ( a 0 , a 1 ) + ( b 0 , b 1 ) = ( a 0 + b 0 , a 1 + b 1 + a 0 b 0 ) ( a 0 , a 1 ) · ( b 0 , b 1 ) = ( a 0 · b 0 , b 0 2 a 1 + b 1 a 0 2 ) . Comp onen t wise appliation of the F rob enius automorphism F r : k → k a 7→ a p 15 4 Examples indues a ring homomorphism F : W 2 ( k ) → W 2 ( k ) ( a 0 , a 1 ) 7→ ( a 0 p , a 1 p ) that is indeed an isomorphism sine k is p erfet. If w e no w tak e k to b e of  harateristi 2 , then w e immediately obtain a omm utativ e example for (ii). F or a non omm utativ e one w e tak e k of  harateristi 2 , but dieren t from F 2 , and set R = W 2 ( k )(( X, F )) , the sk ew Lauren t series ring asso iated to W 2 ( k ) and F . Elemen ts in R are of the form a = P ∞ i = − n a i X i , where n ∈ N dep ends on a and the a i 's are elemen ts of W 2 ( k ) . A ddition is done omp onen t wise, m ultipliation is giv en b y the form ula ∞ X i = − n a i X i · ∞ X i = − m b i X i = ∞ X k = − n − m  X i + j = k a i F i ( b j )  X k . In other w ords, w e tak e the usual m ultipliation in the Lauren t series ring exept for the fat that elemen ts of W 2 ( k ) do not omm ute with X , but are m ultiplied b y the rule X a i = F ( a i ) X . One easily sees that this mak es R in to a unital non omm utativ e ring. W e laim that R is lo al with maximal ideal generated b y 2 : F or this purp ose w e sho w that a = P ∞ i = − n a i X i ∈ R is a unit if and only if one of the a i 's is a unit in W 2 ( k ) . One diretion is lear: If all o eien ts are non units, i.e., are in the maximal ideal ¯ m of W 2 ( k ) , w e ha v e ∞ X i = − n a i X i · ∞ X i = − n a i X i = ∞ X k = − 2 n  X i + j = k a i F i ( a j )  X k = 0 , sine F maps the maximal ideal ¯ m to itself, and therefore for ev ery k ∈ N holds P i + j = k a i F i ( a j ) ∈ ¯ m 2 = { 0 } . Hene, a annot b e a unit. No w supp ose a − n is a unit. W e an indutiv ely dene the o eien ts b j of b = P ∞ j = n b j X j in a w a y su h that b is a righ t in v erse for a , and in the same manner one an nd a left in v erse for a . W e on- lude that a is a unit. If j > − n is minimal with a j ∈ W 2 ( k ) × , set c = P j − 1 i = − n a i X i and d = P i ≥ j a i X i . So a equals c + d with c 2 = 0 and d a unit. Note that ( uc ) 2 equals 0 for an y u ∈ W 2 ( k )(( X, F )) sine the o eien ts of uc are still in ¯ m . W e ha v e: a ( d − 1 c − 1 ) = ( c + d )( d − 1 c − 1 ) = = d ( d − 1 c + 1 )( d − 1 c − 1) = = d (( d − 1 c ) 2 − d − 1 c + d − 1 c − 1 ) = = − d Therefore, a is righ t in v ertible. Similarly , one sho ws that it is also left in v ertible, and so the pro of of the laim is done. No w ¯ m = 2 W 2 ( k ) implies, that the non units of R form a righ t ideal, namely m = 2 R whi h is of ourse also a left ideal sine 2 is en tral, so w e onlude that R is lo al. Finally , w e note that w e already sa w that m 2 is trivial. It is also lear that there are no other non trivial left or righ t ideals sine this w as the ase b y W 2 ( k ) . Therefore, m = xR = Rx for an y elemen t x of m \{ 0 } . In the equi harateristi ase w e use a onstrution similar to the sk ew Lauren t series: the sk ew p olynomials. F or a ring R ′ and a ring homomorphism τ : R ′ → R ′ , the asso iated 16 4 Examples sk ew p olynomials R ′ [ X , τ ] dier from the usual p olynomials R ′ [ X ] in the denition of the m ultipliation: It is giv en, as in the ase of the sk ew Lauren t series, b y n X i =0 a i X i · m X i =0 b i X i = n + m X k =0  X i + j = k a i τ i ( b j )  X k . T ak e R ′ to b e the eld F 8 and τ again the F rob enius automorphism. Then w e ha v e τ 3 = id . No w set R = F 8 [ X , τ ] / ( X 2 ) , where ( X 2 ) denotes the ideal generated b y X 2 . Similar to the previous example one sees that preisely the elemen ts of the form a 0 + a 1 X mo d ( X 2 ) , with a 0 6 = 0 , are the units of R . Therefore, there is a unique ideal m in R , that is generated b y the lass of X : m = R [ X ] = ([ X ]) = [ X ] R . Using the notation from Theorem 3.5 , for x = [ X ] w e ha v e σ x = τ : F 8 → F 8 . F urther, σ 3 x is the iden tit y map, and one an tak e r x = 1 , so all the required onditions are satised. Sine in this partiular example 0 and 1 are the only xed p oin ts of σ x , w e ha v e uniqueness of the triangulation. If w e ex hange F 8 b y F 2 6 and τ b y τ 2 w e obtain another example, this time though, the xed p oin ts of σ x = τ 2 are exatly the elemen ts of F 4 understo o d as elemen ts of F 2 6 via the anonial inlusion. Therefore, w e an tak e r x to b e an y elemen t of F × 4 . Ea h of them denes a dieren t triangulation but all these triangulations are isomorphi b y Corollary 3.8 , sine ev ery su h r x has a lift in the en ter of R giv en b y the lass [ r x + 0 · X ] . Lo oking ba k one an ask whether there are dieren t triangulations in the rst example. The ondition on an elemen t r ∈ d \{ 0 } for dening a triangulation is r to b e in the en ter of d = W 2 ( k )(( X, F )) / 2 W 2 ( k )(( X, F )) ∼ = k (( X, F r )) . Using the ab o v e isomorphism w e view elemen ts of d as elemen ts in k (( X, F r )) . One an alulate that in the ase F r i 6 = id for ev ery i ∈ Z \{ 0 } the en ter onsists only of the elemen ts 0 and 1 . If there exists q ∈ N with the prop ert y F r q = id , then the elemen ts in the en ter of d are of the form P ∞ i = − n a i X iq with n ∈ N , a i ∈ { 0 , 1 } and q is minimal with the required prop ert y . Corollary 3.7 implies w e ha v e dieren t triangulations in this seond ase, but for ev ery P ∞ i = − n a i X iq in Z ( d ) × w e an nd a lift P ∞ i = − n ¯ a i X iq in the en ter of R . Here, for a ∈ k = W 2 ( k ) / 2 W 2 ( k ) w e tak e ¯ a = ( a, 0 ) . Th us, also here all the triangulations are isomorphi. Rema rks In this artile w e ha v e treated a sp eial kind of rings, namely lo al ones. In the omm utativ e ase this is not a ma jor restrition sine omm utativ e QF rings are pro duts of lo al QF rings, and so, using the fat that for R ∼ = R ′ × R ′′ holds Mo d R ∼ = Mo d R ′ × Mo d R ′′ , one an  haraterize all the rings R for whi h the ategory of pro jetiv e mo dules admits a triangu- lation. The situation of non omm utativ e rings is dieren t. Surely w e an also generalize our result for pro duts of lo al rings. Using the Morita equiv alene b et w een a ring R and the full matrix ring M n ( R ) for n ∈ N one an also generalize the result for a pro dut of matrix rings o v er lo al rings. Ho w ev er, this still do esn't o v er all the p ossibilities of ho w a QF ring ould lo ok lik e in the general ase, so the question if a omplete  haraterization an b e done, remains op en. A t the end w e w an t to remark that our mixed- harateristi ase pro vides us with some addi- tional examples of the so alled exoti triangulated ategories in tro dued b y Muro, S h w ede and Stri kland in [MSS ℄. These are ategories that are neither algebrai nor top ologial 17 Referenes triangulated (see [Ke06 ℄ and [S h℄). They are  haraterized b y the fat that ev ery ob jet in the ategory is, as named in [MSS ℄, exoti . This means that for ev ery ob jet M in the ategory , there is an exat triangle of the form M · 2 / / M · 2 / / M / / Σ M . Hene, it is ob vious that in the mixed- harateristi ase w e are dealing with su h exoti triangulated ategories. A tually , the examples onsidered in [MSS ℄ are exatly the ate- gories of nitely generated pro jetiv e mo dules o v er omm utativ e rings R , that satisfy the onditions for the mixed- harateristi ase. Referenes [Coh℄ I. S. Cohen, On the struture and ideal theory of omplete lo al rings, T r ans. A m. Math. So . 59 (1946), 54-106. [Co℄ R. R. Colb y , Rings whi h ha v e at injetiv e Mo dules, J. A lgebr a 35 (1975), 239-252. [F℄ C. F aith, A lgebr a 2, R ing The ory , Springer V erlag, 1976 [F2℄ C. F aith, R ings and Things and a Fine A rr ay of Twentieth Century Asso ia- tive A lgebr a , Seond Edition, Amerian Mathematial So iet y , 2004 [Hop℄ C. Hopkins, Rings with minimal ondition for left ideals, A nn. of Math. 40 (1939), 712-730. [HL℄ M. Ho v ey , K. Lo  kridge, Semisimple Ring Sp etra, New Y ork J. Math. , 15 (2009), 219-243. [Ka℄ I. Kaplansky , Pro jetiv e Mo dules, A nn. of Math. 68 (1958), 372-377. [Ke94℄ B. Keller, Deriving DG ategories, A nn. Si. E ole Norm. Sup.(4) 27 (1994), no.1, 63-102. [Ke06℄ B. Keller, On dieren tial graded ategories, International Congr ess of Math- ematiians. V ol. II, 151-190, Eur. Math. So ., Züri h, 2006. [Kr℄ H. Krause, Deriv ed ategories, resolutions, and Bro wn represen tabilit y , Inter- ations b etwe en homotopy the ory and algebr a, 101-139, Con temp. Math., 436 Amer. Math. So ., Pro videne, RI, 2007. [Lam1℄ T. Y. Lam, A First Course in Non ommutative R ings , Springer V erlag, 1991 [Lam2℄ T. Y. Lam, L e tur es on Mo dules and R ings , Springer V erlag, 1999 [Lo℄ F. Lorenz, Einführung in die A lgebr a 2 , 2. Auage, Sp ektrum, 1997 [MSS℄ F. Muro, S. S h w ede, N. Stri kland, T riangulated ategories without mo dels, Invent. math 139 (2007), no.2, 231-241. [Nee℄ A. Neeman, T riangulate d Cate gories , A nn. Math. Stud. v ol. 148 Prineton Univ ersit y Press, Prineton, NJ (2001) 18 Referenes [Nee2℄ A. Neeman, Some new axioms for triangulated ategories, J. A lgebr a 139 (1991), 221-255. [Ni℄ W. K. Ni holson, M. F. Y ousif, Quasi-F r ob enius R ings , Cam bridge Univ ersit y Press, 2003 [S h℄ S. S h w ede, A lgebr ai versus top olo gi al triangulate d  ate gories , [Se℄ J.-P . Serre, L o  al Fields , Springer V erlag, 1979 19