K-Theory of a Local Ring of Finite Cohen-Macaulay Type

K ′ -THEOR Y OF A LOCAL RING OF FINITE COHEN-MACA ULA Y TYPE VIRAJ NA VKAL Abstract. W e study the K ′ -theory of a CM Henselian lo cal ring R of finite Cohen-Macaula y t yp e. W e first describ e a l ong exact sequence inv olving the groups K ′ i ( R ) and the K - groups of certain other rings, i ncluding the Auslan- der algebra. By examining the terms and maps i n the s eque nce, we obtain information about K ′ ( R ). 1. Introduction This pap er gener alizes the following result of Ausla nder and Reiten; s ee [ AR86 , § 2, Pr op. 2.2] for the o riginal sta temen t o r [ Y os90 , 1 3 .7] for the statement in this form. Theorem 1 .1. L et R b e a Henselian CM lo c al ring of finite re pr esentation typ e. Denote by H the fr e e ab elian gr oup on t he set of isomorphism classes of inde c om- p osable maximal Cohe n -Mac aulay R -mo dules. Then t he map H − → K ′ 0 ( R ) sending [ M ] to [ M ] is surje ctive, and its kernel is the su b gr oup h [ N ] − [ E ] + [ M ] | ∃ a n Ausla nder-Reiten sequence 0 − → N − → E − → M − → 0 i . Thu s if one knows the Auslander-Reiten sequences in the ca tegory of maxi- mal Cohen-Macaulay R -mo dules, the ab ove theore m allows explicit co mputation of K ′ 0 ( R ). Using Auslander and Reiten’s techniques, we are a ble to pro duce a long exact sequence inv olv ing the groups K ′ i ( R ) (see ( 3.18 )). Theorem 1.2. L et R b e a Hen selia n CM lo c al ring of finite Cohen-Mac aulay t yp e. L et I b e the set of isomorphism classes of inde c omp osable maximal Cohen-Mac aulay R -mo dules, and let I 0 = I \ { [ R ] } . Then ther e is a long exact se quen c e (1.3) · · · → M [ M ] ∈ I 0 K i ( κ M ) → K ′ i (Λ) → K ′ i ( R ) → M [ M ] ∈ I 0 K i − 1 ( κ M ) → · · · wher e Λ = End R ( L [ M ] ∈ I M ) op and κ M = (End R M ) op / rad((End R M ) op ) . The long exact se quenc e ends in a pr esentation L [ M ] ∈ I 0 K i ( κ M ) / / K ′ 0 (Λ) / / K ′ 0 ( R ) / / 0 Z I 0 Z I of K ′ 0 ( R ) ; it is exactly the one describ e d in The or em 1.1 . This w ork was partially supported by the Na tional Science F oundation Award DMS-0966 821. 1 2 VIRAJ NA VKAL In the case when R = S/ ( w ) is a h yp ersurfa ce s ingularity o f finite CM type, we derive a similar long exact se quence inv olving the algebraic K -gr o ups of the category MF o f matr ix factorizations with p oten tial w ; see Remark 3.21 . In section 4 , we study the maps α i : L K i ( κ M ) → K ′ i (Λ) appea ring in ( 1.3 ). Not surprisingly , we find a re la tionship b etw een these higher maps and the matrix T defining the map α 0 : Z I 0 = L K 0 ( κ M ) → K ′ 0 (Λ) = Z I . This ma trix T was fir st studied in [ AR86 ]. In Section 5 , we a pply the results of the pr e vious s e ctions to understand K ′ 1 ( R ) and K 1 ( MF ) explicitly when R is a one-dimensio nal sing ula rit y o f type A 2 n . F or this computation, we need to know mor e a bout the term K ′ 1 (Λ) in the sequence; to this end we s tudy K 1 of a Krull- Schmidt additive ca tegory . O ur work in this direction app ears in App endix A . I w ould like to thank my adv isor Christian Haes e meyer for his g uidance a nd suppo rt; without him this work would not hav e b een p ossible. I am grateful to the r eferee for c a tc hing a serious er ror in the origina l version o f this pa per. Also I thank Daniel Gr a ys on for his assistance . 2. Preliminaries All categories in this pap er happen to b e a dditiv e, and a ll functors betw een them are assumed to b e additive. All mo dules ov e r a ring are left mo dules. All sp ectra are viewed as ob jects in the stable ho motop y catego ry Ho ( Sp ), and maps b et ween sp ectra ar e viewed as morphisms in this categor y . This is useful beca use Ho( Sp ) is a triangulated category , and t he stable ho motop y functor Ho( Sp ) → (ab elian groups) is a homolo gical functor. If A is a n additive catego ry and M 1 , . . . , M n ∈ A , add ( M 1 , . . . , M n ) will de- note the full subca tegory o f A consisting of direct summands of direct sums of M 1 , . . . , M n ; in particular add ( M 1 ⊕ · · · ⊕ M n ) = add ( M 1 , . . . , M n ). rad( A ) will denote the Jacobson r adical o f A ; this is a t wo-sided ideal of A and ca n b e defined by Hom rad( A ) ( X, Y ) = { f ∈ Hom A ( X, Y ) | f g ∈ rad(End A Y ) for all g ∈ Hom A ( Y , X ) } . Given a collection of a dditiv e categories {A i } i ∈ I , we define their direct sum L A i to b e the additive category whos e ob jects are I - tuples X = ( X i ) i ∈ I with X i in A i for all i and X i 6 = 0 for only finitely many i . Mor phisms are defined by Hom L A i ( X, Y ) = L Hom A i ( X i , Y i ). Definition 2.1. A nonzero o b ject A of an a dditiv e categor y A is called inde c om- p osable if whenever A ∼ = A ′ ⊕ A ′′ , A ′ ∼ = 0 or A ′′ ∼ = 0. Denote by ind( A ) the set of isomorphism classe s of indecomp osable ob jects in A . Definition 2.2. W e call the additive category A Krul l-S chmid t if a n y ob ject of A is a finite dir ect sum of o b jects with lo cal endo mo rphism rings. By the Krull-Schmidt theorem, every ob ject in a Krull-Schmidt categor y can be written uniquely a s a finite direct sum o f indeco mposa bles [ Kra12 , 3.2.1 ]. Definition 2. 3. An ex act c ate gory is a n additive catego ry E toge ther with a distin- guished clas s of s equences M ′ / / / / M / / / / M ′′ , called c onfl ations , suc h that there is a fully faithful additive functor f fro m E in to an ab elian catego r y A satis- fying the following tw o prop erties: K ′ -THEOR Y OF A LOCAL RING OF FINITE COHE N-MA CAULA Y TYPE 3 1. f reflects ex actness; that is, M ′ / / / / M / / / / M ′′ is a confla tion in E if and only if 0 → f ( M ′ ) → f ( M ) → f ( M ′′ ) → 0 is a short ex a ct sequence in A . 2. E is closed und er ex tensions in A ; that is , if 0 → f ( M ′ ) → A → f ( M ′′ ) → 0 is exa ct in A , then A ∼ = f ( M ) for some M in E . W e say E ′ is a n exact sub c ate gory of E if E ′ and E are exact categor ies, E ′ is a sub c ategory of E , the inclusion functor E ′ → E r eflects exactness, a nd E ′ is closed under extensions in E . If each E i is an exact ca tegory , we endow L E i with the structure of exact category by setting the confla tio ns to b e the seq uences which are co ordinate-wise conflations. This direct sum is then the co pr oduct with resp ect to exa ct functors betw een exact categ ories. If R is a ring, the categ ory mo d ( R ) of finitely prese n ted R - mo dules is an exact category in which the conflations ar e the short exa ct sequences. Similarly , the category proj ( R ) o f finitely generated pr o jective R -mo dules is an exact ca teg ory with short exact sequences for confla tions. Note that every co nflation in p ro j ( R ) is split. A subcateg ory w E ⊂ E of an exact category E is called a sub c ate gory of we ak e quivalenc es if it con tains all ob jects of E and all iso morphisms in E and sa tisfies W a ldhausen’s Gluing Lemma [ W a l83 , 1.2]. W e define the K -theo ry of an exact cat- egory E r elativ e to a s ubcatego ry of weak equiv ale nce s w E ⊂ E using W a ldhausen’s S · -construction [ W a l83 , 1.3]. It is denoted s imply b y K ( E ), a nd it is an Ω-sp ectrum whose n th space is K ( E ) n = Ω | wS n · E | . If no sub categor y of weak equiv a lences is sp ecified, the K -theor y of a n exact ca te- gory E is ta k en r elativ e to the sub categor y i E of isomor phisms. Set K n ( E ) = π n ( K ( E )) . F or a ring R , set K ( R ) := K ( pr o j ( R )) K i ( R ) := π i ( K ( R )) K ′ ( R ) := K ( mo d ( R )) K ′ i ( R ) := π i ( K ′ ( R )) Then K 0 ( R ) is the usual Grothendieck g roup o f R . If R is a commutativ e loca l ring, K 1 ( R ) ∼ = R × . 3. The Long Exact Sequ ence Fix a Henselian Cohen-Macaulay lo cal r ing R with maximal ideal m . Assume also that R has a ca nonical mo dule, and that R is an isolated singular it y , in the sense that R p is regular for nonmax ima l primes p . As R is Henselian lo cal, mo d ( R ) is a Kr ull- Sc hmidt catego ry ([ V ´ 90 , Lemma 13], [ Sid90 ]). Definition 3. 1. A finitely g enerated R -mo dule M is called maximal Coh en - Mac aulay if it sa tisfies the following tw o equiv alent co nditions: 1. depth M = dim R 2. Ext i R ( R/ m , M ) = 0 for i < dim( R ) Denote by C the exact sub category of mo d ( R ) consisting of the maximal Cohen- Macaulay mo dules. Let C ⊕ be the same catego ry as C but with a different exact structure: the conflations in C ⊕ are the split short exa ct s e quences. 4 VIRAJ NA VKAL Since C is clo sed under summands and mo d ( R ) is Krull-Schmidt, C a lso is Krull- Schm idt. In a dditio n, us ing 3.1 .2, one can easily pro ve tha t C satisfies the following. Lemma 3.2. Supp ose 0 / / M ′ / / M / / M ′′ / / 0 is a short exact se quenc e of R -mo dules. 1. If M ′ and M ′′ ar e in C , then so is M . 2. If M and M ′′ ar e in C , then so is M ′ . Definition 3.3. Let e C b e the a belian categ ory of co ntrav aria n t additive functors from C to the categor y of ab elian groups. Let b C b e the full subc a tegory of e C consisting of functor s F which fit into an exact seq uence Hom C ( − , M ) / / Hom C ( − , M ′ ) / / F / / 0 with M and M ′ in C ; such functors are calle d finitely pr esente d . b C is an a belian category [ Y os90 , 4 .19]; k er nels and cokernels are computed p oint wise, as in e C . F o r the next pr opos ition we will need Quillen’s r esolution theorem. F or conve- nience we repro duce it here, in a form that is useful for us. Theorem 3. 4 ([ Qui73 , Cor. 1 to Thm. 3]) . L et E b e an exact c ate gory and P ⊂ E an ex act sub c ate gory. Supp ose t hat 1. for any c onflation M / / / / P / / / / P ′ with P and P ′ in P , M is iso- morphic to an obje ct in P ; and 2. every obje ct of E has a P - r esolut ion of finite lengt h. Then the inclusion funct or induc es a homotopy e qu ival enc e K ( P ) ≃ K ( E ) . Prop osition 3.5. The Y one da functor h : C ⊕ → b C is a K - t he ory e quivalenc e. Pr o of. W e verify the hypotheses of the resolution theorem. W e nee d to show that the essential image of h is close d under extensions. Supp o se 0 / / h ( A ) / / F p / / h ( B ) / / 0 is exact in b C , and choose a lift x ∈ F ( B ) of id B ∈ h ( B )( B ). Define i : h ( B ) = Hom C ( − , B ) → F by i C ( f ) = F ( f )( x ), for f : C → B . Then i splits p , so F is isomorphic to h ( A ) ⊕ h ( B ) ∼ = h ( A ⊕ B ). W e need a lso to show that if 0 / / F / / h ( A ) p / / h ( B ) / / 0 is exact in b C , then F is in the essential imag e of h . As befor e, one can find a section of p , which m ust be of th e form h ( i ) for some split mono morphism i : B → A . Then F ∼ = h (coker( i )). Last, we must show that every functor in b C has a finite resolution by represe ntable functors. By Auslander-Buch weitz a pproximation [ AB89 ], for every X ∈ mo d ( R ) there is a map f : M → X with M in C such that Hom( − , f ) : Hom( − , M ) → Hom( − , X ) is a n epimorphism b et ween functors in e C . By rep eatedly taking ap- proximations to syzygies, we find that any F in b C has a resolution · · · h ( f 2 ) / / h ( M 1 ) h ( f 1 ) / / h ( M 0 ) / / F / / 0 K ′ -THEOR Y OF A LOCAL RING OF FINITE COHE N-MA CAULA Y TYPE 5 with M i in C for i ≥ 0. Ev aluating this sequence at R pro duces an exact seq uence of R -mo dules · · · f 2 / / M 1 f 1 / / M 0 f 0 / / F ( R ) / / 0 . By the exa ct sequences 0 / / im f n +1 / / M n / / im f n / / 0 it follows that whenever n ≥ dim R , E xt i R ( R/ m , im f n ) = 0 for i < dim R . Ther efore im f dim( R ) is maxima l C o hen-Macaulay and 0 / / h (im f dim( R ) ) / / h ( M dim( R ) − 1 ) / / · · · / / h ( M 0 ) / / F / / 0 is a reso lution of F b y repr esen table functors.  Recall ([ Swa68 , I.2]) that if A is a well-pow e red a b elian c ategory and B ⊂ A a Serre sub category , there is an abelia n ca tegory A // B (usually denoted A / B , but we reserve this notatio n for the additive quotient), together with an exa ct functor A → A // B which is universal for exact functors out of A which v anis h on B . The following criter io n for re cognizing the quotient catego ry is fro m [ AR86 , I.1.6]; w e omit the pro of. Lemma 3.6. L et e : A → D b e an exact funct or b etwe en wel l-p ower e d ab elian c ate gories, and let p : A → A / / k er e b e the c anonic al functor. Su pp ose ther e is an additive functor s : D → A such that e s = id D . Th en ps : D → A // ker e is an exact functor and an inverse isomorphism to the functor e : A // k er e → D induc e d by e . F o r the next theo rem we will need Q uillen’s lo c alization theorem; we repro duce it here. Theorem 3. 7 ([ Qui73 , Theore m 5]) . Le t A b e an ab elian c ate gory with a set of isomorphi sm classes of obje cts. Le t B ⊂ A b e a Serr e su b c ate gory, i : B → A t he inclusion functor, and p : A → A // B t he c anonic al functor. Then K ( B ) K ( i ) / / K ( A ) K ( p ) / / K ( A // B ) is a homotopy fib er se quenc e. Let e : b C → mo d ( R ) b e the e v alua tion functor F 7→ F ( R ). Let b C 0 = ker( e ) ⊂ b C , i.e. b C 0 is the categor y of finitely presented functors F satisfying F ( R ) ∼ = 0. b C 0 is an ab elian ca tegory ([ Y o s90 , 4.1 7]); kernels and co k er nels in b C 0 are computed po in twise. Theorem 3.8 ([ AR86 , I.1.5]) . L et i : b C 0 → b C b e the inclusion fun ctor. Then K ( b C 0 ) K ( i ) / / K ( b C ) K ( e ) / / K ′ ( R ) is a homotopy fib er se quenc e. Pr o of. Since b C 0 = ker e , it s uffice s by Lemma 3.6 and the lo calization theorem 3.7 to find an additive functor s which is r igh t inv e r se to e . F o r each M in mo d ( R ), cho ose a pro jective presentation P 1 → P 0 → M → 0 of M . Set s ( M ) = coker (Hom C ( − , P 1 ) → Hom C ( − , P 0 )). This ass ig nmen t extends to a right exact functor s : mod ( R ) → b C suc h that es = id mo d ( R ) , as de s ired.  6 VIRAJ NA VKAL Remark 3. 9 . One can a lternatively deduce T he o rem 3.8 from a theo rem of Schlic ht- ing. Acco rding to [ Sc h0 6 , Prop osition 2], K ( b C 0 ) is the homotop y fib er o f K (id) : K ( C ⊕ ) → K ( C ). Now Theorem 3 .8 follows from the fact that h : C ⊕ → b C and the inclusion C → mo d ( R ) a re K -theo ry equiv alences (the for mer by Pr opo sition 3.5 , the latter by the res o lution theorem). Next we classify the simple ob jects o f e C . F or eac h indecompos able maximal Cohen-Macaulay mo dule M , let R M = End R ( M ) op . L e t κ M be the quotient of R M by its Jacobso n ra dica l; κ M is a div ision ring. Since End R M is a loca l ring, the functor C ( − , M ) ∈ e C has a unique maximal subfunctor F M , which coincides with r ad( C )( − , M ). F or a n y indeco mp osable N 6 ∼ = M , F M ( N ) = C ( N , M ), a nd F M ( M ) = rad( End R M ). Let S M = C ( − , M ) / F M . As a n additive functor , S M is determined uniquely up to isomorphis m by the following prop erties. 1. S M ( N ) = 0 if N is indecomp osable and not isomor phic to M . 2. S M ( M ) = κ op M . 3. F or f : M − → M , S M ( f )( α ) = α · f , where f is the image of f in κ op M . Prop osition 3.10. L et S b e a functor in e C . The n the fol lowing ar e e qu ival en t: 1. S is simple in e C , and S ( R ) ∼ = 0 . 2. S ∼ = S M for some inde c omp osable R -mo dule M which is not isomorphic to R . Pr o of. 2 ⇒ 1 is clear, s ince F M ⊂ C ( − , M ) is a maximal subfunctor and F M ( R ) = C ( R , M ). W e prov e 1 ⇒ 2. Supp o se S is a simple ob ject in e C . S cannot v anish on all indecomp osables, so let M b e an indeco mposable in C suc h that S ( M ) 6 = 0. Cho ose nonzero x ∈ S ( M ). x defines a nonzero mor phism x ∗ : C ( − , M ) − → S by ( x ∗ ) N ( f ) = S ( f )( x ) for f : N − → M in C . As S is simple, x ∗ m ust be an epimo rphism, and its kernel is a maximal subfunctor of C ( − , M ). Therefore S ∼ = C ( − , M ) /F M = S M .  Recall that a map f : E → M in C is called right almost split if im( C ( − , f )) = rad( C )( − , M ). If M is indecomp osable and no t isomorphic to R , this is equiv alent to saying tha t C ( − , E ) C ( − ,f ) / / C ( − , M ) / / S M / / 0 is exact. f is ca lled minimal right almost split if f is right almost split and, fo r any g ∈ End C E such that f g = f , g is a n automorphism. The follo w ing prop osition was or ig inally proved by Ausla nder (see, e .g ., [ Aus86 ]) under the as sumption that R is complete; as observed in [ Y os90 , 3.2 ], o ne ne e ds only assume R is Henselian. W e omit the pro of. Prop osition 3.11. F or any inde c omp osable M in C which is not isomorphi c to R , ther e is a m inimal right almost split morphism E − → M . It follows that each functor S M is finitely presented. The next theore m follows e asily from results of Auslander; o ur proo f is taken from [ Y o s90 ]. Theorem 3.12. Su pp ose R is of finite Cohen-Mac aulay typ e. Then every functor F in b C 0 admits a filtr ation 0 = F 0 ⊂ F 1 ⊂ · · · ⊂ F n − 1 ⊂ F n = F K ′ -THEOR Y OF A LOCAL RING OF FINITE COHE N-MA CAULA Y TYPE 7 with F i in b C 0 and F i /F i − 1 simple in e C , for al l i . Pr o of. F has a presentation of the for m C ( − , N ) C ( − ,f ) / / C ( − , M ) / / F / / 0 for s ome epimo rphism f : N / / / / M in C . Set K = k er( f ), so F is a subfunctor of Ext 1 R ( − , K ). F or an y maximal Cohen- Macaulay mo dule L and prime p 6 = m , L p is a maximal Co hen-Macaulay R p -mo dule; since R p is regular lo cal, L p is in fact a free R p -mo dule. Therefore (Ext 1 R ( L, K )) p = Ext 1 R p ( L p , K p ) = 0. Since Ext 1 R ( L, K ) is suppo rted o nly at m , it must be a finite length R -mo dule. Therefore the submodule F ( L ) is finite length a s w ell. Now let L b e the direct sum of all non-free indecomp o sables in C . The pro of pro ceeds by induction on the length of F ( L ). If length( F ( L )) = 0, F v anishes o n all indecomp osable maximal Co hen-Macaulay mo dules, s o F ∼ = 0 and F triv ially admits the desired filtratio n. Suppo se, then, that length( F ( L )) > 0, and assume that any functor G in b C 0 with length( G ( L )) < le ng th( F ( L )) a dmits a filtratio n a s ab o ve. Cho o se an indecomp osable M with F ( M ) 6 = 0; then c ho ose a n epimorphism of R M -mo dules p : F ( M ) − → κ op M . p ex tends to a natural trans formation π : F − → S M . Let G = k er( π ), so there is an exa ct sequence 0 / / G / / F / / S M / / 0 . Therefore length( G ( L )) < length( F ( L )). Since b C 0 is ab elian, G is in b C 0 , so G admits the desired filtr ation. F r om the exa ct sequence ab o ve, it follo ws that F a dmits suc h a filtration a s w ell.  Definition 3.1 3. Let b C s 0 be the full sub category of b C 0 consisting of ob jects which are semisimple in e C . In other words, b C s 0 consists of those functors which ar e isomo rphic to finite direct sums of the functors S M , with M 6 ∼ = R . Definition 3. 1 4. W e say R is of finite Cohen-Mac aulay typ e if there a re only finitely many isomorphism clas ses of indecomp osable maximal Cohen- Ma caulay R - mo dules. Theorem 3.12 will imply the inclusion b C s 0 → b C 0 is a K -theo ry equiv alence ; to prov e this we first reca ll Q uillen’s D ´ evissage theorem. Theorem 3.15 ([ Qui73 , Thm. 4]) . L et A b e an ab elian c ate gory and B a nonempty ful l sub c ate gory close d under sub obje cts, quotients, and fi nite pr o ducts in A . Su pp ose that every obje ct F in A admits a fi ltr ation 0 = F 0 ⊂ F 1 ⊂ · · · ⊂ F n − 1 ⊂ F n = F with F i /F i − 1 in B for e ach i . Then the inclus ion functor induc es a homotopy e quivalenc e K ( B ) ≃ K ( A ) . Corollary 3.16. If R is of fin ite Cohen-Mac aulay typ e, the inclusion b C s 0 − → b C 0 is a K -t he ory e quivalenc e. Pr o of. Since sub ob jects, quotients, and pro ducts of s emisimple ob jects in a n ab elian category are ag ain semisimple, b C s 0 is closed under ta k ing subob jects, quotient s, and 8 VIRAJ NA VKAL pro ducts. Theor em 3.12 a llows us to apply D´ evissage to the sub category b C s 0 ⊂ b C 0 . The conclusion follows.  Assume R is of finite Cohen- Ma caulay type. Let ind( C ) denote the set of iso- morphism classes of indecomp o sable ob jects in C , and put ind 0 ( C ) = ind( C ) \ { [ R ] } . b C s 0 is semisimple, a nd its simple ob jects a re the functors S M for [ M ] ∈ ind 0 ( C ). Therefore the eq uiv ale nc e s proj (End ( S M ) op ) ≃ add ( S M ) induce an equiv alence M [ M ] ∈ ind 0 ( C ) pr oj (End( S M ) op ) ∼ − → b C s 0 . Let L = L [ M ] ∈ ind( C ) M and put Λ = (End R L ) op . Λ is sometimes ca lled the Aus- lander algebr a . Since L is an additive generator for C ⊕ , the hor izon ta l functors in the diagr am b elow a re equiv alences: C ⊕ Hom C ⊕ ( L, − ) ≃ / / h   pr oj (Λ)  _   b C F 7→ F ( L ) ≃ / / mo d (Λ) Combining everything, we end up with a diagr am (3.17) L ind 0 ( C ) pr oj ( κ M ) ≃   pr oj (Λ) ≃ } } ④ ④ ④ ④ ④ ④ ④ ④ ④ ④ ≃ K   b C s 0 ≃ K   C ⊕ ≃ K   mo d (Λ) ≃ | | ② ② ② ② ② ② ② ② ② ② b C 0 / / b C / / mo d ( R ) in which ar rows lab eled ≃ K induce eq uiv alences in K -theory . T o gether with the K -theor y e q uiv alences in this diagram, Theo rem 3.8 implies there are homo top y fiber sequences (3.18) W [ M ] ∈ ind 0 ( C ) K ( κ M ) / / K ′ (Λ) / / K ′ ( R ) (3.19) W [ M ] ∈ ind 0 ( C ) K ( κ M ) α / / K ( C ⊕ ) β / / K ′ ( R ) T a king ho motop y groups in ( 3.18 ) yields the long ex act sequence of Theo rem 1 .2 . Remark 3.20 . By Nak ay ama’s Lemma, the image of m in R M is cont a ined in the maximal ideal of R M . So we may view κ M as a division algebra over R/ m . As R M is a finitely g enerated R -mo dule, κ M is a finite dimensio nal v ec tor space ov er R/ m . In particular, if R/ m is algebra ically close d, κ M = R/ m . Remark 3.21 . Suppose R is of the for m S/ ( w ) for some r egular lo cal ring S and w ∈ S . Then we may apply the techniques ab o ve to obta in a decomp osition of the K -theor y of the categor y MF o f matrix factorizations in S with po ten tial w . An ob ject of this categ ory is a Z / 2 Z -gr a ded finitely generated free S -mo dule X with K ′ -THEOR Y OF A LOCAL RING OF FINITE COHE N-MA CAULA Y TYPE 9 a degree- one endomorphism d X such that d 2 X = w · id. A morphism f : X → Y in MF is a degree zero ma p s atisfying d Y f = f d X . MF is a F ro benius catego ry whose s ubcatego ry prinj ( MF ) of pr o jective-injectiv e ob jects co nsists o f the contractible matrix factorizations – that is, those ob jects X for which there is a degree-o ne endomo r phism t : X → X such that td X + d X t = id X . Any F ro benius category F defines a n exa ct catego ry with weak equiv a- lences w F cons isting of those morphisms beco ming in vertible in the stable category F / p rinj ( F ). W e shall take the K -theo ry of MF relative to this sub category w MF of weak equiv alences. The categor y Ch b ( E ) of b ounded chain complexe s in an exact categor y E is a F r obenius categor y; its conflations ar e the sequences which a re degr ee-wise split, and its pro jective-injective ob jects ar e the contractible co mplexes. Denote by p erf ( R ) the ca tegory of p erfect complexe s of R -mo dules, i.e. the exa c t subcate- gory o f Ch b mo d ( R ) co nsisting of complexes quasi-is omorphic to a complex of free R -mo dules. T he r e is a map o f F r obenius pa irs Ω : ( MF , prinj ( MF )) / / ( Ch b mo d ( R ) , p erf ( R )) taking a matrix facto rization X 1 d 1 / / X 0 d 0 o o to coker( d 1 ), consider ed as a complex concentrated in degree zero . (Since coker( d 1 ) is a nnihilated b y w , w e may view it as an R - module.) B y [ O rl04 , Theorem 3.9], Ω induces an e q uiv alence on de- rived catego ries, so in particula r it is an equiv alence in K -theory ([ Sch06 , Pr o posi- tion 3]). Since the degree-zer o inclusions mo d ( R ) → Ch b mo d ( R ) and proj ( R ) → Ch b pr oj ( R ) a re K - theory equiv alences ([ TT90 , 1 .11.7]), and sinc e the inclus ion Ch b pr oj ( R ) → p erf ( R ) is a derived equiv alence, it follows that K ( MF ) is eq uiv a- lent to the homotopy cofiber of the map K ( R ) − → K ′ ( R ) induced by the inclusion pr oj ( R ) → mod ( R ). Let r : proj ( R ) → C ⊕ be the inclusion, and set X = co ne( K ( r )). Co nsider the following exac t tria ngles of sp ectra. (3.22) K ( R ) K ( r ) / / K ( C ⊕ ) ρ / / X / / P K ( R ) K ( C ⊕ ) β / / K ′ ( R ) / / P W ind 0 ( C ) K ( κ M ) P α / / P K ( C ⊕ ) K ( R ) β ◦ K ( r ) / / K ′ ( R ) / / K ( MF ) / / P K ( R ) The midd le sequence is the triangle from ( 3.19 ), rotated once. Using t he octahedr al axiom to co mpare the cones o f β , K ( r ), and β ◦ K ( r ), we obtain an ex a ct triangle (3.23) W ind 0 ( C ) K ( κ M ) α ′ / / X / / K ( MF ) / / P W ind 0 ( C ) K ( κ M ) . W e will study α ′ in the next section. 10 VIRAJ NA VKAL 4. The Auslander-Reiten Ma trix Assume now that R is of finite C M type, that k = R/ m is a lgebraically clo s ed, and that R contains k . W e wish to under stand the map (4.1) α : W ind 0 ( C ) K ( k ) / / K ( C ⊕ ) which app ears in ( 3.19 ). Let M be an indeco mp osable in C whic h is not is omorphic to R . A sho rt exact sequence (4.2) 0 / / N g / / E f / / M / / 0 in C is called the A uslander-R eiten se quenc e ending in M if f is minimal righ t almost split. This is the sa me as saying that 0 / / C ( − , N ) C ( − ,g ) / / C ( − , E ) C ( − ,f ) / / C ( − , M ) is a minimal pro jective resolution in b C of S M . The Auslander- Reiten sequence ending in M is unique up to is omorphism. By Prop osition 3.11 , a n y non-pro jective indecomp osable M in C is part of an Auslander-Reiten s equence as in ( 4.2 ). Let M 0 , . . . , M n be the indeco mposa ble maximal Cohen-Macaulay R -modules, with M 0 = R . F o r j > 0, set 0 / / N j / / E j / / M j / / 0 to b e the Auslander- Reiten sequence ending in M j . Given any Q in C , let #( j, Q ) be the num b er of M j -summands app earing in a decomp osition of Q int o indecom- po sables. Denote by k j the ob ject o f L ind 0 ( C ) mo d ( k ) which is k in the M j co ordinate and 0 in the others. Note that to define a k -linea r functor out of L ind 0 ( C ) mo d ( k ), one needs only to sp ecify the imag e of each ob ject k j . Define k -linear functor s a 0 , a 1 , a 2 : M ind 0 ( C ) mo d ( k ) → C ⊕ by a 0 ( k j ) = M j a 1 ( k j ) = E j a 2 ( k j ) = N j Set a : L ind 0 ( C ) mo d ( k ) → b C to b e the k -linear functor sending k j to S M j , and as befo re let h : C ⊕ → b C b e the Y o neda functor. T rac ing through the functors in ( 3.17 ), one s ees that K ( a ) = K ( h ) ◦ α . Since there is an exa ct sequence of functor s 0 / / h ◦ a 2 / / h ◦ a 1 / / h ◦ a 0 / / a / / 0 , we hav e b y the additivity theorem ([ Qui73 , § 3 , Cor. 2]) K ( ha 0 ) − K ( ha 1 ) + K ( ha 2 ) = K ( a ) so that (4.3) α = K ( h ) − 1 K ( a ) = K ( a 0 ) − K ( a 1 ) + K ( a 2 ) . K ′ -THEOR Y OF A LOCAL RING OF FINITE COHE N-MA CAULA Y TYPE 11 Let m l : mo d ( k ) → C ⊕ be the k -linea r functor which sends k to M l . F or m an ( n + 1) × n int eger matrix T whose l j -entry is #( l, M j ) − #( l , E j ) + #( l , N j ). ( T has a 0th row but no 0th column.) Applying the additivity theorem to a 0 , a 1 , and a 2 , we conclude from ( 4.3 ) tha t the j th comp onent of α is (4.4) ( α ) j = X l T lj K ( m l ) . Let m = L m l : L ind( C ) mo d ( k ) → C ⊕ . Then we ma y rewrite ( 4.4 ) as (4.5) α = K ( m ) ◦ ( T · id K ( k ) ) In this sense α is defined by the matrix T . Ident ifying K 0 ( C ⊕ ) with Z n +1 via the basis { [ M 0 ] , . . . , [ M n ] } , we see that K 0 ( m l ) : K 0 ( mo d ( k )) = Z → Z n +1 is just the inclus io n into the l th co ordinate. Therefore π 0 ( α ), as a map b etw een free abelian g roups, is defined b y T . This is the d escription of π 0 ( α ) orig inally given in [ AR86 , § 4 .3]. Let T ′ be the n × n integer matrix obtained f r om T b y deleting its top ro w, whic h corres p onds to the indecomp osable M 0 = R . Just as T descr ib ed α , T ′ describ es the map α ′ = ρ ◦ α : W ind 0 ( C ) K ( k ) → X from ( 3.23 ). Recall from ( 3.22 ) th e homotop y fiber sequence K ( R ) K ( r ) / / K ( C ⊕ ) ρ / / X Since m 0 : mo d ( k ) → C ⊕ factors through r , ρ ◦ K ( m 0 ) is nulhomotopic and therefor e the j th comp onent of α ′ is given b y ( α ′ ) j = ρ ◦ α j = ρ ◦ X l ≥ 0 T lj K ( m l ) = ρ ◦ X l> 0 T ′ lj K ( m l ) (4.6) Let m ′ = L l> 0 m l : L ind 0 ( C ) mo d ( k ) → C ⊕ . Then we ma y rewrite ( 4.6 ) as (4.7) α ′ = ρ ◦ K ( m ′ ) ◦ ( T ′ · id K ( k ) ) 5. An Example Let R b e a 1-dimensional singula rit y of t y pe A 2 n , i.e. R = k [[ t 2 , t 2 n +1 ]] with k an alge braically closed field. The MCM R -modules a re the mo dules M i = k [[ t 2 , t 2( n − i )+1 ]], i = 0 , . . . , n , o n which R acts by mult iplica tion. The Auslander - Reiten quiver of R is then (5.1) [ R ] / / [ M 1 ] t 2 o o / / · · · t 2 o o / / [ M n ] t 2 o o t p p (each righ t arrow is the inc lus ion x 7→ x ). In this section we will try to descr ibe, as explicitly as p ossible, the groups K ′ 1 ( R ) and K 1 ( MF ), using the techniques devel- op ed elsewhere in this pap er. Thes e descriptions app ear in Prop osition 5.7 . Let B i = add ( M 0 , . . . , M i ) ⊂ C ⊕ , a nd le t B − 1 = { 0 } ⊂ C ⊕ . Let f i : B i − 1 → B i denote the inclusion functor and p i : B i → B i / B i − 1 the quotient functor; let F i 12 VIRAJ NA VKAL denote the image in K 1 ( C ⊕ ) o f K 1 ( B i ). Then the so lid diagr am b elo w co mm utes and has exa ct r o ws ; the to p row is exact by Theorem A.12 . (5.2) K 1 ( B i − 1 ) K 1 ( f i ) / /     K 1 ( B i ) K 1 ( p i ) / /     K 1 ( B i / B i − 1 ) / /   0 0 / / F i − 1 / / F i / / 8 8 ♣ ♣ ♣ ♣ ♣ ♣ F i /F i − 1 / / 0 Moreov er, in the diagr am b elow, the right vertical arrow is an equiv alence when i > 0. B i / /   B i / B i − 1 ≃   C ⊕ / / C ⊕ / add ( L l 6 = i M l ) Therefore fo r i > 0, the map K 1 ( B i ) → K 1 ( B i / B i − 1 ) factors through F i as indicated in ( 5.2 ). It follows that the right vertical map in ( 5.2 ) is an is o morphism. So there are short exa ct s equences (5.3) 0 / / F i − 1 / / F i / / K 1 ( B i / B i − 1 ) / / 0 F o r each i , B i / B i − 1 has one nonzero indeco mposa ble M i . The r ing homomor - phism k [[ t 2 , t 2( n − i )+1 ]] → E nd R ( M i ), sending f to the multiplication-by- f endo- morphism, is an isomor phism for each i . Using this and the AR quiver ( 5.1 ), w e see that for i > 0, End B i / B i − 1 M i = (End R M i ) / ( t 2 ) =  k if 0 < i < n k [ t ] / ( t 2 ) if i = n so tha t B i / B i − 1 ≃ mo d ( k ) if 0 < i < n , and B n / B n − 1 ≃ proj ( k [ t ] / ( t 2 )). Let k + be the a dditiv e abelia n gr oup of k . Then ( k [ t ] / ( t 2 )) × ∼ = k × ⊕ k + via the identification α (1 + β t ) 7→ ( α, β ). Therefore K 1 ( B i / B i − 1 ) = K 1 (End B i / B i − 1 M i ) =  k × if 0 < i < n k × ⊕ k + if i = n So, a ccording to the sequences ( 5.3 ) and the descriptions ab o ve of the g roups K 1 ( B i / B i − 1 ), ther e is a filtratio n 0 ⊂ F 0 ⊂ · · · ⊂ F n = K 1 ( C ⊕ ) such that 1. F 0 is a quotient of K 1 ( R ) = R × . 2. F i /F i − 1 ∼ = k × for i = 1 , . . . , n − 1. 3. F n /F n − 1 ∼ = k × ⊕ k + . The group k × app ears in this filtration as a sub quotien t o f K 1 ( C ⊕ ) n + 1 times: it app ears as a sub ob ject of F 0 (w e sha ll so on see that the co mp osition k × → R × → F 0 is monic); it app ears n − 1 times a s F i /F i − 1 , 0 < i < n ; a nd it app ears as a summand of F n /F n − 1 . W e next argue that ea ch of these copies o f k × is in fa c t a summand of K 1 ( C ⊕ ). Let m i : mod ( k ) → C ⊕ and j i : mod ( k ) → B i K ′ -THEOR Y OF A LOCAL RING OF FINITE COHE N-MA CAULA Y TYPE 13 be the k -linea r functor s which (both) send k to M i , and let m = L m i : ( mod ( k )) ⊕ n +1 → C ⊕ and m ′ = L i> 0 m i : ( mo d ( k )) ⊕ n → C ⊕ . Let q : C ⊕ → C ⊕ / rad( C ⊕ ) b e the quo tien t functor and pro j i : C ⊕ / rad( C ⊕ ) ≃ ( mod ( k )) ⊕ n +1 → mod ( k ) the i th pro jection. Then the diagra m b elow commutes. mo d ( k ) = / / j i   m i   mo d ( k ) B i p i / /  _   B i / B i − 1   C ⊕ q / / C ⊕ / rad( C ⊕ ) pro j i ^ ^ F r om this we deduce the fo llowing. 1. The map k × → F 0 induced by j 0 has a left inv er se which factors thr o ugh K 1 ( C ⊕ ). Therefo re k × is embedded in F 0 in such a wa y tha t is a summand of K 1 ( C ⊕ ). 2. F or 0 < i < n , K 1 ( j i ) em b eds k × as a summand of K 1 ( C ⊕ ) which is contained in F i and pro jects isomo r phically o n to F i /F i − 1 . 3. K 1 ( j n ) embeds k × as a summand of K 1 ( C ⊕ ), a nd K 1 ( p n j n ) : k × → k × ⊕ k + is the inclusion into the first c o or dinate. W e compile all of this data in the following comm uting diagr am, in whic h a ll rows ar e split short exa ct s equences and all c olumns are exact. (5.4) 0 / / k × / /   K 1 ( m 0 ) % % ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ R × / / K 1 ( r )   R × /k × / /   0 0 / / ( k × ) n +1 K 1 ( m ) / /   K 1 ( C ⊕ ) / / π 1 ( ρ )   coker K 1 ( m ) / /   0 ( k × ) n   K 1 ( X )   k +   0 0 0 Note that π 1 ( ρ ) is surjective b e cause there is an exa ct s equence K 1 ( C ⊕ ) π 1 ( ρ ) / / K 1 ( X ) / / K 0 ( R ) / / K 0 ( C ⊕ ) Z   / / Z n +1 Therefore the ter ms in the third row of ( 5.4 ) fit into an exa ct sequence ( k × ) n π 1 ( ρ ) ◦ K 1 ( m ′ ) / / K 1 ( X ) / / k + / / 0 . W e next arg ue that the fir st map in this sequence is injectiv e. F or this it suf- fices to show that im K 1 ( m ) ∩ im K 1 ( r ) ⊂ im K 1 ( m 0 ). F or i 6 = 0, the co mposition 14 VIRAJ NA VKAL pr oj ( R ) r / / C ⊕ pro j i ◦ q / / mo d ( k ) is zero, so im K 1 ( r ) ⊂ ker K 1 (pro j i ◦ q ) a nd there- fore im K 1 ( m ) ∩ im K 1 ( r ) ⊂ im K 1 ( m ) ∩ ( \ i> 0 ker K 1 (pro j i ◦ q )) = K 1 ( m )( \ i> 0 ker K 1 (pro j i ◦ q ◦ m )) = im K 1 ( m 0 ) , as desired. Now using ( 3.19 ) and ( 4 .5 ) one obtains a n exact sequence (5.5) ( k × ) n K 1 ( m ) ◦ ( T · id k × ) / / K 1 ( C ⊕ ) / / K ′ 1 ( R ) / / ( K 0 ( k )) n / / K 0 ( C ⊕ ) Z n T / / Z n +1 Similarly , by ( 3.23 ) and ( 4.7 ) there is an exact se q uence (5.6) ( k × ) n π 1 ( ρ ) ◦ K 1 ( m ′ ) ◦ ( T ′ · id k × ) / / K 1 ( X ) / / K 1 ( MF ) / / ( K 0 ( k )) n / / K 0 ( X ) Z n T ′ / / Z n The matrices T and T ′ can b e computed directly from the Auslander-Reiten quiv er ( 5.1 ); k eeping in m ind our conv ent ion that T has a zer oth r ow but no zeroth column, these matrices a re T =             − 1 0 0 2 − 1 0 − 1 2 − 1 · · · 0 − 1 2 0 0 0 − 1 − 1 0 0 0 0 . . . 2 − 1 . . . − 1 1             ; explicitly , T lj =        − 1 if j = l ± 1 2 if j = l < n 1 if j = l = n 0 otherwise T ′ =           2 − 1 0 − 1 2 − 1 · · · 0 − 1 2 0 0 0 − 1 − 1 0 0 0 0 . . . 2 − 1 . . . − 1 1           . One prov es e asily by induction that det T ′ > 0 , so the last map in each sequence ( 5.5 ) and ( 5.6 ) is injective. Since these s equences a re exact, it follows that K ′ 1 ( R ) ∼ = coker [ K 1 ( m ) ◦ ( T · id k × )] K 1 ( MF ) ∼ = coker [ π 1 ( ρ ) ◦ K 1 ( m ′ ) ◦ ( T ′ · id k × )] T o gether with the data fr om ( 5.4 ), we obtain the following decomp ositions. K ′ -THEOR Y OF A LOCAL RING OF FINITE COHE N-MA CAULA Y TYPE 15 Prop osition 5.7. 1. Ther e is an ab elian gr oup G (= coker K 1 ( m )) such that K 1 ( C ⊕ ) ∼ = coker ( T · id k × ) ⊕ G and G fi ts into an exact se quenc e R × /k × / / G / / k + / / 0 . 2. Ther e is a short exact se quenc e 0 / / coker ( T ′ · id k × ) / / K 1 ( MF ) / / k + / / 0 . Appendix A. K 1 Localiza tion for a Krull-Schmidt Ca tegor y K 1 of an Additiv e Category. Let A be a n additive ca tegory . W e shall consider A to b e an exact ca tegory with t he split exact structure, in which the conflations a re the sequences isomorphic to a direct sum seque nce A / / / / A ⊕ B / / / / B . Set Aut( A ) to b e the catego ry whose ob jects are pair s ( A, φ ) with A in A and φ ∈ Aut A A , a nd whose morphisms ar e defined by Hom Aut( A ) (( A, φ ) , ( A ′ , φ ′ )) = { f ∈ Hom A ( A, A ′ ) | φ ′ f = f φ } . Aut( A ) has an exact structur e in which a seq uence is a co nflation iff it is a co nflation in A . Recall from ([ She8 2 , § 3]) that there is a natural s ur jection K 0 (Aut( A )) / / / / K 1 ( A ) whose kernel is g enerated b y elements of the form [( A, αβ )] − [( A, α )] − [( A, β )]. Denote b y [ A, φ ], or just [ φ ], the ima ge in K 1 ( A ) of the K 0 -class of the o b ject ( A, φ ) of Aut( A ). Lemma A.1. L et f : A → A ′ b e a morphism in A , and let φ =  1 A 0 f 1 A ′  ∈ Aut A ( A ⊕ A ′ ) . Then in K 1 ( A ) , [ φ ] = 0 . Similarly, given any g : A ′ → A , h 1 A g 0 1 A ′ i = 0 . Pr o of. There is a conflation in Aut( A ) ( A ′ , 1 A ′ ) / / / / ( A ⊕ A ′ , φ ) / / / / ( A, 1 A ) so that [ φ ] = [1 A ′ ] + [1 A ] = 0. The se cond statemen t is proved the sa me w ay .  Remark A.2. Suppo se φ , ψ ∈ Aut A ( L A i ) are row- or column-equiv alent – that is, the matr ices defining φ a nd ψ differ only up to le ft- or rig h t-multiplication by elementary matr ices, which are iden tity along the diagonal a nd z ero off the diagona l except in one ent ry . Then using the lemma ab ov e, one sees eas ily that [ φ ] = [ ψ ]. Automorphisms in a Krull-Sc hmidt Category. The following prop erties of rad( A ) are ea sy to prov e. 1. Hom rad( A ) ( X, X ) is the Jacobso n ra dic a l ra d (End A ( X )) of End A ( X ). 2. If X = L X i , Y = L Y j , then Ho m rad( A ) ( X, Y ) = L i,j Hom rad( A ) ( X i , Y j ). 3. If A is Krull- Sc hmidt and X a nd Y are nonisomor phic indecomp osables, Hom rad( A ) ( X, Y ) = Hom A ( X, Y ). The nex t lemma gives us an eas y criterion for reco gnizing automor phisms in a Krull-Schmidt a dditiv e ca teg ory . 16 VIRAJ NA VKAL Lemma A.3. Supp ose A is K rul l-Schmidt and A = d L i =1 A n i i with A 1 , . . . , A d p air- wise nonisomorphi c inde c omp osables in A . Supp ose φ ∈ End A A , and denote by φ ij the induc e d morphism A n j j → A n i i . Then φ is invertible iff for al l i , φ ii is invertible. Pr o of. Let R = End A A and R i = End A ( A n i i ). Using the abov e prop erties of rad( A ), one sees that the J acobson radical o f R is rad( R ) = Hom rad( A ) ( ⊕ A n i i , ⊕ A n j j ) = { φ ∈ R | φ ii ∈ rad( R i ) for a ll i } , so that R/ ra d ( R ) = Q R i / rad( R i ). Therefor e φ is inv er tible iff φ is inv ertible in R/ ra d ( R ), iff φ ii is inv ertible in R i / rad( R i ) for all i , iff φ ii is inv ertible for all i .  Using this lemma we can deduce the following. Corollary A.4. Assu m e A is Krul l-Schmidt, and let B ⊂ A b e a ful l additive c ate gory close d under summands. L et A b e an obje ct of A which has no nonzer o summand in B , and let B b e an obje ct of B . L et φ =  a b c d  ∈ Aut A ( A ⊕ B ) . The n a and d ar e invertible. Lo calization of an Additive Category. Let A b e a n additive catego ry , and let B ⊂ A b e a full additiv e s ubcatego ry closed und er summa nds. Let e : B → A b e the inclusion and s : A → A / B the quotient of A by the ideal consisting of morphisms factoring thro ugh B . L et w ⊂ A be the multiplicativ e set consisting of maps which are a comp osition of the form A i / / A ⊕ B ∼ = / / A ′ ⊕ B ′ p / / A ′ with A and A ′ in A , B a nd B ′ in B , i and p the cano nic a l inclusion and pro jection, and the middle map an isomo rphism. It is easy to check that w is closed under sums and co mpositions . An additiv e functor out of A s e nds a ll morphisms in w to isomorphisms iff it sends all ob jects in B to zero, so s is initial among a dditive functors inv e rting w . In this sense A / B is simultaneously the quotient of A by B and the lo calizatio n of A at w . Consider the following conditio ns on a morphism f : C → C ′ : 1. F or an y choice of isomor phisms α : C ∼ − → A ⊕ B , α ′ : C ′ ∼ − → A ′ ⊕ B ′ , with B , B ′ in B a nd A, A ′ not having an y nonz e ro summands in B , the comp osition (A.5) A i A / / A ⊕ B α ′ f α − 1 / / A ′ ⊕ B ′ p ′ A / / A ′ is an is omorphism. 2. F or so me choice o f isomor phisms α : C ∼ − → A ⊕ B , α ′ : C ′ ∼ − → A ′ ⊕ B ′ , with B , B ′ in B and A, A ′ not having any nonzero summands in B , ( A.5 ) is an isomorphism. 3. f is in w . Lemma A.6. Among t he ab ove c onditions, 1 ⇒ 2 ⇒ 3 . If A is Krul l-S chmid t , then 3 ⇒ 1 . Pr o of. 1 ⇒ 2 T rivial. K ′ -THEOR Y OF A LOCAL RING OF FINITE COHE N-MA CAULA Y TYPE 17 2 ⇒ 3 Say α ′ f α − 1 : A ⊕ B → A ′ ⊕ B ′ is given by the matrix  φ a b c  . Then A ⊕ B α ′ f α − 1 / / i A ⊕ B   A ′ ⊕ B ′ A ⊕ B ⊕ B ′  φ a 0 0 1 0 b c 1  / / A ′ ⊕ B ⊕ B ′ p A ′ ⊕ B ′ O O commutes, and the low er horizontal map is a n iso morphism with inverse  φ − 1 − φ − 1 a 0 0 1 0 − bφ − 1 bφ − 1 a − c 1  . So α ′ f α − 1 is in w , and therefore f is in w . 3 ⇒ 1 Assume A is K rull-Sc hmidt and f is in w . Given decomp ositions α : C ∼ − → A ⊕ B , α ′ : C ′ ∼ − → A ′ ⊕ B ′ , there is by hypothes is a commuting diagram as below. A / / i A   A ′ A ⊕ B α ′ f α − 1 / / i A ⊕ B   A ′ ⊕ B ′ p A ′ O O A ⊕ B ⊕ B 1 ∼ = / / A ′ ⊕ B ′ ⊕ B ′ 1 p A ′ ⊕ B ′ O O By Co r ollary A.4 , the top ho rizont al map is an is omorphism.  Remark A.7. If A is Kr ull-Sc hmidt, it follo ws fro m the ab o ve characterizatio n of maps in w that w sa tisfies the 2-out-o f-3 pro perty: if t wo o ut of f , g , and f ◦ g are in w then all three a re. In this cas e w is exactly the class of maps in A which bec ome in vertible in A / B . The Exact Sequence. Now we pro ceed to our destination, T heo rem A.12 . W e adopt the notation of the previous s e ction, with the a dded a ssumption that A is Krull-Schmidt. In this case B a nd A / B are a utomatically Krull-Schmidt. Lemma A.8. Supp ose A and A ′ ar e obje cts of A with n o nonzer o summands in B . Supp ose ψ ∈ A ( A, A ′ ) , and assume the image ψ of ψ in A / B is an isomorphi sm . Then ψ is itself an isomorphism. Pr o of. First we show that ker( E nd A A → E nd A / B A ) ⊂ rad(End A A ) . That is, w e show that g iv en morphis ms A f / / B g / / A with B in B , g f ∈ rad(End A A ). It suffices , using the characterization of rad(End A A ) in the pro of of Lemma A.3 , to show that for ea c h indecomp osable s ummand A i ⊂ A , the induced map A i f i / / B g i / / A i is in rad(End A A i ). But this follows from the fa ct that End A A i is lo cal and A i is not a summand of B . Since ψ is inv er tible, there is φ ∈ A ( A ′ , A ) such that φψ − id A ∈ ker (E nd A A → End A / B A ) a nd ψ φ − id A ′ ∈ ker(End A A ′ → End A / B A ′ ). Ther efore φψ − id A ∈ rad(End A A ) and (by the same a r gumen t) ψ φ − id A ′ ∈ rad(End A A ′ ), s o φψ and ψ φ are b oth inv ertible. It follows that ψ is inv er tible.  18 VIRAJ NA VKAL Lemma A.9. Supp ose A is an obj e ct of A with no nonzer o su mmand in B , B is an obje ct of B , and α =  φ a b c  ∈ Aut A ( A ⊕ B ) . Then [ α ] − [ φ ] ∈ im K 1 ( e ) . Pr o of. Note that φ is an automorphism by Corollary A.4 . Using Remark A.2 , compute:  A ⊕ B ,  φ a b c  = h A ⊕ B ,  φ a 0 c − bφ − 1 a i = h A ⊕ B ,  φ 0 0 c − bφ − 1 a i = [ A, φ ] + [ B , c − bφ − 1 a ] ≡ [ A, φ ] (mo d im K 1 ( e ))  Lemma A.10. Su pp ose ( A, φ ) and ( A, ψ ) ar e obje cts of Aut( A ) such t hat φ = ψ in A / B . Then [ φ ] − [ ψ ] ∈ im K 1 ( e ) . Pr o of. Since φ = ψ , φ − ψ factors as a co mposition A f / / B g / / A thro ugh some ob ject B of B . Using Remark A.2 , co mpute: [ A, φ ] =  A ⊕ B ,  φ 0 0 1 B  = h A ⊕ B ,  φ 0 f 1 B i = h A ⊕ B ,  φ − gf − g f 1 B i ≡ [ A, ψ ] (mo d im K 1 ( e )) , by Lemma A.9 Therefore [ A, φ ] ≡ [ A, ψ ] (mo d im K 1 ( e )).  Lemma A.1 1. Supp ose ( C, α ) and ( C ′ , α ′ ) ar e obj e ct s of Aut( A ) such that ( C , α ) ∼ = ( C ′ , α ′ ) in Aut( A / B ) . Then [ α ] − [ α ′ ] ∈ im K 1 ( e ) . Pr o of. W e may assume there a re dec ompositio ns ( C, α ) =  A ⊕ B ,  φ a b c  ( C ′ , α ′ ) =  A ′ ⊕ B ′ ,  φ ′ a ′ b ′ c ′  such that A and A ′ hav e no no nzero summand in B and B and B ′ are in B . By assumption ther e is β : A ⊕ B → A ′ ⊕ B ′ such that β is an iso morphism in A / B and β α = α ′ β . Say β is g iv en by a matrix of the form ( ψ ∗ ∗ ∗ ). By Lemma A.8 , ψ m ust be an isomorphis m. No w, β α = α ′ β ⇒ ψ φ = φ ′ ψ ⇒ φ = ψ − 1 φ ′ ψ s o [ C, α ] ≡ [ A, φ ] (mo d im K 1 ( e )) , by Lemma A.9 ≡ [ A, ψ − 1 φ ′ ψ ] (mo d im K 1 ( e )) , by Lemma A.10 = [ A ′ , φ ′ ] ≡ [ C ′ , α ′ ] (mo d im K 1 ( e )) , by Lemma A.9  Theorem A.12. Supp ose A is Krul l- S chmid t . Then t he se quen c e K 1 ( B ) K 1 ( e ) / / K 1 ( A ) K 1 ( s ) / / K 1 ( A / B ) ED BC GF 0 @A / / K 0 ( B ) K 0 ( e ) / / K 0 ( A ) K 0 ( s ) / / K 0 ( A / B ) / / 0 K ′ -THEOR Y OF A LOCAL RING OF FINITE COHE N-MA CAULA Y TYPE 19 is exact. Pr o of. W e show only that the sequence is exact at K 1 ( A / B ) and K 1 ( A ); the res t is easy . T o s ho w K 1 ( s ) is surjective, take [ A, φ ] ∈ K 1 ( A / B ); after replacing ( A, φ ) by an isomorphic ob ject of Aut ( A / B ), we may assume A has no no nzero summands in B . Using Lemma A.8 , w e see that any lift e φ of φ to A is automatically an automorphism of A . Hence K 1 ( s )([ e φ ]) = [ φ ], as desir ed. Next we show the sequence is exact at K 1 ( A ). Since K 1 ( s ) is surjective, any element of ker K 1 ( s ) may be written a s a sum of elements o f the for m 1. [ A, φ ] − [ A ′ , φ ′ ], f o r s ome φ ∈ Aut A A a nd φ ′ ∈ Aut A A ′ with ( A, φ ) ∼ = ( A ′ , φ ′ ) in Aut( A / B ) 2. [ A, φ ] − [ A, α ] − [ A, β ] for some φ, α, β ∈ Aut A A with αβ = φ in A / B 3. [ A, φ ] − [ A ′ , φ ′ ] − [ A ′′ , φ ′′ ], fo r some ( A, φ ), ( A ′ , φ ′ ), and ( A ′′ , φ ′′ ) in Aut( A ) such that there is a conflatio n in Aut ( A / B ) ( A ′ , φ ′ ) / / / / ( A, φ ) / / / / ( A ′′ , φ ′′ ) W e need to chec k that any such element is in im K 1 ( e ). Since a n ele ment of the first form is a lso of the third form, we skip the check for elemen ts of the first form. F o r elemen ts of the second form, o bserve that [ A, φ ] − [ A, α ] − [ A, β ] = [ A, φ ] − [ A, αβ ] ∈ im K 1 ( e ) by Lemma A.10 . So it remains to show any element of the third form is in im K 1 ( e ). Given ( A, φ ), ( A ′ , φ ′ ), and ( A ′′ , φ ′′ ) as in 3 . a bov e, there is a n is omorphism in Aut( A / B ) ( A, φ ) ∼ =  A ′ ⊕ A ′′ ,  φ ′ h 0 φ ′′  for some h : A ′′ → A ′ . Since  φ ′ h 0 φ ′′  is inv er tible, it follows b y Lemma A.11 that [ A, φ ] ≡ h A ′ ⊕ A ′′ ,  φ ′ h 0 φ ′′ i (mo d im K 1 ( e )) = [ A ′ , φ ′ ] + [ A ′′ , φ ′′ ] and therefore [ A, φ ] − [ A ′ , φ ′ ] − [ A ′′ , φ ′′ ] ∈ im K 1 ( e ).  Remark A.13. There is a one-to- o ne corre s pondence b etw een equiv alence classes of Krull-Schmidt categories with finitely many indecomp osables and Morita cla s ses of s emiperfect rings. The co r resp o ndence is defined by assigning to a catego ry A the ring (E nd A ( L [ M ] ∈ ind( A ) M )) op , and b y assig ning to a ring R the ca teg ory proj ( R ). Using this co rresp ondence we may resta te Theorem A.12 as follows. Let R be a semip erfect ring and e ∈ R an idemp oten t. Let S = { x ∈ R | (1 − e ) x (1 − e ) ∈ R × } . Then R/ReR coincides with the lo calization S − 1 R of R at S ; that is, the pro jection p : R → R/R eR is initial among S -inv erting ring homomorphisms. Let f : R → eRe be the r ing homo morphism x 7→ exe . Then the following sequence is exac t. K 1 ( eRe ) f ∗ / / K 1 ( R ) p ∗ / / K 1 ( S − 1 R ) ED BC GF 0 @A / / K 0 ( eRe ) f ∗ / / K 0 ( R ) p ∗ / / K 0 ( S − 1 R ) / / 0 20 VIRAJ NA VKAL Remark A.14. Theor em A.12 does not seem to fo llow from kno wn lo calization theorems in K -theory . In particular , since the functor D b ( s ) : D b ( A ) → D b ( A / B ) betw een b ounded derived categories may no t b e full, it may not induce an equiv a- lence b et ween the V erdier quotien t D b ( A ) / D b ( B ) (or even its idemp otent comple- tion) and D b ( A / B ). And Theorem A.12 do es not follo w from [ NR04 , Theo rem 0.5]: the T or-condition in that theorem do es not seem to b e satisfied in this situation. References [AB89] M aur i ce A usl ander and Ragna r -Olaf Buch weitz, The homo lo gic al the ory of maximal Cohen-Mac aulay appr oximations , Soci´ et ´ e Math ´ ematique de F rance (1989) , no. 38, 5–37. 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