Delooping of the K -theory of strictly derivable Waldhausen categories
DDelooping of the K -theory of Waldhausencategories with factorizations Satoshi Mochizuki
Introduction
In this article, we will provide new delooping methods of the K -theory of certainWaldhausen categories (for precise conditions, see §
2) and abelian categories.A specific feature of our delooping is that a suspension of an abelian categoryis again an abelian category. By utilizing this delooping method, we will showthat negative K -groups of an abelian category is trivial (see Corollary 3.3)which is conjectured by Marco Schlichting in [Sch06, Conjecture 9.7] and asits consequence, we will obtain a generalization of a theorem of Auslander andSherman in [She89] which says that negative direct sum K -groups and negative K -groups are isomorphisms for any small exact categories. (see Corollary 3.4.)Now we give a guide for the structure of this article. In section 1, we giveone to one correspondence between the class of admissible classes and the classof Serre subcategories in a Waldhausen category with factorizations. (for moreprecise statement, see Proposition 1.8.) In section 2, we will define the deloop-ing of K -theory for certain Waldhausen categories by using unbounded filteredobjects. (see Definition 2.10.) In the final section, by combining with the resultsin previous sections, we will prove negative K -groups of an abelian category istrivial and we obtain a generalization of a theorem of Auslander and Shermanas mentioned above. Acknowledgement
I wish to express my deep gratitude to Marco Schlichting,Toshiro Hiranouchi and Sho Saito for stimulative discussion in the early stageof the work in this article. w -Serresubcategories In this section, let C be a category with cofibrations and assume that C isenriched over the category of abelian groups. Since C is closed under finitecoproducts, in this case C is an additive category. We say that a class w of morphisms in C is admissible if w satisfies the following three conditions.1 a r X i v : . [ m a t h . K T ] M a r w contains all isomorphisms. • w satisfies the two out of three property. Namely for a pair of composablemorphisms x f → y g → z , if two of gf , f and g are in w , then the third oneis also in w . • In the commutative diagram of cofibration sequences below x (cid:47) (cid:47) j (cid:47) (cid:47) a (cid:15) (cid:15) y q (cid:47) (cid:47) (cid:47) (cid:47) b (cid:15) (cid:15) y/x c (cid:15) (cid:15) x (cid:48) (cid:47) (cid:47) j (cid:48) (cid:47) (cid:47) y (cid:48) q (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) y (cid:48) /x (cid:48) , (1)if two of a , b and c are in w , then the third one is also in w . If w is an admissible class of morphisms in C , then w satisfiesthe gluing axiom of [Wal85] . In particular the pair ( C , w ) is a category withcofibrations and weak equivalences.Proof. In the commutative diagram below, assume that a , b and c are in w . y b (cid:15) (cid:15) x (cid:111) (cid:111) (cid:47) (cid:47) (cid:47) (cid:47) a (cid:15) (cid:15) z c (cid:15) (cid:15) y (cid:48) x (cid:48) (cid:111) (cid:111) (cid:47) (cid:47) (cid:47) (cid:47) z (cid:48) . Then there are commutative diagrams of cofibrations sequences. y (cid:47) (cid:47) (cid:47) (cid:47) b (cid:15) (cid:15) y ⊕ z (cid:47) (cid:47) (cid:47) (cid:47) (cid:18) b c (cid:19) (cid:15) (cid:15) z c (cid:15) (cid:15) y (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) y (cid:48) ⊕ z (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) z (cid:48) , x (cid:47) (cid:47) (cid:47) (cid:47) a (cid:15) (cid:15) y ⊕ z (cid:18) b c (cid:19) (cid:15) (cid:15) (cid:47) (cid:47) (cid:47) (cid:47) y (cid:116) x z b (cid:116) a c (cid:15) (cid:15) x (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) y (cid:48) ⊕ z (cid:48) (cid:47) (cid:47) (cid:47) (cid:47) y (cid:48) (cid:116) x (cid:48) z (cid:48) . By the left diagram above, it turns out that (cid:0) b c (cid:1) is in w and therefore b (cid:116) a c is also in w by the right diagram above.We fix an admissible class w of morphisms in C . w -Serre subcategory). We say that a full subcategory S of C is a w -Serre subcategory of C if S satisfies the following two conditions. • For a cofibration sequence x (cid:26) y (cid:16) y/x in C , if two of x , y and y/x arein S , then the third one is also in S . • S is w -closed. Namely for an object x in C , if there exists an object y in S and if there exists a zig-zag sequence of morphisms in w which connects x and y , then x is also in S . 2or an admissible class u of morphisms in C which contains w , we write C u forthe full subcategory of C consisting of those objects x such that the canonicalmorphism 0 → x is in u . Then by the lemma below, C u is a w -Serre subcategoryof C . For an admissible class u of morphisms in C which contains w , C u is a w -Serre subcategory of C .Proof. For a cofibration sequence x (cid:26) y (cid:16) y/x , by applying the axiom ofadmissible classes to the commutative diagram below, it turns out that if twoof x , y and y/x are in C u , then the third one is also in C u . (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) x (cid:47) (cid:47) (cid:47) (cid:47) y (cid:47) (cid:47) (cid:47) (cid:47) y/x. Next let x f → y be a morphism in w and assume that x (resp. y ) is in C u ,then y (resp. x ) is also in C u by the two out of three property of u . x f (cid:47) (cid:47) y (cid:95) (cid:95) (cid:63) (cid:63) The rest of this section, we assume that the pair ( C , w ) satisfies the following factorization axiom in [Sch06, A.5]. For any morphism x f → y in C , there exists acofibration i f : x (cid:26) u f and a morphism p f : u f → y in w such that f = p f i f .In this case, we call a triple ( i f , p f , u f ) a factorization of f . S -weak equivalences). Let S be a w -Serre subcategory of C . Then for a morphism x f → y in C , the following two conditions are equivalentby Lemma 1.7 below and in this case, we say that f is an S -weak equivalence . • There exists a factorization ( i f , p f , u f ) of f such that u f /x is in S . • For any factorization ( i f , p f , u f ) of f , u f /x is in S .We denote the class of all S -weak equivalences in C by w S , C or shortly w S . Let S be a w -Serre subcategory of C . Then (1) Let x i (cid:26) y j (cid:26) z be a pair of composable cofibrations in C such that z/y isin S . Then y/x is in S if and only if z/x is in S . If x i (cid:26) y is a cofibration and a morphism in w , then y/x is in S . (3) In the commutative diagram below, assume that both i and i (cid:48) are cofibra-tions and both p and p (cid:48) are morphisms in w , x (cid:47) (cid:47) i (cid:47) (cid:47) (cid:15) (cid:15) i (cid:48) (cid:15) (cid:15) z p (cid:15) (cid:15) z (cid:48) p (cid:48) (cid:47) (cid:47) y, then z/x is in S if and only if z (cid:48) /x is in S .Proof. (1) By considering a cofibrations sequence y/x (cid:26) z/x (cid:16) z/y , we obtainthe result.(2) By the commutative diagram below, it turns out that the canonical mor-phism y/x → w by the gluing axiom. x (cid:47) (cid:47) i (cid:47) (cid:47) (cid:15) (cid:15) i (cid:15) (cid:15) y (cid:47) (cid:47) (cid:47) (cid:47) y/x (cid:15) (cid:15) y y (cid:47) (cid:47) . Then since S is w -closed, it turns out that y/x is in S .(3) By the factorization axiom, there exists a factorization ( i p (cid:116) p (cid:48) , q p (cid:116) p (cid:48) , u ) of p (cid:116) p (cid:48) : z (cid:116) z (cid:48) → y . Then by the two out of three property for w , the composition z → z (cid:116) z (cid:48) i p (cid:116) p (cid:48) → u is a cofibration and a morphism in w . Thus by (2), u/z is in S . Then by (1), z/x is in S if and only if u/x is in S . By symmetry, it is alsoequivalent to the condition that z (cid:48) /x is in S . Let S be a w -Serre subcategory of C and let u be an admis-sible class of morphisms in C which contains w . Then (1) w S is an admissible class of morphisms in C which contains w . (2) We have the equalities C w S = S , (2) w C u = u. (3)To show Proposition, we will utilize the following lemma and a proof isstraightforward. Let S be a w -Serre subcategory of C and let x f → y and x (cid:48) f (cid:48) → y (cid:48) be a pair of morphisms in C and let ( i h , p h , u h ) be a factorization of h for h ∈ { f, g, (cid:0) f f (cid:48) (cid:1) } . Then the triple (cid:16)(cid:16) if if (cid:48) (cid:17) , (cid:16) pf pf (cid:48) (cid:17) , u f ⊕ u f (cid:48) (cid:17) is a factor-ization of (cid:0) f f (cid:48) (cid:1) . In particular u (cid:16) f f (cid:48) (cid:17) / ( x ⊕ x (cid:48) ) is in S if and only if u f /x and u f (cid:48) /x (cid:48) are in S . roof of Proposition 1.8. (1) For a morphism f : x → y in w , the triple (id x , f, x )is a factorization of f such that x/x ∼ → S . Thus f is in w S . In particular, w S contains all isomorphisms.We consider the commutative diagram of cofibration sequences (1). Let( i a , p a , u a ) be a factorization of a and let ( i (cid:48) , p b , u b ) be a factorization of j (cid:48) p a (cid:116) x b : u a (cid:116) x y → y (cid:48) . We denote the composition y (cid:26) u a (cid:116) x y i (cid:48) (cid:26) u b by i b andwe set u c := u b /u a the quotient of u b by u a (cid:26) u a (cid:116) x y (cid:26) u b . Then theinduced morphism i b /i a : y/x → u b /u a = u c is a cofibration by [Wal85, 1.1.2]and p b /p a : u c → y (cid:48) /x (cid:48) is in w by the gluing axiom. x (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) y (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) y/x (cid:15) (cid:15) (cid:15) (cid:15) u a (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) u b (cid:47) (cid:47) (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) u c (cid:15) (cid:15) (cid:15) (cid:15) u a /x (cid:47) (cid:47) (cid:47) (cid:47) u b /y (cid:47) (cid:47) (cid:47) (cid:47) u c / ( y/x ) . Thus the triple ( i b /i a , p b /p a , u c ) is a factorization of c and the sequence u a /x (cid:26) u b /y (cid:16) u c / ( y/x ) is a cofibration sequence by [Wal85, 1.1.2] again. Hence if twoof u a /x , u b /y and u c / ( y/x ) are in S , then the third one is also in S . Namely iftwo of a , b and c are in w S , then the third one is also in w S .Next let x f → y g → z be a pair of composable morphisms in C . By applying theargument in the previous paragraph to the commutative diagram of cofibrationsequences below x (cid:47) (cid:47) (cid:18) id xf (cid:19) (cid:47) (cid:47) f (cid:15) (cid:15) x ⊕ y ( − f id y ) (cid:47) (cid:47) (cid:47) (cid:47) (cid:18) gf
00 id y (cid:19) (cid:15) (cid:15) y g (cid:15) (cid:15) y (cid:47) (cid:47) (cid:18) g id y (cid:19) (cid:47) (cid:47) z ⊕ y ( − id z g ) (cid:47) (cid:47) (cid:47) (cid:47) z, we obtain a factorization ( i h , p h , u h ) of h for h ∈ { f, g, gf, id y , (cid:0) gf
00 id y (cid:1) } and acofibration sequence u f /x (cid:26) u (cid:16) gf
00 id y (cid:17) / ( x ⊕ y ) (cid:16) u g /y . Since id y : y → y is in w S , u id y /y is in S . Thus by Lemma 1.9, u (cid:16) gf
00 id y (cid:17) / ( x ⊕ y ) is in S if and onlyif u gf /x is in S . Hence if two of u f /x , u gf /x and u g /y are in S , then the thirdone is also in S . Namely w S satisfies the two out of three property.(2) We will show the equality (2). Let x be an object in S . Then (0 → x, id x , x )is a factorization of the canonical morphism 0 → x such that x/ ∼ → x is in S .Thus 0 → x ∈ w S . Namely x is in C w S . Next let y be an object in C w S . Thatis, the canonical morphism 0 → y is in w S . Then there exists a factorization(0 → z, z p → y, z ) such that z/ ∼ → z is in S . Since p is in w and S is w -closed, y is in S .Next we will prove the equality (3). Let x f → y be a morphism in C and let( i f , p f , u f ) be a factorization of f . Then f is in w C u if and only if u f /x is in5 u . In the commutative diagram below, x (cid:47) (cid:47) i f (cid:47) (cid:47) f (cid:15) (cid:15) u f (cid:47) (cid:47) (cid:47) (cid:47) p f (cid:15) (cid:15) u f /x (cid:15) (cid:15) y y (cid:47) (cid:47) , f is in u if and only if the canonical morphism u f /x → u and the lastcondition is equivalent to the condition that 0 → u f /x is in u by the two out ofthree property of u . Assume C is an abelian category and let S be a w -Serresubcategory of C . Then the quotient functor C → C / S induces a homotopyequivalence of spectra K ( C ; w ) → K ( C / S ) on K -theory.Proof. Notice that since S is w -closed, we have an equality S w = C w and thepair ( S , w | S ) is a Waldhausen category with factorization. Thus by fibrationtheorem in [Sch06, A.3], there exists a left commutative diagram of fibrationsequences below K ( S w ) (cid:15) (cid:15) K ( C w ) (cid:15) (cid:15) K ( S ) (cid:47) (cid:47) (cid:15) (cid:15) K ( C ) (cid:47) (cid:47) (cid:15) (cid:15) K ( C / S ) I (cid:15) (cid:15) K ( S ; w ) (cid:47) (cid:47) K ( C ; w ) (cid:47) (cid:47) K ( C ; w S ) , iS · ( C / S ) ∼ (cid:47) (cid:47) iS · S · (0 ⊂ C / S ) iS · S · ( S ⊂ C ) (cid:111) (cid:79) (cid:79) (cid:15) (cid:15) w S S · C (cid:79) (cid:79) ∼ (cid:47) (cid:47) w S S · S · ( S ⊂ C ) where the map I is a zig-zag sequence of morphisms which makes the rightdiagram above commutative. Thus by 3 × I is a homotopyequivalence of spectra and it is an inverse map of the induced map K ( C ; w S ) → K ( C / S ). In this section, let C be an essentially small category with cofibrations such that C is an additive category and let w be an admissible class of morphisms in C (see 1.1) such that the pair C := ( C , w ) satisfies the factorization axiom (see1.5). Let N be the linearly ordered set of allnatural numbers with the usual linear order. As usual we regard N as a categoryand we denote the category of functors and natural transformations from N to C by F C and call an object in F C a filtered object ( in C ). For a filtered object x and for a natural number n , we write x n and i xn for an object x ( n ) in C anda morphism x ( n ≤ n + 1) in C respectively.6et f : x → y be a morphism in F C . We say that f is a level cofibration (resp. level weak equivalence ) if for each natural number n , f n is a cofibration(resp. f n is in w ). We denote the class of all level weak equivalences by lw F C or simply lw . We can make F C into a category of cofibrations by declaring theclass of all level cofibrations to be the class of cofibrations in F C . (1) lw is an admissible class of morphisms in F C . (2) The pair ( F C , lw ) satisfies the factorization axiom.Proof. A proof of assertion (1) is straightforward. We will give a proof of asser-tion (2). Let n be a natural number and assume that σ ≤ n f : σ ≤ n x → σ ≤ n y ad-mits a factorization ( i σ ≤ n f , p σ ≤ n f , u σ ≤ n f ). Then let a triple ( i (cid:48) , p f n +1 , u f n +1 ) be afactorization of a morphism f n +1 (cid:116) x n i yn p f n : x n +1 (cid:116) x n u f n → y n +1 and we denotethe compositions x n +1 (cid:26) x n +1 (cid:116) x n u f n i (cid:48) (cid:26) u f n +1 and u f n → x n +1 (cid:116) x n u f n i (cid:48) (cid:26) u f n +1 by i f n +1 and i u f n respectively. Then the pair of triples ( i σ ≤ n f , p σ ≤ n f , u σ ≤ n f )and ( i f n +1 , p f n +1 , u f n +1 ) give a factorization of σ ≤ n +1 f : σ ≤ n +1 x → σ ≤ n +1 . Byproceeding induction on n , we finally obtain a factorization of f . Let a ≤ b be a pair of natural numbers and let x be a filtered object in C . We say that x has amplitude contained in [ a, b ] if for any 0 ≤ k < a , x k = 0 and for any b ≤ k , x k = x b and i xk = id x b . In this case we write x ∞ for x b = x b +1 = · · · . Similarlyfor any morphism f : x → y in F C between objects which have amplitudecontained in [ a, b ], we denote f b = f b +1 = · · · by f ∞ . We denote the fullsubcategory of F C consisting of those objects having amplitude contained in[ a, b ] by F [ a,b ] C . We also set F b C := (cid:91) a
10 in [Sch06], by virtue of Proposition 10.1 in [Sch06],Corollary 3.3 implies the following result which is a generalization of a theoremof Auslander and Sherman in [She89].
Let E be an essentially small exact category. We write E ⊕ foran exact category with split exact sequences whose underlying additive categoryis E . Then the identity functor E ⊕ → E induces an isomorphism of K -groups K n ( E ⊕ ) → K n ( E ) for any negative integer n . References [Gil81] H. Gillet,
Riemann-Roch theorems for higher algebraic K -theory ,Adv. in Math. (1981), p.203-289.[Sch06] M. Schlichting, Negative K -theory of derived categories , Math. Z. (2006), p.97-134.[She89] C. Sherman, On the homotopy fiber of the map BQ A ⊕ → BQ A (after M. Auslander) , In Algebraic K -theory and algebraic numbertheory (Honolulu, HI, 1987), vol. 83 of Contemp. Math., Amer. Math.Soc. (1989), p.343-348.[Wal85] F. Waldhausen, Algebraic K -theory of spaces , In Algebraic and geo-metric topology, Springer Lect. Notes Math. (1985), p.318-419.SATOSHI MOCHIZUKI DEPARTMENT OF MATHEMATICS, CHUO UNIVERSITY, BUNKYO-KU, OKYO, JAPAN. e-mail: [email protected]@gug.math.chuo-u.ac.jp