An Algebraic Proof of Quillen's Resolution Theorem for K_1
aa r X i v : . [ m a t h . K T ] M a r An Algebraic Proof ofQuillen’s Resolution Theorem for K ∗† November 1, 2018
Abstract
In his 1973 paper [4] Quillen proved a resolution theorem for the K -Theory of an exact category; his proof was homotopic in nature.By using the main result of Nenashev’s paper [3], we are able to givean algebraic proof of Quillen’s Resolution Theorem for K of an exactcategory. This represents an advance towards the goal of giving anessentially algebraic subject an algebraic foundation. Mathematics Subject Classification:
Keywords:
Resolution Theorem, K -Theory, Exact Category Introduction
This paper presents an algebraic proof of Quillen’s Resolution Theorem of K of an exact category. The original proof of Quillen’s Resolution Theorem,[4] uses homotopic techniques. This research was done with the eventual aimof giving algebraic proofs of most of the major homotopic results in the areaof K -Theory of exact categories. This would provide a new, and hopefullyinsightful way, to work in the field.The paper is divided into three sections. The first reviews the resultsnecessary for this work from Nenashev’s three papers, [1, 2] and [3] as wellas making an important observation about the proof of his main result. Thesecond section presents Quillen’s Resolution Theorem and gives the algebraicproof for the case of K of an exact category. We leave the proof of a lemmato the third and last section as it is long, computational and distracts fromthe main result. ∗ [email protected] † Centre for Gravitational Physics, Department of Physics, Faculty of Science, TheAustralian National University, Canberra, ACT 0200, Australia K of an Exact Category Between 1996 and 1998 Nenashev published three papers [1, 2] and [3].He showed how it was possible to construct a group D ( M ) from an exactcategory M such that there was an isomorphism D ( M ) m / / K ( M ) . Webegin by giving an overview of the material that we shall need from thesepapers. Let M be an exact category. A double short exact se-quence in M , is two short exact sequences on the same objects, a ′ , a, a ′′ ∈ M .That is, a double short exact sequence, / / a ′ f i / / / / a g i / / / / a ′′ / / where i = 1 , , is really two short exact sequences, / / a ′ f / / a g / / a ′′ / / and / / a ′ f / / a g / / a ′′ / / . .1.2 Definition. A × commutative diagram is a diagram / / a ′ f ai / / / / O O a g ai / / / / O O a ′′ O O / / / / b ′ f bi / / / / β ′ i O O O O b g bi / / / / β i O O O O b ′′ β ′′ i O O O O / / / / c ′ f ci / / / / α ′ i O O O O c g ci / / / / α i O O O O c ′′ α ′′ i O O O O / / O O O O O O where each row and column is a double short exact sequence such that mor-phisms of the same subscript commute. Where no subscript is given we shallassume that right maps commute with top maps and left maps with bottommaps. When writing a × commutative diagram we shall leave out the objects. Given an exact category M , let D ( M ) be the group withgenerators < d > for all double short exact sequences d in M , and with therelations,1. Given < d > = < / / a ′ f i / / / / a g i / / / / a ′′ / / > if f = f and g = g then < d > = 0 .2. If there exists a × commutative diagram, a ′ / / / / a / / / / a ′′ b ′ / / / / O O O O b / / / / O O O O b ′′ O O O O c ′ / / / / O O O O c / / / / O O O O c ′′ O O O O here, < H T > = < / / a ′ / / / / a / / / / a ′′ / / >< H M > = < / / b ′ / / / / b / / / / b ′′ / / >< H B > = < / / c ′ / / / / c / / / / c ′′ / / >< V L > = < / / c ′ / / / / b ′ / / / / a ′ / / >< V M > = < / / c / / / / b / / / / a / / >< V R > = < / / c ′′ / / / / b ′′ / / / / a ′′ / / > then we have the relation, < H T > − < H M > + < H B > = < V L > − < V M > + < V R > in D ( M ) . D ( M ) This result was first proved by Nenashev in [3]. We will use it implicitlythroughout the rest of this paper.
Consider the double short exact sequence d = 0 / / a i r / / i l / / a ⊕ a − p l / / p r / / a / / then < d > = 0 in D ( M ) . This theorem is the foundation for our proof of Quillen’s Resolution Theoremas it allows us to work with the elements of K algebraically. For any exact category M , there exists an isomorphism m : D ( M ) / / K ( M ) Proof.
Refer to the papers [1, 2] and [3].4 .4 The Generating Relations of D ( M ) In [3] Nenashev constructs an inverse to the group homomorphism m . Whenhe does this he only uses a few 3 × D ( M ) is generated by thefew 3 × a ′ f ′ i / / / / a g ′ i / / / / a ′′ a ′ f i / / / / a g i / / / / α i O O O O a ′′ O O O O O O a ab ′ f b / / b g b / / β i O O O O b ′′ β ′′ i O O O O b ′ α ′ i O O O O f c / / c α i O O O O g c / / c ′′ α ′′ i O O O O and a f / / f / / b g / / g / / ca ⊕ a − p l O O p r O O f ⊕ f / / f ⊕ f / / b ⊕ b − p l O O p r O O g ⊕ g / / g ⊕ g / / c ⊕ c − p l O O p r O O a f / / f / / i r O O i l O O b g / / g / / i r O O i l O O c i r O O i l O O This observation shall be important in our proof of the Resolution The-orem. K using Nenashev’s Isomorphism We shall prove Quillen’s Resolution Theorem for K of an exact categoryusing the algebraic ‘description’ of K given by theorem 1.3.1. Specifically,we shall prove the following theorem: (Quillen’s Resolution Theorem for K ) . Let M be an exactcategory and F a full subcategory of M such that for all short exact sequences / / a ′ / / a / / a ′′ / / , . If a ′ , a ′′ ∈ F then a ∈ F ,
2. If a ∈ F then a ′ ∈ F ,3. For any a ′′ ∈ M there exists a short exact sequences, as above, so that a ∈ F .Then the inclusion functor F (cid:31) (cid:127) / / M induces an isomorphism i ∗ : K ( F ) / / K ( M ) . Quillen’s original proof of this theorem may be found in [4].
A Few Remarks
For the rest of this section we shall assume that M and F satisfy the hy-potheses of the Resolution Theorem. We shall use results from homologicalalgebra, such as the snake lemma, throughout the rest of this paper. Thisis justified by the Gabriel-Quillen Embedding Theorems [5, pp 399]. Also,we shall draw a ring around the objects in a 3 × F . We shall say that a double short exact sequence, / / a ′ / / / / a / / / / a ′′ / / is of type 0 if there are no restrictions on a ′ , a and a ′′ , type 1 if a ′ ∈ F , type2 if a ′ , a ∈ F and of type 3 if a ′ , a, a ′′ ∈ F . For all j = 0 , , , let F j be the free abelian group withgenerators [ d ] j , where d is a double short exact sequence of type j . Whereunambiguous we shall drop the j and write [ d ] . From definition 2.1.1, we see that we have inclusion homomorphismsbetween the F j , F (cid:31) (cid:127) i / / F (cid:31) (cid:127) i / / F (cid:31) (cid:127) i / / F . For all j = 0 , , , , let T j be the quotient of F j by therelation [ 0 / / a ′ f / / f / / a g / / g / / a ′′ / / j = 0 and the relations given by the × commutative diagrams below. ase
1. For j = 0 we include all relations [ V L ] − [ V M ] + [ V R ] = [ H T ] − [ H M ] + [ H B ] given by specializations of the × commutative diagram a ′ / / / / a / / / / a ′′ b ′ / / / / O O O O b / / / / O O O O b ′′ O O O O c ′ / / / / O O O O c / / / / O O O O c ′′ O O O O Thus T ≃ K ( M ) , by theorem 1.3.1. Case
2. For j = 1 we restrict our relations to those given by specializa-tions of the following × commutative diagrams; for convenience we writeunder each × commutative diagram the relation that it gives in T . a ′ / / / / a / / / / a ′′ a ′ / / / / a / / / / a ′′ O O O O O O O O O O ED@AEDGF@A BC [ H T ] − [ H M ] = [ V R ] a ′ f / / f / / a g / / g / / a ′′ a ′ ⊕ a ′ f ⊕ f / / p r O O − p l O O a ⊕ a g ⊕ g / / p r O O − p l O O a ′′ ⊕ a ′′ p r O O − p l O O a ′ f / / f / / i l O O i r O O a g / / g / / i l O O i r O O a ′′ i l O O i r O O @A BCEDGF [ H T ] + [ H B ] = 0 a ′ / / a / / a ′′ a ′ O O O O / / b O O O O / / b ′′ O O O O c O O O O c O O O O BCEDGF@A [ V L ] − [ V M ] + [ V R ] = 0 a ′ / / / / a / / / / a ′′ a ′ ( ) / / a ′ ⊕ p O O O O ( 0 1 ) / / p O O O O k O O O O k O O O O ED@AEDGF@A BC [ H T ] = [ V R ] − [ V M ]7 ase
3. If j = 2 we restrict our relations to those given by specializationsof the following × commutative diagrams; for convenience we write undereach × commutative diagram the relation that it gives in T . a ac ′ / / / / b O O O O / / / / b ′′ O O c ′ / / / / c / / / / O O O O c ′′ O O EDGF@A BC [ H B ] − [ H M ] = − [ V M ] a ′ / / / / a / / / / a ′′ a ′ / / / / a / / / / O O O O a ′′ O O O O O O EDGF@A BC [ H T ] − [ H M ] = − [ V M ] a ′ f / / f / / a g / / g / / a ′′ a ′ ⊕ a ′ f ⊕ f / / p r O O − p l O O a ⊕ a g ⊕ g / / p r O O − p l O O a ′′ ⊕ a ′′ p r O O − p l O O a ′ f / / f / / i l O O i r O O a g / / g / / i l O O i r O O a ′′ i l O O i r O O @A BCEDGF [ H T ] + [ H B ] = 0 a aa (cid:16) − (cid:17) / / a ⊕ a − p l O O p r O O ( 1 1 ) / / aa i r O O i l O O a EDGF@A BC [ V M ] = 08 ′ / / / / a / / / / a ′′ a ′ ( ) / / a ′ ⊕ p O O O O ( 0 1 ) / / p O O O O k O O O O k O O O O ED@AEDGF@A BC [ H T ] = [ V R ] − [ V M ] Case
4. If j = 3 then we allow all relations [ V L ] − [ V M ] + [ V R ] = [ H T ] − [ H M ] + [ H B ] given by specializations of the × commutative diagram a ′ / / / / a / / / / a ′′ b ′ / / / / O O O O b / / / / O O O O b ′′ O O O O c ′ / / / / O O O O c / / / / O O O O c ′′ O O O O EDGF@A BC
Thus T ≃ K ( F ) , by theorem 1.3.1.In all cases we shall denote the equivalence class of [ d ] j by < d > j .When unambiguous we shall drop the j and simply write < d > . From thedefinition of T j it is clear that there exists surjective group homomorphisms θ j : F j / / T j , for all j = 0 , , , . We shall say that a × commutative diagram D is oftype j if it is a specialization of one of the diagrams given in the definitionof T j . Lemma 1.2.1 revisited
The 3 × j , thus we can conclude that < / / a i r / / i l / / a ⊕ a − p l / / p r / / a / / > j = 0 . .2 The Group Homomorphisms φ j In order to prove the Resolution Theorem we shall construct a group homo-morphism that is an inverse to i ∗ . To do this we show how we may constructhomomorphisms φ j +1 : F j / / F j +1 that induce functions φ ∗ j +1 between T j and T j +1 . Before we do this, we present a construction that will allow usto define φ j +1 . The φ -construction2.2.1 Construction. For all double short exact sequences d / / a ′ f i / / / / a g i / / / / a ′′ / / there exists p ∈ F such that we have the commutative triangle p g i η i / / / / (cid:31) (cid:31) (cid:31) (cid:31) >>>>>>>> η i (cid:31) (cid:31) (cid:31) (cid:31) >>>>>>>> a ′′ a > > > > ~~~~~~~~ g i > > > > ~~~~~~~~ Proof.
We construct the pullback of the diagram a g / / / / a ′′ a g O O O O to get the object a × a ′′ a . By property (3) of theorem 2.0.1, we can find p ∈ F and ψ : p / / / / a × a ′′ a , an admissible epimorphism. This gives us thecommutative diagram a g / / / / a ′′ a × a ′′ a γ O O O O γ / / / / a g O O O O p ψ ; ; ; ; Let η i = γ i ψ . Then we have the commutative triangle p g i η i / / / / η i (cid:31) (cid:31) (cid:31) (cid:31) >>>>>>>> (cid:31) (cid:31) (cid:31) (cid:31) >>>>>>>> a ′′ a > > > > ~~~~~~~~ g i > > > > ~~~~~~~~
10s required. (The φ -construction) . Given a type j double shortexact sequence d / / a ′ f i / / / / a g i / / / / a ′′ / / and p ∈ F such that we have the commutative triangle p g i η i / / / / (cid:31) (cid:31) >>>>>>>> η i (cid:31) (cid:31) (cid:31) (cid:31) >>>>>>>> a ′′ a > > > > ~~~~~~~~ g i > > > > ~~~~~~~~ we can construct a type j + 1 double short exact sequence, φ ( p, d ) , such that < d > j = − < φ ( p, d ) > j .Proof. The commutative triangle allows us to form the 3 × a ′ / / f i / / a / / g i / / a ′′ a ′ ( ) / / a ′ ⊕ p O O ω a ′′ i O O ( 0 1 ) / / p g i η i O O k O O ν a ′′ i O O k τ O O where ω a ′′ i = ( f i , η i ). By property (2) of theorem 2.0.1 we see that k ∈ F ,thus the double short exact sequence V M is of type j +1. We shall denote thisdouble short exact sequence by φ ( p, d ). Note that the 3 × j thus we have the relation, < d > j = − <φ ( p, d ) > j as required.We shall often wish to apply construction 2.2.1 to a double short exactsequence d , 0 / / a ′ f i / / / / a g i / / / / a ′′ / / p ∈F , given by construction 2.2.1, to apply the φ -construction to d . When wedo this we shall simply say that we have formed the φ -construction over themorphisms g i . 11 he Definition of φ j As a result of construction 2.2.1, for each double short exact sequence d wemay choose p ∈ F so that the φ -construction may be applied to d using p .We shall denote this p by p d . Define φ j +1 : F j / / F j +1 by φ j +1 ([ d ] j ) = [ φ ( p d , d )] j +1 then extend by linearity to the whole group, F j . So far we have constructed a number of groups and group homomorphismsso that we have the diagram, F (cid:31) (cid:127) i / / θ (cid:15) (cid:15) F (cid:31) (cid:127) i / / θ (cid:15) (cid:15) φ l l F (cid:31) (cid:127) i / / θ (cid:15) (cid:15) φ l l F θ (cid:15) (cid:15) φ l l T ≃ K ( F ) i ∗ T T T ≃ K ( M )It is possible to see that there are a number of relations between thehomomorphisms above. We shall only need two of these relations for ourproof. For all j = 1 , , , let θ j , φ j , i j and θ be defined as above.Then we have the two equations,1. θ φ φ φ i i i = − θ ,2. θ i i i φ φ φ = − θ .Proof. The proof follows from the fact that < φ ( p d , d ) > j = − < d > j .The next lemma is the result that our proof rests on. The proof is longand computational in nature, hence we leave it to the next section. The functions φ j +1 : F j / / F j +1 induce functions φ ∗ j +1 : T j / / T j +1 such that the diagram F θ (cid:15) (cid:15) F θ (cid:15) (cid:15) φ o o F θ (cid:15) (cid:15) φ o o F θ (cid:15) (cid:15) φ o o T ≃ K ( F ) T φ ∗ o o T φ ∗ o o T ≃ K ( M ) φ ∗ o o (1) commutes. .3.3 Definition. Define ϕ : K ( M ) / / K ( F ) , by ϕ = φ ∗ φ ∗ φ ∗ . We can now prove that the Resolution Theorem follows from Lemma2.3.2.
Proof of Theorem 2.0.1.
We observe that ϕθ = θ φ φ φ by the commuta-tivity of diagram (1) and that i ∗ θ = θ i i i trivially. Hence by lemma2.3.1) we know that i ∗ ϕθ = θ i i i φ φ φ = − θ and that ϕi ∗ θ = θ φ φ φ i i i = − θ . Therefore as θ and θ are both epimorphisms weknow that i ∗ ϕ = ϕi ∗ = − K ( M ) . Thus, − ϕ is an inverse to i ∗ , and so the in-clusion functor i ∗ : K ( F ) / / K ( M ) is an isomorphism, as required. In preparation for the results needed to prove the key lemma, we presentfive constructions.
Given a specialization of the × commutative dia-gram D a ab ′ f bi / / / / b g bi / / / / β i O O O O b ′′ β ′′ i O O O O b ′ / / f ci / / α ′ i O O O O c / / g ci / / α i O O O O c ′′ α ′′ i O O O O we may construct a specialization of the × commutative diagram D ′ b ′ f bi / / / / b g bi / / / / b ′′ b ′ O O O O (cid:16) f ci (cid:17) / / / / c ⊕ p O O O O (cid:16) g ci
00 1 (cid:17) / / / / c ′′ ⊕ p O O O O k O O O O k O O O O BCEDGF@A where V ′ M = φ ( p, V M ) and V ′ R = φ ( p, V R ) . roof. We take the φ -construction over the two maps β i , which allows us toconstruct the two commutative triangles p β i η i / / / / η i (cid:30) (cid:30) (cid:30) (cid:30) ======= (cid:30) (cid:30) (cid:30) (cid:30) ======= ab β i @ @ @ @ (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) @ @ @ @ (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) p β ′′ i g bi η i / / / / g bi η i (cid:31) (cid:31) (cid:31) (cid:31) ??????? (cid:31) (cid:31) (cid:31) (cid:31) ??????? ab ′′ ? ? ? ? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) β ′′ i ? ? ? ? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) Letting k = ker( β i η i ) we find the 3 × D ′ b ′ f bi / / / / b g bi / / / / b ′′ b ′ O O O O (cid:16) f ci (cid:17) / / / / c ⊕ p O O O O (cid:16) g ci
00 1 (cid:17) / / / / c ′′ ⊕ p O O O O k O O O O k O O O O BCEDGF@A where V ′ M = φ ( p, V M ) and V ′ R = φ ( p, V R ) as required. Given a specialization of the × commutative dia-gram D a ′ f a / / a g a / / a ′′ a ′ β ′ i O O O O f bi / / / / b β i O O O O g bi / / / / b ′′ β ′′ i O O O O c α i O O O O c α ′′ i O O O O we may construct a specialization of the × commutative diagram D ′ b ′′ b ′′ k a / / / / c ⊕ p O O O O / / / / b g bi O O O O k a / / k a ′′ / / / / O O O O a ′ f bi O O O O ED@AEDGF@A BC where H ′ M = φ ( p, V M ) and V ′ M = φ ( p, V R ) and if V L is of type j then, < H ′ B > j +1 = < φ ( k a ′′ , V L ) > j +1 . roof. We form the φ -construction over the maps β i , to get the two com-mutative triangles p β i η i / / / / η i (cid:30) (cid:30) (cid:30) (cid:30) ======= (cid:30) (cid:30) (cid:30) (cid:30) ======= ab β i @ @ @ @ (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) @ @ @ @ (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) p β ′′ i g bi η i / / / / g bi η i (cid:31) (cid:31) (cid:31) (cid:31) >>>>>>>> (cid:31) (cid:31) (cid:31) (cid:31) >>>>>>>> a ′′ b ′′ > > > > }}}}}}} β ′′ i > > > > }}}}}}} We can construct the 3 × b ′′ b ′′ k a ν ai / / / / c ⊕ p ω a ′′ i O O O O ω ai / / / / b g bi O O O O k a µ i / / / / k a ′′ χ i / / / / ν a ′′ i O O O O a ′ f bi O O O O ED@AEDGF@A BC
By considering, however, the projections onto p we see that µ = µ . Let µ = µ i . We now need to compute the map χ i . Note that we have the twocommutative diagrams, with short exact rows and columns a ′ (cid:15) (cid:15) k a / / µ (cid:15) (cid:15) p / / a (cid:15) (cid:15) k a ′′ / / δ (cid:15) (cid:15) p / / a ′′ a ′ a ′ (cid:15) (cid:15) (cid:15) (cid:15) k a / / / / µ (cid:15) (cid:15) c ⊕ p / / / / b (cid:15) (cid:15) (cid:15) (cid:15) k a ′′ / / / / χ i (cid:15) (cid:15) (cid:15) (cid:15) c ⊕ p / / / / b ′′ a ′ The right diagram maps onto the left diagram, by the obvious maps. Thesediagrams give us two commutative squares from which it is possible to seethat χ i = β ′ − i δ . Therefore we have the commutative triangle k a ′′ δ / / / / AAAAAAAA χ i AAAAAAAA a ′ a ′ ? ? ? ? ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) ? ? β ′ i ? ? ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) . V L is of type j then the double short exact sequence φ ( k a ′′ , V L ) is of type j + 1 and we have the type j + 1 3 × / / a ′ a ′ ) / / ⊕ k a ′′ O O ( 0 χ i ) O O ( 0 1 ) / / k a ′′ O O χ i O O k a (cid:16) µ (cid:17) O O k aµ O O which gives us the relation < H ′ B > j +1 = < φ ( k a ′′ , V L ) > j +1 as required. Given a specialization of the × commutative dia-gram D a ′ f / / f / / a g / / g / / a ′′ a ′ ⊕ a ′ f ⊕ f / / p r O O − p l O O a ⊕ a g ⊕ g / / p r O O − p l O O a ′′ ⊕ a ′′ p r O O − p l O O a ′ f / / f / / i l O O i r O O a g / / g / / i l O O i r O O a ′′ i l O O i r O O we may construct a specialization of the × commutative diagram D ′ k ν / / ν / / a ′ ⊕ p ω / / ω / / ak ⊕ k ν ⊕ ν / / p r O O − p l O O a ′ ⊕ p ⊕ a ′ ⊕ p ω ⊕ ω / / p r O O − p l O O a ⊕ a p r O O − p l O O k ν / / ν / / i l O O i r O O a ′ ⊕ p ω / / ω / / i l O O i r O O a i l O O i r O O EDBC@AGF where H ′ T = φ ( p, H T ) and H ′ B = φ ( p, H B ) .Proof. Form the φ -construction over the maps g i to get the commutativediagram p g i η i / / / / η i (cid:31) (cid:31) (cid:31) (cid:31) >>>>>>>> (cid:31) (cid:31) (cid:31) (cid:31) >>>>>>>> a ′′ a g i > > > > ~~~~~~~~ > > > > ~~~~~~~~
16e can then find the 3 × k ν / / ν / / a ′ ⊕ p ω / / ω / / ak ⊕ k ν ⊕ ν / / p r O O − p l O O a ′ ⊕ p ⊕ a ′ ⊕ p ω ⊕ ω / / p r O O − p l O O a ⊕ a p r O O − p l O O k ν / / ν / / i l O O i r O O a ′ ⊕ p ω / / ω / / i l O O i r O O a i l O O i r O O EDBC@AGF as required.
Given a × commutative diagram which is a spe-cialization of the × commutative diagram Da ac ′ f bi / / / / b β i O O O O g bi / / / / b ′′ β ′′ O O c ′ f ci / / / / c g ci / / / / α i O O O O c ′′ α ′′ O O we may construct a specialization of the × commutative diagram D ′ k a / / / / c ⊕ p / / / / bk a / / k a ⊕ c ′ ⊕ p / / O O O O c ′ ⊕ p O O O O k b ′′ O O O O k b ′′ O O O O BC@AGFED EDBC@AGF where H ′ T = φ ( p, V M ) , V ′ R = φ ( p, H M ) and if H B if of type 2, then < V ′ M > = < φ ( k a , H B ) > . Proof.
We form the φ -construction over the morphisms g bi , which allows us17wo construct the two commutative triangles, p g bi η i / / / / η i (cid:30) (cid:30) (cid:30) (cid:30) ======== (cid:30) (cid:30) (cid:30) (cid:30) ======== b ′′ b ? ? ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) g bi ? ? ? ? (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) p β i η i / / / / (cid:30) (cid:30) (cid:30) (cid:30) ======= η i (cid:30) (cid:30) (cid:30) (cid:30) ======= ab @ @ @ @ (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) β i @ @ @ @ (cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) Thus we may also construct the 3 × k a ν ai / / / / c ⊕ p ω ai / / / / bk a / / k a ⊕ c ′ ⊕ p / / (cid:18) p L ν ai f ci p R ν ai (cid:19) O O O O c ′ ⊕ p ω b ′′ i O O O O k b ′′ O O O O k b ′′ ν b ′′ i O O O O BC@AGFED EDBC@AGF
Now suppose that H B is of type 2. We may construct the followingcommutative diagram c ′′ k aξ O O τ a / / p β i η i / / ak b ′′ τ a ′′ / / µ O O p g bi η i / / b ′′ β ′′ O O c ′′ α ′′ O O which gives us the relation α ′′ ξ = g bi η i τ a . Combined with ω ai ν ai = 0 we canshow that ξ = − g ci p L ν ai and hence we know that we have the commutativetriangle, k a ξ / / / / @@@@@@@@ − p L ν ai @@@@@@@@ c ′′ c ? ? ? ? ~~~~~~~ g ci ? ? ? ? ~~~~~~~ . This triangle allows us to construct φ ( k a , H M ) and thus the two type 318 × k b ′′ / / / / k a ⊕ c ′ ⊕ p / / / / c ⊕ pk b ′′ / / / / p ⊕ c ′ ⊕ k a / / / / (cid:18) (cid:19) O O c ⊕ p O O O O O O EDGF@A BC p ( ) / / c ⊕ p ( 1 0 ) / / cp (cid:18) (cid:19) / / p ⊕ c ′ ⊕ k a O O O O (cid:16) − (cid:17) / / c ′ ⊕ k a O O O O k b ′′ O O O O k b ′′ O O O O EDGF@A BC give the relation < V ′ M > = < φ ( k a , H B ) > as required. Given a specialization of the × commutative dia-gram D a ′ f i / / / / a g i / / / / a ′′ a ′ ( ) / / a ′ ⊕ p ′ ( 0 1 ) / / ( f i η ′ i ) O O O O p ′ g i η ′ i O O O O k ′ ν ′ i O O O O k ′ τ ′ i O O O O EDBC@AGF we may construct a specialization of the × commutative diagram D ′ k ′ ν ′ i / / / / a ′ ⊕ p ′ ( f i η ′ i ) / / / / ak ′ (cid:18) (cid:19) / / k ′ ⊕ a ′ ⊕ p O O O O ( ) / / a ′ ⊕ p O O O O k O O O O k O O O O BC@AGFED EDBC@AGF where V ′ R = φ ( p, H T ) and if D is of type j then < V ′ M > j +1 = < φ ( p, V R ) > j +1 . Proof.
Form the φ -construction over the maps g i η ′ i to get the two commu-19ative triangles, p g i η ′ i γ i / / / / (cid:30) (cid:30) (cid:30) (cid:30) ======== γ i (cid:30) (cid:30) (cid:30) (cid:30) ======== a ′′ p ′ g i η ′ i ? ? ? ? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) ? ? ? ? (cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127)(cid:127) p g i η ′ i γ i / / / / (cid:31) (cid:31) >>>>>>>> η ′ i γ i (cid:31) (cid:31) >>>>>>>> a ′′ a > > > > ~~~~~~~~ g i > > > > ~~~~~~~~ We may construct the type j + 1 3 × k ′ ν ′ i / / / / a ′ ⊕ p ′ ( f i η ′ i ) / / / / ak ′ (cid:18) (cid:19) / / k ′ ⊕ a ′ ⊕ p O O O O ( ) / / a ′ ⊕ p O O O O k O O O O k O O O O BC@AGFED EDBC@AGF
Now, let D be of type j . By using the first commutative triangle we canconstruct φ ( p, V R ), since ker( γ i ) = ker( g i η ′ i ) = k ′ . Then we have the twotype j + 1 3 × k / / / / k ′ ⊕ a ′ ⊕ p / / / / a ′ ⊕ p ′ k / / / / a ′ ⊕ k ′ ⊕ p / / / / (cid:18) (cid:19) O O a ′ ⊕ p ′ O O O O O O BC@AGFED a ′ ( ) / / a ′ ⊕ p ′ ( 0 1 ) / / p ′ a ′ (cid:18) (cid:19) / / a ′ ⊕ k ′ ⊕ p ( ) / / O O O O k ′ ⊕ p O O O O k O O O O k O O O O EDBC@AGF which together give us the relation < V ′ M > j +1 = < φ ( p, V R ) > j +1 , as re-quired. Given a specialization of the × commutative dia-gram D a ′ f ′ i / / / / a g ′ i / / / / a ′′ a ′ f i / / / / β ′ i O O O O a g i / / / / β i O O O O a ′′ β ′′ i O O O O O O O O O O EDBC@AGF e may construct specializations of the two × commutative diagrams D and D respectively k / / / / a ′ ⊕ p / / / / ak / / ˜ ν i / / a ′ ⊕ p / / ˜ ω i / / (cid:16) β ′ i
00 1 (cid:17) O O O O a β i O O O O O O O O O O ED@AEDGF@A BC a ′ f i / / / / a g i / / / / a ′′ a ′ / / a ′ ⊕ p / / ˜ ω i O O O O p O O O O k ˜ ν i O O O O k O O BCGFEDBC@AGF where H T = φ ( p, H T ) and V R = φ ( p, V R ) .Proof. We form the φ -construction over the maps g ′ i , which gives us thethree commutative triangles, p g ′ i η i / / / / η i (cid:31) (cid:31) (cid:31) (cid:31) >>>>>>>> (cid:31) (cid:31) (cid:31) (cid:31) >>>>>>>> a ′′ a g ′ i > > > > ~~~~~~~~ > > > > ~~~~~~~~ p g ′ i η i / / / / (cid:31) (cid:31) (cid:31) (cid:31) ???????? β ′′ i − g ′ i η i (cid:31) (cid:31) (cid:31) (cid:31) ???????? a ′′ a ′′ β ′′ i > > > > }}}}}}}} > > > > }}}}}}}} and p g i β − i η i / / / / / / / / (cid:31) (cid:31) (cid:31) (cid:31) >>>>>>>> β − η i (cid:31) (cid:31) (cid:31) (cid:31) >>>>>>>> a ′′ a g i > > > > ~~~~~~~~ > > > > ~~~~~~~~ . Note that ker( g i β − i η i ) = ker( g ′ i η i ) as g i β − i η i differs from g ′ i η i by the auto-morphism β ′′ i − . Hence from these triangles we form the 3 × k / / / / a ′ ⊕ p / / / / ak / / ˜ ν i / / a ′ ⊕ p / / ˜ ω i / / (cid:16) β ′ i
00 1 (cid:17) O O O O a β i O O O O O O O O O O ED@AEDGF@A BC a ′ f i / / / / a g i / / / / a ′′ a ′ / / a ′ ⊕ p / / ˜ ω i O O O O p O O g i β − i η i O O k ˜ ν i O O O O k O O BCGFEDBC@AGF as required. 21 .2 The Homomorphisms θ j φ j are Independent of Choice We now give a proof that, for all j = 1 , , θ j φ j is independent of the choiceof p d . This result is used implicitly throughout the rest of this paper. Let d = 0 / / a ′ f i / / / / a g i / / / / a ′′ / / be the type j double short exact sequence, and suppose we have the choice of p or p ′ for p d . Then < φ ( p, d ) > j +1 = < φ ( p ′ , d ) > j +1 . Proof.
Let p, p ′ ∈ F be such that either are valid choices for p d . Then wehave two commutative triangles, p g i η i / / / / η i (cid:31) (cid:31) >>>>>>>> (cid:31) (cid:31) >>>>>>>> a ′′ a > > > > ~~~~~~~~ g i > > > > ~~~~~~~~ p ′ g i η ′ i / / / / η ′ i (cid:31) (cid:31) ???????? (cid:31) (cid:31) ???????? a ′′ a ? ? ? ? ~~~~~~~~ g i ? ? ? ? ~~~~~~~~ . We may form the pullback square p g i η i / / / / a ′′ p × a ′′ p ′ p L O O O O p R / / / / p ′ g i η ′ i O O O O and by property (1) of theorem 2.0.1 we know that p × a ′′ p ′ ∈ F . For ease ofnotation, let ˜ p = p × a ′′ p ′ . We will show that < φ ( p, d ) > j +1 = < φ (˜ p, d ) > j +1 .Note that ker( p L ) = ker( g i η ′ i ), so again for ease of notation let k ′ = ker( p L ). Case
1. Suppose that j = 0. We may construct the 3 × a ak ′ (cid:16) γ (cid:17) / / a ′ ⊕ ˜ p ˜ ω i O O O O (cid:16) p L (cid:17) / / a ′ ⊕ p ω i O O O O k ′ / / µ i / / ˜ k / / χ i / / ˜ ν i O O O O k ν i O O O O ED@AEDGF@A BC
However by considering the projections onto p and ˜ p we see that χ = χ and µ = µ . For consistency of notation, let χ = χ i and µ = µ i . Thus we22ave really constructed the 3 × Da ak ′ (cid:16) γ (cid:17) / / a ′ ⊕ ˜ p ˜ ω i O O O O (cid:16) p L (cid:17) / / a ′ ⊕ p ω i O O O O k ′ µ / / ˜ k χ / / ˜ ν i O O O O k ν i O O O O ED@AEDGF@A BC
We apply construction 3.1.1 to D to get the relation < φ ( p , V M ) > = <φ ( p , V R ) > . As V M and V R are both of type 1 we know that < V M > =
2. Suppose that j = 1 ,
2. We have the commutative triangle, a ′ ⊕ ˜ p / / / / ˜ ω i / / / / (cid:16) p L (cid:17) $ $ $ $ HHHHHHHHH aa ′ ⊕ p = = = = {{{{{{{{{ ω i = = = = {{{{{{{{{ with which we can construct the type j + 1 3 × Dk / / ν i / / a ′ ⊕ p / / ω i / / ak (cid:18) (cid:19) / / k ⊕ a ′ ⊕ ˜ p ( ) / / ξ i O O O O a ′ ⊕ ˜ p O O ˜ ω i O O ˜ k O O ρ i O O ˜ k O O ˜ ν i O O ED@AEDGF@A BC
By considering the morphisms χ and p L we can construct the type j + 13 × k k ˜ k χ O O ρ i / / / / k ⊕ a ′ ⊕ ˜ p ( − O O ξ i / / / / a ′ ⊕ pk ′ µ O O (cid:16) γ (cid:17) / / a ′ ⊕ ˜ p (cid:18) (cid:19) O O (cid:16) p l (cid:17) / / a ′ ⊕ p EDGF@A BC × < φ ( p, d ) > j +1 = < φ (˜ p, d ) > j +1 . Hence in either case we see that < φ ( p, d ) > j +1 = < φ (˜ p, d ) > j +1 as wasrequired. For all j = 1 , , , the function θ j φ j is independent of thechoice of p d .Proof. Let p and p ′ be two choices for p d . Then the lemma tells us that < φ ( p, d ) > j +1 = < φ ( p ′ , d ) > j +1 . Hence we have our result.Given a double short exact sequence d we shall now denote < φ ( p d , d ) > j by < φ ( d ) > j , since < φ ( p d , d ) > j is independent of p d . φ j Induce Group Homomorphisms
We now need to show that if P ni =1 a i < d i > j = 0 , a i ∈ Z is a relation then P ni =1 a i < φ ( d i ) > j +1 = 0 , a i ∈ Z is also a relation. To do so we need onlycheck this equation for the generating relations of each group. The Relations for T From the comments and diagrams in subsection 1.4, we have three relationsto check.
Let < H T > − < H M > = − < V M > be the relation givenby the type 0 × commutative diagram Da ′ f ′ i / / / / a g ′ i / / / / a ′′ a ′ f i / / / / a g i / / / / α i O O O O a ′′ O O O O O O EDBC@AGF
Then < φ ( H T ) > − < φ ( H M ) > = − < φ ( V M ) > is a relation in T .Proof. We apply construction 3.1.6 to D to get the relation < φ ( H T ) > − <φ ( H M ) > = < V M > , but we also know that < V M > = − < φ ( V M ) > ,and so we have the relation < φ ( H T ) > − < φ ( H M ) > = − < φ ( V M ) > , as required. 24 .3.2 Lemma. Let < V L > − < V M > + < V R > = 0 be the relation givenby the type 0 × commutative diagram D , a ab ′ f b / / b g b / / β i O O O O b ′′ β ′′ i O O O O b ′ α ′ i O O O O f c / / c α i O O O O g c / / c ′′ α ′′ i O O O O Then < φ ( V L ) > − < φ ( V M ) > + < φ ( V R ) > = 0 is a relation in T .Proof. We apply construction 3.1.1 to D and get the relation < V ′ L > − <φ ( V M ) > + < φ ( V R ) > = 0 and as we have to two type 1 3 × b ′ / / / / b ′ / / b ′ ( ) / / b ′ ⊕ O O O O ( 0 1 ) / / O O O O / / b ′ b ′ / / b ′ ⊕ O O O O / / b ′ O O O O O O O O we can see that < V ′ L > = < φ ( V L ) > . Therefore we have the relation <φ ( V L ) > − < φ ( V M ) > + < φ ( V R ) > = 0 as required. Let < H T > + < H B > = 0 be the relation given by the type0 × commutative diagram Da f / / f / / b g / / g / / ca ⊕ a − p l O O p r O O f ⊕ f / / f ⊕ f / / b ⊕ b − p l O O p r O O g ⊕ g / / g ⊕ g / / c ⊕ c − p l O O p r O O a f / / f / / i r O O i l O O b g / / g / / i r O O i l O O c i r O O i l O O Then < φ ( H T ) > + < φ ( H B ) > = 0 is a relation in T .Proof. We apply construction 3.1.3 to D to get the relation < φ ( H T ) > + < φ ( H B ) > = 0 as required. 25 he Relations for T For T we need to check the four relations given by the 3 × Let < H T > − < H M > = < V R > be the relation given bythe type 1 × commutative diagram Da ′ f ′ i / / / / a g ′ i / / / / a ′′ a ′ f i / / / / a g i / / / / a ′′ O O O O O O O O O O ED@AEDGF@A BC
Then < φ ( H T ) > − < φ ( H M ) > = < φ ( V R ) > is a relation in T .Proof. Apply construction 3.1.6 to D to get two 3 × D ′ and D ′′ respectively k / / ν i / / a ′ ⊕ p / / ω i / / ak / / ˜ ν i / / a ′ ⊕ p / / ˜ ω i / / a O O O O O O ED@A EDBC@AGF a ′ f i / / / / a g i / / / / a ′′ a ′ / / a ′ ⊕ p / / ˜ ω i O O O O p O O O O k ν i O O O O k O O ED@AEDGF@A BC where H ′ T = φ ( H T ) and V ′′ R = φ ( V R ). The 3 × D ′ tells us that ω i = ˜ ω i and ν i = ˜ ν i , so when we apply construction 3.1.5 to D ′′ we get relation < φ ( H T ) > = < φ ( H M ) > − < φ ( V R ) > . Now, as φ ( V R )is of type 2 we know that < φ ( V R ) > = − < φ ( V R ) > . Hence we get therelation < φ ( H T ) > − < φ ( H M ) > = < φ ( V R ) > as required. Let < H T > + < H B > = 0 be the relation given by the type × commutative diagram Da ′ f / / f / / a g / / g / / a ′′ a ′ ⊕ a ′ f ⊕ f / / p r O O − p l O O a ⊕ a g ⊕ g / / p r O O − p l O O a ′′ ⊕ a ′′ p r O O − p l O O a ′ f / / f / / i l O O i r O O a g / / g / / i l O O i r O O a ′′ i l O O i r O O @A BCEDGF Then < φ ( H T ) > + < φ ( H B ) > = 0 is a relation in T .Proof. We apply construction 3.1.3 to D to get the relation < φ ( H T ) > + < φ ( H B ) > = 0 as required. Let < V L > − < V M > + < V R > = 0 be the relation givenby the type 1 × commutative diagram Da ′ / / a / / a ′′ a ′ O O O O / / b O O O O / / b ′′ O O O O c O O O O c O O O O BCEDGF@A
Then the relation < φ ( V L ) > − < φ ( V M ) > + < φ ( V R ) > = 0 is a relationin T .Proof. We apply construction 3.1.2 to D to get the relation < φ ( V L ) > − < φ ( V M ) > + < φ ( V R ) > = 0 as required. Let < H T > = < V R > − < V M > be the relation given bythe type 1 × commutative diagram Da ′ / / / / a / / / / a ′′ a ′ ( ) / / a ′ ⊕ p O O O O ( 0 1 ) / / p O O O O k O O O O k O O O O ED@AEDGF@A BC hen < φ ( H T ) > = < φ ( V R ) > − < φ ( V M ) > is a relation in T .Proof. We apply construction 3.1.5 to D to get the relation < V M > = <φ ( H T ) > − < φ ( V R ) > . However, since V M is of type 2 we know that < V M > = − < φ ( V M ) > . Thus we get the relation < φ ( H T ) > = <φ ( V R ) > − < φ ( V M ) > as required. The Relations for T For T we need to check the relations given by the 3 × Let < H B > − < H M > = − < V M > be the relation givenby the type 2 × commutative diagram Da ac ′ / / / / b O O O O / / / / b ′′ O O c ′ / / / / c / / / / O O O O c ′′ O O EDGF@A BC
Then < φ ( H B ) > − < φ ( H M ) > = − < φ ( V M ) > is a relation in T .Proof. We apply construction 3.1.4 to D to get the relation < φ ( V M ) > = <φ ( H M ) > − < φ ( H B ) > as required. Let < H T > − < H M > = − < V M > be the relation givenby the type 2 × commutative diagram Da ′ / / / / a / / / / a ′′ a ′ / / / / a / / / / O O O O a ′′ O O O O O O EDGF@A BC
Then < φ ( H T ) > − < φ ( H M ) > = − < φ ( V M ) > is a relation in T .Proof. Note that D is also of type 3, hence we have the relation < H T > − < H M > = − < V M > . Also, H T , H M and V M are all of type 3 so weknow that < φ ( H T ) > − < φ ( H M ) > = − < φ ( V M ) > as required.28 .3.10 Lemma. Let < H T > + < H M > = 0 be the relation given by thetype 2 × commutative diagram Da ′ f / / f / / a g / / g / / a ′′ a ′ ⊕ a ′ f ⊕ f / / p r O O − p l O O a ⊕ a g ⊕ g / / p r O O − p l O O a ′′ ⊕ a ′′ p r O O − p l O O a ′ f / / f / / i l O O i r O O a g / / g / / i l O O i r O O a ′′ i l O O i r O O @A BCEDGF Then < φ ( H T ) > + < φ ( H M ) > = 0 is a relation in T .Proof. We apply construction 3.1.3 to D to get the relation < φ ( H T ) > + < φ ( H M ) > = 0 as required. Let < V M > = 0 be the relation given by the × commu-tative diagram D a aa (cid:16) − (cid:17) / / a ⊕ a − p l O O p r O O ( 1 1 ) / / aa i r O O i l O O a EDGF@A BC
Then < φ ( V M ) > = 0 is a relation in T .Proof. The 3 × D is also of type 3, therefore wehave the relation < V M > = 0. However, V M is of type 3 thus we know that < φ ( V M ) > = 0 as required. Let < H T > = < V R > − < V M > be the relation given bythe type 2 × commutative diagram Da ′ / / / / a / / / / a ′′ a ′ / / a ′ ⊕ p O O O O / / p O O O O k O O O O k O O O O ED@AEDGF@A BC hen < φ ( H T ) > = < φ ( V R ) > − < φ ( V M ) > is a relation in T .Proof. We apply construction 3.1.5 to D to get the relation < V M > = <φ ( H T ) > − < φ ( V R ) > , but V M is of type 3 and so we know that <φ ( V R ) > − < φ ( V M ) > = < φ ( H T ) > as required. The Proof of the Key Lemma
The proof of the key lemma, lemma 2.3.2, follows from the comments at thebeginning of this section and from lemmas 3.3.1 to 3.3.12.
References [1] A. Nenashev. K by generators and relations. J. Pure Appl. Algebra ,131(2):195–212, 1998.[2] Alexander Nenashev. Double short exact sequences produce all elementsof Quillen’s K . In Algebraic K -theory (Pozna´n, 1995) , volume 199of Contemp. Math. , pages 151–160. Amer. Math. Soc., Providence, RI,1996.[3] Alexander Nenashev. Double short exact sequences and K of an exactcategory. K -Theory , 14(1):23–41, 1998.[4] Daniel Quillen. Higher algebraic K -theory. I. In Algebraic K -theory, I:Higher K -theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash.,1972) , pages 85–147. Lecture Notes in Math., Vol. 341. Springer, Berlin,1973.[5] R. W. Thomason and T. Trobaugh. Higher algebraic K -theory ofschemes and of derived categories. In The Grothendieck Festschrift,Vol. III , volume 88 of