K -theoretic torsion and the zeta function
aa r X i v : . [ m a t h . K T ] S e p K -THEORETIC TORSION AND THE ZETA FUNCTION JOHN R. KLEIN AND CARY MALKIEWICH
Abstract.
We generalize to higher algebraic K -theory an iden-tity (originally due to Milnor) that relates the Reidemeister torsionof an infinite cyclic cover to its Lefschetz zeta function. Our iden-tity involves a higher torsion invariant, the endomorphism torsion, of a parametrized family of endomorphisms as well as a higher zetafunction of such a family. We also exhibit several examples of fam-ilies of endomorphisms having non-trivial endomorphism torsion. Contents
1. Introduction 12. Preliminaries 53. Higher endomorphism torsion 104. The higher zeta function 135. A generalization of Milnor’s identity 156. Chain complexes 177. The non-linear setting 218. Families of endomorphisms 24Appendix A. Fundamental theorems of endomorphism K -theory 28Appendix B. Endomorphism torsion and boundary map 30Appendix C. The sphere theorem for End A and End SA Introduction
Algebraic K -theory is a natural setting for algebraic torsion invari-ants such Reidemeister and Whitehead torsion. In this paper we con-sider a closely related invariant that we call endomorphism torsion .Endomorphism torsion provides a means of conceptually relating Rei-demeister torsion to dynamical systems. Specifically, it is related to the“counting” of periodic orbits in a dynamical system by work of Milnor [23], and subsequently partially developed in the papers of Fried [8],[9] and Geoghegan and Nicas [10].Milnor considers infinite cyclic coverings e X → X with generatingdeck transformation T : e X → e X . Assume that e X has finite rationalhomology. One can then consider two kinds of invariants associatedwith ( e X, T ).The first invariant associated with ( e X, T ) is the algebraic torsion τ ( t ) ∈ Q ( t ) × of the contractible chain complex C ∗ ( e X ) ⊗ Z [ t,t − ] Q ( t ).This is the Reidemeister torsion of X with respect to the homomor-phism π ( X ) → Z → Q ( t ) × coming from the cyclic cover e X → X and the map n t n . Normally τ ( t ) would only be defined up to anelement of Z [ t, t − ] × = {± t n } n ∈ Z , but in this paper we re-interpret τ ( t )as the endomorphism torsion of the complex C ∗ ( e X ). As a result, it isan element of Q ( t ) × with no indeterminacy.The other invariant associated with ( e X, T ) is the Lefschetz zetafunction ζ ( t ) := exp X k ≥ L ( T ◦ k ) t k k ! ∈ Q ( t ) × , which encodes counting the Lefschetz numbers of the iterates of T taken collectively. Here L ( T ◦ k ) is the Lefschetz number of the k -foldcomposite T ◦ k , which is in some sense a homological count of the k -foldperiodic points of T .Milnor’s remarkable formula is the functional equation(1) ζ ( t − ) τ ( t ) = t χ ( e X ) , where χ ( e X ) is the Euler characteristic. In this paper we show thatthis equation (1) is a consequence of another equation that holds inthe higher algebraic K -theory of endomorphisms.Let End A denote the endomorphism category of an associative unitalring A . The objects of End A are pairs ( P, f ) such that P is a finitelyprojective right A -module, and f : P → P is an endomorphism. Amorphism ( P , f ) → ( P , f ) is a homomorphism of A -modules g : P → P such that f g = gf .Grayson [15] considers a variant of the endomorphism category thatis based on a localizing parameter. Recall that a polynomial p ∈ A [ t ]is centric or central if it lies in Z ( A [ t ]) = Z ( A )[ t ], where Z ( A ) is thecenter of A . Let S ⊂ A [ t ] be any multiplicative subset of monic centricpolynomials. Define the full subcategory of S -torsion endomorphismsEnd SA ⊂ End A -THEORETIC TORSION AND THE ZETA FUNCTION 3 whose objects are those ( P, f ) such that p ( f ) = 0 for some p ∈ S . Let A [ t ] S := S − A [ t ]denote the localization of A [ t ] with respect to S . Example . Suppose A is commutative and S ⊂ A [ t ] is the set of allmonic polynomials. Then End SA = End A . If A is an integral domain,then A [ t ] S = A ( t ) is the ring of rational functions. Example . Let S = { t n | n ≥ } . Then End SA = Nil A is the categoryof nilpotent endomorphisms, and A [ t ] S = A [ t, t − ] is the ring of Laurentpolynomials.In § K -theory spaces τ ( t ) : K (End SA ) −→ Ω K ( A [ t ] S )called the higher endomorphism torsion map. Note that in contrastto more familiar kinds of torsion, this induces a map on all homotopygroups, not just π , and has no indeterminacy. Remark . Throughout this paper, we regard higher K -theory as aspace rather than a spectrum. This is only for simplicity – the argu-ments also work for spectra, we just have to include the iterations ofthe S. construction and the compatibility between them.For p ∈ S of degree d , set ˜ p ( t ) := p (1 /t ) t d . Define T ⊂ A [ t ] tobe the set { ˜ p | p ∈ S } . Then T is a multiplicative subset of centricpolynomials with leading term 1. In § higher K -theoryzeta function ζ ( t ) : K (End SA ) −→ Ω K ( A [ t ] T ) , following [19]. This zeta function is related to the more classical onesvia the trace map to the topological de Rham-Witt space TR( A ), seeRemark 4.4 and the recent preprints [4, 6].Assume t ∈ S . Then there is a canonical ring homomorphism A [ t − ] T → A [ t ] S , that is induced by t − /t . Therefore, if we substitute t by t − , thezeta function ζ ( t − ) can also be regarded as a map into Ω K ( A [ t ] S ).Informally, our main result is the following statement. Theorem A.
The loop product of the maps ζ ( t − ) , τ ( t ) : K (End SA ) −→ Ω K ( A [ t ] S ) is homotopic to the map that sends each endomorphism ( P, f ) to theloop defined by multiplication by t : P [ t ] S → P [ t ] S , where P [ t ] S is themodule P ⊗ A A [ t ] S . JOHN R. KLEIN AND CARY MALKIEWICH
Taking S ⊂ A [ t ] to be all monic centric polynomials and passing to π recovers Milnor’s formula (1). For a more precise statement, seeTheorem 5.1 below.The forgetful functor ( P, f ) P induces a map K (End SA ) → K ( A ).Let e K (End SA ) be its homotopy fiber; then e K (End SA ) → K (End SA ) ishomotopically a retract. We show that t : K (End SA ) −→ Ω K ( A [ t ] S ) ishomotopically trivial when restricted to e K (End SA ). Let˜ τ ( t ) , ˜ ζ ( t − ) : e K (End SA ) −→ Ω K ( A [ t ] S )denote the respective restrictions of τ ( t ) and ζ ( t − ) to e K (End SA ). Corollary B.
As homotopy classes of maps, ˜ τ ( t ) and ˜ ζ ( t − ) are addi-tive inverses. In particular, the homotopy class of ˜ τ ( t ) admits a canon-ical factorization through Ω K ( A [ t − ] T ) . In § p : E → B, f : E → E ) , in which p is fibration with finitely dominated fibers, and f is a self-mapcovering the identity map of B . Hence, we may regard E as fiberwisespace over B equipped with fiberwise N -action. Both the higher endo-morphism torsion τ ( p, f ) and zeta function ζ ( p, f ) are invariant withrespect to N -equivariant fiberwise weak equivalence.For any preselected ring homomorphism Z [ π ( E )] → A and subset S ⊂ A [ t ] as above, the torsion invariant τ ( p, f ) is a composition of theform B c −→ K (End SA ) τ ( t ) −−→ Ω K ( A [ t ] S ) , where the map c is induced by the classifying space data for the familyof endomorphisms.In particular, taking B = S n to be the n -sphere, the torsion producesan invariant in K n +1 ( A [ t ] S ). In Theorem 8.5 we prove the following: Theorem C.
There is a pair ( p : E → S , f : E → E ) for which the component of τ ( p, f ) in K ( Q ( t )) is non-trivial. In the above, the fibration p has fiber S n ∨ S n and the monodromy θ : S n ∨ S n → S n ∨ S n has degree given by the matrix Q := (cid:20) −
11 1 (cid:21) . -THEORETIC TORSION AND THE ZETA FUNCTION 5 As I − Q is invertible over the integers, E is a non-trivial homologycircle. The self-map f : E → E is the map of mapping tori induced by θ . Remark . The example in Theorem C is explicit. In § K ( Q ( t )).In § K n +2 ( Q ( t )) for n > n +1)-spheres equipped with a fiberwise self-map. However, these examplesare not explicit and their existence relies on recent work by AndrewSalch [26].1.1. Further results.
The paper also contains three appendices.Appendix A recalls two “fundamental theorems” of endomorphism K -theory. The first relates K (End SA ) to Ω K ( A [ t ] S ) and can be derivedwithout much effort from Waldhausen’s generic fibration theorem. Wesketch a proof since we could not find it stated in the literature. Theother result, which is due to Grayson and is better known, relates the re-duced K -theory space e K (End SA ) to Ω K ( A [ t − ] T ). We make little claimto originality except to note that the zeta function provides Grayson’sequivalence and the endomorphism torsion provides the other one.In Appendix B we give an explicit description in terms of Wald-hausen’s S. construction of the boundary map ∂ of the K -theory local-ization sequence associated with the localization A [ t ] → A [ t ] S . In thiscase there is a short exact sequence0 → K n +1 ( A [ t ]) → K n +1 ( A [ t ] S ) ∂ −→ K n (End SA ) → A andEnd SA . This is somewhat non-trivial due to the discrepancy betweenthe categories of perfect complexes over the rings A [ t ] and A . Acknowledgements.
The first author was partially supported by SimonsCollaboration Grant 317496. The second author was partially sup-ported by NSF DMS-2005524.2.
Preliminaries K -theory of Waldhausen categories. Let C be a Waldhausencategory, i.e., a category with cofibrations co C and weak equivalences JOHN R. KLEIN AND CARY MALKIEWICH wC . It is customary to denote the cofibrations by and the weakequivalences by ∼ −→ . When the isomorphisms of C are used as the weakequivalences, we denote wC by iC .The K -theory of ( C, co C, wC ) is the based loop spaceΩ | wS.C | , where | wS.C | denotes the realization of a simplicial pointed category wS.C . The objects of wS k C are given by filtered objects of C of theform A • := A A · · · A k together with explicit choices of quotient data A ij := A j /A i ∈ C i ≤ j . See [27] or [29] for more details.In particular, wS C is the trivial category with object ∗ , and wS C = wC . As a result there is a canonical map(2) | wC | −→ Ω | wS.C | which is adjoint to the inclusion of the one-skeleton in the simplicialdirection. Example . Let A be an associative unital ring.Let P ( A ) be the category of finitely generated projective (right) A -modules. A cofibration of P ( A ) is a monomorphism P → Q whosecokernel is projective. A weak equivalence of P ( A ) is an isomorphism.The category of weak equivalences is denoted iP ( A ). Then the K -theory space of A is K ( A ) := Ω | iS.P ( A ) | . This coincides up to equivalence with the Quillen K -theory of P ( A )considered as an exact category [27, § Example . Let End A = End( P ( A ))be the category in which an object is a pair ( P, f ), where • P a finitely generated projective A -module, and • f : P → P is an A -linear endomorphism.A morphism ( P, f ) → ( Q, g ) of End A is a homomorphism h : P → Q such that gh = hf .We equip End A with a Waldhausen category structure induced bythe forgetful functor End A → P ( A ). That is, a morphism h : ( P, f ) → ( Q, g ) is a cofibration/weak equivalence if and only if h : P → Q is cofi-bration/weak equivalence in P ( A ). The K -theory of endomorphisms is K (End A ) := Ω | iS. End A | . -THEORETIC TORSION AND THE ZETA FUNCTION 7 Example S -torsion) . Each endomorphism(
P, f ) is an A [ t ]-module with t acting by f . Here A [ t ] is the polynomialring, the free A -algebra on one generator t subject to the relation that t commutes with A . Given a polynomial p ( t ), we denote its action on P by p ( f ). Note that if p ( t ) is in the center Z ( A [ t ]) = Z ( A )[ t ], then p ( f ) is an A [ t ]-linear map.Let S ⊆ A [ t ] be any multiplicatively closed set of monic centric poly-nomials. Then we define End SA = End( P ( A )) S be the full subcategoryof End A on the objects ( P, f ) such that p ( f ) = 0 for some p ( t ) ∈ S .Equivalently, the localized module S − P is zero.When A is commutative and S consists of all monic polynomials,then End SA = End A . In other words, every endomorphism in End A ismonic-torsion. This is because we could take p ( t ) to be the charac-teristic polynomial of any extension of f to a finite free module. Thisargument does not apply outside the commutative case because wecannot define determinants in the usual way. Example . The full subcategory Aut A ⊂ End A consisting of those objects ( P, f ) such that f is an automorphism iscalled the automorphism category of A . The K -theory of automor-phisms is K (Aut A ) := Ω | iS. Aut A | . Example . Let φ : A → B be a ring homomor-phism. Then B has the structure of an A - A -bimodule. The operation( P, f ) ( P ⊗ A B, f ⊗ K (End A ) → K (End B ) . For the first segment of the paper these examples will be sufficient,but see § Categorical loops.
For a category C let End( C ) be the categorywhose objects are self-maps φ a : a → a of C and whose morphisms( a, φ a ) → ( b, φ b ) are maps u : a → b which fit into a commutativediagram a u / / φ a (cid:15) (cid:15) b φ b (cid:15) (cid:15) a u / / b . There is a “coassembly” map of spaces(3) e : | End( C ) | −→ L | C | , JOHN R. KLEIN AND CARY MALKIEWICH where L | C | is the unbased loop space of | C | . This map commuteswith the projection of each space to | C | , by sending ( a, φ a ) to a or byevaluating a loop at the basepoint.The coassembly map may be described in the following way: each k -simplex in the realization | End( C ) | is represented by a functor f : [1] × [ k ] → C . Here, [ k ] is the category with objects i for 0 ≤ i ≤ k and a unique morphism from i → j whenever i ≤ j . The functor f also has the property that its restrictions to the two objects 0 , ∈ [1]give the same functor [ k ] → C . We take realization to get a map(∆ /∂ ∆ ) × ∆ k −→ | C | whose adjoint is a map ∆ k → L | C | . It is easyto check this rule respects faces and degeneracies, giving a well-definedmap | End( C ) | −→ L | C | . Furthermore, restricting to the basepoint of S gives precisely the map | End( C ) | −→ | C | that forgets the endomorphism φ a .Applying the above to iS k Aut A = End( iS k P ( A )), we obtain a map | iS k Aut A | → L | iS k P ( A ) | which becomes a map of simplicial spaces when k is varied. Taking therealization of the adjoint, we obtain a map S × | iS. Aut A | → | iS.P ( A ) | ,which after looping once defines a map S × K (Aut A ) → K ( A ) . Taking the adjoint again defines a coassembly map on K -theory(4) c : K (Aut A ) → LK ( A ) . Remark . Note that this K -theory coassembly map does not extendto K (End A ); it is necessary to use isomorphisms to actually get a loopin the target space. More generally, we can define such a coassemblymap for any Waldhausen category C , and it takes the form K (Aut C ) → LK ( C ) , where Aut C is the category of objects c ∈ C and maps c ∼ −→ c in wC .2.3. Composition.
Composition of endomorphisms defines a functorEnd( P ( A )) × P ( A ) End( P ( A )) ◦ −→ End( P ( A )) . It is straightforward to check that it is strictly associative and has unitgiven by the functor P ( A ) → End A taking each module to its identitymorphism. Since K -theory commutes with pullbacks, this gives maps K (End A ) × K ( A ) K (End A ) ◦ −→ K (End A )making the space K (End A ) into a fiberwise topological monoid overthe space K ( A ). Restricting further to automorphisms makes K (Aut A )into a fiberwise topological group over K ( A ). -THEORETIC TORSION AND THE ZETA FUNCTION 9 Remark . This composition operation does not induce a composi-tion on the homotopy fiber e K (End A ) because K (End A ) → K ( A ) is nota fibration. This would lead to a contradiction anyway: using the char-acteristic polynomial map e K (End( A )) → (1 + tA [[ t ]]) × , we would getan incorrect identity p ( f ◦ g ) = p ( f ) p ( g ), where p is the characteristicpolynomial of an endomorphism f .Similarly, the free loop space LK ( A ) is a fiberwise grouplike A ∞ -space over K ( A ), with composition ∗ given by composing loops (usingthe little intervals operad). Note that LK ( A ) → K ( A ) is a fibrationand so this does give a composition on the homotopy fiber, namely theusual composition on the based loop space Ω K ( A ). Lemma 2.8.
The following square commutes up to homotopy: K (Aut A ) × K ( A ) K (Aut A ) ◦ (cid:15) (cid:15) / / LK ( A ) × K ( A ) LK ( A ) ∗ (cid:15) (cid:15) K (Aut A ) / / LK ( A ) . Here, the horizontal maps are induced by the coassembly map (4) .Proof.
We first describe a natural homotopy making the following squarecommute. | End( C ) | × | C | | End( C ) | ◦ (cid:15) (cid:15) / / L | C | × | C | L | C | ∗ (cid:15) (cid:15) | End( C ) | / / L | C | , where the horizontal maps are induced by the coassembly map (3).Since realization commutes with pullbacks, each k -simplex in the top-left is described as a pair of functors [1] × [ k ] → C that restrict to thesame functor [ k ] → C on the endpoints of [1]. These together specifya functor [2] × [ k ] → C , which realizes to a map(5) ∆ × ∆ k → | C | . The left-bottom route of the commuting diagram is the restriction of(5) to the face [0 <
2] of ∆ , while the top-right route is the restrictionto the two faces [0 <
1] and [1 < provides a homotopy betweenthem. So if we fix a choice for this homotopy without reference to ∆ k ,we get a homotopy that commutes with face and degeneracy maps andtherefore gives a well-defined homotopy on the entire realization. Now we apply this to C = iS k P ( A ) for each k ≥ | iS k Aut A | × | iS k P ( A ) | | iS k Aut A | ◦ (cid:15) (cid:15) / / L | iS k P ( A ) | × | iS k P ( A ) | L | iS k P ( A ) | ∗ (cid:15) (cid:15) | iS k Aut A | / / L | iS k P ( A ) | , where the horizontal maps arise from (3). Taking realizations, the leftcolumn becomes the left column in the statement of the proposition,while the right column admits a strictly commuting map to the rightcolumn in the statement of the proposition. Along these identifica-tions, the realization of our previous homotopy becomes the desiredhomotopy. (cid:3) Higher endomorphism torsion
Some extensions of the polynomial ring.
Let A be an as-sociative unital ring, and let A [ t ] be the ring of polynomials withcoefficients in A . By convention, this means that the indeterminate t commutes with A . A polynomial is centric if it lies in the center Z ( A [ t ]) = Z ( A )[ t ].Following Grayson [15], fix a multiplicative subset S ⊂ A [ t ] of moniccentric polynomials in A [ t ], containing at least the powers of t . Let A [ t ] S := S − A [ t ]denote the localization of A [ t ] given by the left fractions with respectto S .For each polynomial p ( t ) = t n + a t n − + a t n − + . . . + a n − t + a n ∈ S , let ˜ p ( t ) denote the re-normalized polynomial˜ p ( t ) = 1 + a t + a t + . . . + a n − t n − + a n t n = p (1 /t ) t n . This defines a second multiplicative subset T = { ˜ p ( t ) | p ( t ) ∈ S } ofcentric polynomials having leading coefficient 1. Let A [ t ] T := T − A [ t ] T . The homomorphisms A [ t ] → A [ t ] S and A [ t ] → A [ t ] T are both flatlocalizations [28].The universal property of localization gives canonical, injective ringhomomorphisms(6) A [ t ] T → A [[ t ]] and A [ t − ] T → A [ t ] S → A ( t ) , -THEORETIC TORSION AND THE ZETA FUNCTION 11 where A [[ t ]] is the ring of formal power series over A and A ( t ) is thelocalization of A [ t ] at the set of all central non-zero-divisors. The firstof these homomorphisms is induced by the inclusion A [ t ] → A [[ t ]]. Thesecond is is induced by the homomorphism A [ t − ] → A [ t ] S given by p ( t − ) ( t n p ( t − )) /t n , where n is the degree of p ( t − ). If A is a fieldthen A ( t ) is the field of rational functions; in this case if S consistsof all monic polynomials, then the homomorphism A [ t ] S → A ( t ) is anisomorphism.For a finitely generated projective (right) A -module P we define P [ t ] := P ⊗ A A [ t ] ∼ = ∞ M t =0 P, P [[ t ]] := P ⊗ A A [[ t ]] ∼ = ∞ Y t =0 P. Note that the second isomorphism holds because P is a summand of afinite free module.3.2. Higher endomorphism torsion.
Recall from Example 2.3 thatEnd SA is the category of all endomorphisms of finitely generated projec-tive A -modules that are S -torsion. If ( P, f ) is an object of End SA thenthere is a short exact sequence of of A [ t ]-modules(7) 0 → P [ t ] t − f −−→ P [ t ] → P → , called the characteristic sequence of P . Here t − f is shorthand for1 ⊗ t − f ⊗
1. The action of t on the first two terms is by · t , and onthe third term is by f . Let τ f ( t ) : P [ t ] S t − f −−→ ∼ = P [ t ] S denote the localization of the first map. Note that τ f ( t ) is an isomor-phism, by the assumption that P S = S − P vanishes.The rule ( P, f ) ( P [ t ] S , τ f ( t )) defines an exact functorEnd SA −→ Aut A [ t ] S and therefore a map of K -theory spaces K (End SA ) → K (Aut A [ t ] S ). Wecompose with the coassembly map (4) to obtain the endomorphismtorsion map(8) τ End ( t ) : K (End SA ) −→ LK ( A [ t ] S ) . Notation . When the indeterminate t or the decoration ‘End’ areunderstood, we drop them from that notation, writing τ End = τ End ( t ) = τ ( t ) = τ . Remark . On path components, τ defines a homomorphism τ ∗ : K (End A ) → K ( A [ t ] S ) × K ( A [ t ] S ) . The projection of τ ∗ onto the second factor sends an object ( P, f ) tothe equivalence class of P [ t ] S . The projection onto the first factor isgiven by the equivalence class in K ( A [ t ] S ) of the automorphism t − f .The latter can be viewed as the Reidemeister torsion of the short chaincomplex P [ t ] t − f −−→ P [ t ] with respect to extension P [ t ] → P [ t ] S . Thischain complex is a resolution of the A [ t ]-module P by free modules, sowe can think of this as the Reidemeister torsion of the pair ( P, f ).3.3.
Examples.
Let A be a field and let S ⊂ A [ t ] consist of all monicpolynomials. Then A [ t ] S = A ( t ) is the field of rational functions. Thereis an isomorphism(9) ⊕ p K ∗ ( A [ t ] / ( p )) → K ∗ (End SA ) = K ∗ (End A ) , where p := p ( t ) ranges over the irreducible monic polynomials [25],[14, p. 233], [11, th. 1.4], [22, th. 2.3]. The restriction to the p -thsummand is the map K ∗ ( A [ t ] / ( p )) → K ∗ (End A ) induced by the exactfunctor which sends a finite dimensional vector space V over A [ t ] / ( p )to the pair ( V, t · ).Fix an irreducible monic polynomial p and let u ∈ A [ t ] / ( p ) bea nontrivial unit. Then u ∈ K ( A [ t ] / ( p )) is non-trivial. Its imageˆ u ∈ K (End A ) is the equivalence class of the automorphism u of theendomorphism t of A [ t ] / ( p ). Since (9) is an isomorphism, ˆ u is non-trivial.Let τ ∗ : K (End A ) → K ( A ( t )) be the homomorphism induced bythe endomorphism torsion map τ . Proposition 3.3.
The element τ ∗ (ˆ u ) ∈ K ( A ( t )) is non-trivial.Proof. As ˆ u is non-trivial, the result is an immediate consequence ofTheorem B.1 below which says that the endomorphism torsion map τ : K (End A ) → Ω K ( A ( t )) induces a section to the boundary map ∂ inthe localization sequence0 → K ( A [ t ]) → K ( A ( t )) ∂ −→ K (End A ) −→ . (cid:3) Examples in higher dimensions.
Let A = Q , and p ( t ) ∈ Q [ t ]be an irreducible monic polynomial. Then Q [ t ] / ( p ) is a number field.Let r denote its number of distinct embeddings into R and let r itsnumber of distinct conjugate pairs of embeddings into C . Then by a -THEORETIC TORSION AND THE ZETA FUNCTION 13 theorem of Borel [3], for n ≥ K n ( Q [ t ] / ( p )) ⊗ Q ∼ = n even; Q r + r n ≡ Q r n ≡ . For example, let p ( t ) = t + bt + c have negative discriminant D := b − c . Then Q [ t ] / ( p ) = Q ( √ D ) is a quadratic number field with r = 0 and r = 1. Hence, the abelian group K n +1 ( Q [ t ] / ( p )) has rankone for n ≥
1. Let x ∈ K n +1 ( Q [ t ] / ( p )) be an element of infinite orderand let ˆ x ∈ K n +1 (End Q )be its image with respect to the homomorphism K n +1 ( Q [ t ] / ( p )) → K n +1 (End Q ). Then ˆ x also has infinite order (cf. (9)). By Theorem B.1below, we infer that the endomorphism torsion τ ∗ (ˆ x ) ∈ K n +2 ( Q ( t )) hasinfinite order. 4. The higher zeta function
If (
P, f ) is an object of End SA then we can define a different endo-morphism of finitely generated projective A [ t ]-modules(10) P [ t ] − ft −−−→ P [ t ] . Recall the multiplicative subset T ⊂ A [ t ] of § Lemma 4.1.
After localizing at T , the map − f t is an isomorphism.Proof. Before localization, the map is injective with cokernel f − P ,with t acting by f − . Therefore after localizing at T , it is still injectiveand its cokernel is T − f − P . So it suffices to show that some polyno-mial in T annihilates f − P . By assumption some p ( t ) ∈ S annihilates P , so we take ˜ p ( t ) = t n p ( t − ) ∈ T . The action of this polynomial on f − P is by ˜ p ( f − ) = f − n p ( f ), which vanishes because p ( f ) vanishes on P . (cid:3) Let ζ f ( t ) = (1 − f t ) − : P [ t ] T ∼ = −→ P [ t ] T denote the inverse map. Again the rule ( P, f ) ( P [ t ] T , ζ f ( t )) definesan exact functor End SA −→ Aut A [ t ] T and therefore a map of K -theory spaces K (End SA ) → K (Aut A [ t ] T ). Wecompose with the coassembly map (4) to obtain(11) ζ ( t ) : K (End SA ) → LK ( A [ t ] T ) . Definition 4.2.
The map (11) is called the K -theoretic zeta function .The operation t t − defines an involution A ( t ) → A ( t ) whichrestricts to a ring isomorphism A [ t ] T → A [ t − ] T . As t ∈ S , we also havethe homomorphism A [ t − ] T → A [ t ] S . Assembling these, we obtain amap on K -theory that we call ζ ( t − ): ζ ( t − ) : K (End SA ) ζ ( t ) −−→ LK ( A [ t ] T ) t t − −−−→ LK ( A [ t − ] T ) → LK ( A [ t ] S ) . Remark . The term “zeta function” is motivated by the observationthat if we extend scalars along the embedding A [ t ] T → A [[ t ]] we getthe expression ζ f ( t ) = ∞ X k =0 f k t k : P [[ t ]] ∼ = −→ P [[ t ]] , where f k t k is shorthand for f ◦ k ⊗ t k . Note that this further variantrequires no torsion assumptions, defining a map(12) ζ ( t ) : K (End A ) → LK ( A [[ t ]]) . The zeta function is therefore more general than endomorphism torsion.
Remark . The zeta function is related to the topological de Rham-Witt complex, studied by Hesselholt [16], Betley and Schlichtkrull [1]and Lindenstrauss and McCarthy [19]. In particular, the latter authorsexamine the map ζ ( t ) : e K (End A ) −→ Ω K ( A [[ t ]]) , where e K (End A ) is the homotopy fiber of the forgetful map K (End A ) → K ( A ). Generalizing this to coefficients, this induces an equivalenceof Taylor towers, whose homotopy limits are W ( A ) = TR( A ). Wetherefore get a commutative diagram up to homotopy e K (End A ) ζ / / trc % % ▲▲▲▲▲▲▲▲▲▲ Ω K ( A [[ t ]]) “ det ” x x qqqqqqqqqqq TR( A ) . where the lower-left map is the trace to TR as in [4, 19]. Howeverthere doesn’t appear to be an interesting lift of Milnor’s identity tothis setting, because we don’t have an obvious ring map from A [ t ] S or A [ t − ] S to A [[ t ]]. -THEORETIC TORSION AND THE ZETA FUNCTION 15 The map t . To an object (
P, f ) we can also associate the endo-morphism t : P [ t ] → P [ t ] . Note that this does not depend on f , because it is the torsion of thezero endomorphism. As t ∈ S , it becomes an isomorphism once weextend scalars to A [ t ] S . So the assignment ( P, f ) ( P [ t ] S , t ) defines amap t : K (End SA ) → K ( A ) → K (Aut A [ t ] S ) c −→ LK ( A [ t ] S ) . A generalization of Milnor’s identity
We are now in a position to generalize Milnor’s identity on the levelof K -theory. Recall that ∗ denotes the operation LX × X LX → X that composes loops with the same basepoint. Theorem 5.1.
Assume that A is an associative unital ring and S ⊂ A [ t ] is a multiplicative subset of monic centric polynomials that contains t . Then the composition K (End SA ) ζ ( t − ) ∗ τ ( t ) −−−−−−→ LK ( A [ t ] S ) is homotopic to the map t , as maps over K ( A ) → K ( A [ t ] S ) .Proof. We first rewrite ζ ( t − ). By the naturality of the coassemblymap (4), it is equal to the composite K (End SA ) ζ ( t ) −−→ K (Aut A [ t ] T ) t t − −−−→ K (Aut A [ t ] S ) c −→ LK ( A [ t ] S ) . The first map sends (
P, f ) to ( P [ t ] T , (1 − f t ) − ). The second map re-interprets this as ( P [ t − ] T , (1 − f t − ) − ) and then extends scalars to A [ t ] S . On the underlying module this gives A [ t ] S ⊗ A [ t − ] T ( A [ t − ] T ⊗ A P ) ∼ = A [ t ] S ⊗ A P = P [ t ] S . Along this identification, the endomorphism (1 − f t − ) − is sent to theinverse of 1 ⊗ (1 − f t − ) = t − ⊗ ( t − f ). This inverse is t ⊗ ( t − f ) − .With this modification, the three maps τ ( t ) , ζ ( t − ) and t are alldescribed as maps K (End SA ) → K (Aut A [ t ] S ), followed by coassembly.After applying the forgetful map K (Aut A [ t ] S ) → K ( A [ t ] S ), all threesend ( P, f ) to P [ t ] S , up to canonical isomorphism. On K -theory spacesthey are therefore canonically homotopic. In fact, by fixing models forthe extension of scalars of each module, we can modify them up tohomotopy so that their projections to K ( A [ t ] S ) strictly coincide. Wetherefore have three maps of the form K (End SA ) → K (Aut A [ t ] S ) overthe same map K (End SA ) → K ( A [ t ] S ). As a result of Lemma 2.8, it is now enough to prove that ζ ( t − ) ◦ τ ( t ) = t as fiberwise maps K (End SA ) −→ K (Aut A [ t ] S )over K ( A [ t ] S ). For an object ( P, f ) ∈ End SA , we have shown earlier inthe proof that ζ f ( t − ) produces the endomorphism( t − f ) − ⊗ t : P ⊗ A A [ t ] S → P ⊗ A A [ t ] S . On the other hand, τ f ( t ) produces the endomorphism ( t − f ) ⊗ ⊗ t as desired. (cid:3) We now explain how Theorem 5.1 is a lift of Milnor’s original iden-tity. Note that Milnor only considers the case when A is a field and S consists of all monic polynomials. In this case the identity lies in theunits of the ring of rational functions A ( t ).We argue as follows: projecting from free loops to based loops, ourresult implies the same identity ζ ( t − ) ∗ τ ( t ) ∼ t as maps K (End SA ) −→ Ω K ( A [ t ] S ) . Taking π gives the identity ζ ( t − ) ∗ τ ( t ) = t as maps of K -groups K (End SA ) −→ K ( A [ t ] S ) , where ∗ is the abelian group structure on K . Applying the determinantmap K ( R ) → R × makes our definitions of ζ ( t − ) and τ ( t ) agree withtheir original definitions, namely the determinant of (1 − f t − ) − and t − f , respectively. We recover the identity ζ ( t − ) τ ( t ) = t χ ( P ) as maps K (End SA ) −→ ( A [ t ] S ) × . Remark . The notation t χ ( P ) = det( t ) is slightly misleading becausein general, P is not necessarily a free module, so the determinant of · t on P [ t ] might not be a monomial of the form t χ ( P ) . However it isguaranteed to be a monomial as soon as A is an integral domain andtherefore has no nontrivial idempotents; see [12].For a different simplification, recall that t : K (End A ) → LK ( A [ t ] S )factors through the forgetful functor K (End A ) → K ( A ). Consequently,if we take the identity from Theorem 5.1 and pass to homotopy fibersover K ( A ) → K ( A [ t ] S ), the map t becomes trivial (constant map tothe basepoint) and so we get the simpler identity ˜ ζ ( t − ) ∗ ˜ τ ( t ) ∼ ζ and ˜ τ refer to the restrictions of ζ and τ to maps e K (End SA ) −→ Ω K ( A [ t ] S ) . -THEORETIC TORSION AND THE ZETA FUNCTION 17 Corollary 5.3. ˜ ζ ( t − ) and ˜ τ ( t ) are additive inverses in the homotopycategory. Chain complexes
In this section we describe a chain complex model for K (End SA ).We introduce this model in order to construct endomorphism torsionfor topological families of endomorphisms in §
8. The construction andproof of Milnor’s identity for the chain complex model is essentiallyidentical to the constructions given above, but passing to chain com-plexes gives us the ability to relate the higher torsion to the boundarymap of the K -theory localization sequence; see Appendix B.For simplicity, we consider chain complexes in non-negative degreesonly, though the results of this section also hold for chain complexes inall degrees.Again let A be an associative unital ring. Let Ch( A ) denote the cate-gory of chain complexes of A -modules and chain maps. Let Ch A -proj ( A )denote the subcategory of degreewise projective complexes. This hasthe structure of a Waldhausen category: a cofibration is a levelwiseinjective map with projective cokernel, and a weak equivalence is aquasi-isomorphism.Given a map f : P. → Q. of A -chain complexes, we let T ( f ) . denoteits mapping cylinder. It is characterized by the property that maps T ( f ) . → D. correspond to pairs of maps g : P. → D. , h : Q. → D. and a chain homotopy g ∼ hf . Similarly let C ( f ) . = T ( f ) ./P. be themapping cone.If f : P. → P. is an endomorphism, let T + ( f ) . denote its map-ping telescope. Let T − ( f ) . denote the reverse mapping telescope, con-structed as T + ( f ) but appending each new segment to the front ofthe cylinder, rather than the back. A map T + ( f ) → D. consists ofmaps g i : P. → D. for i ≥ g i ∼ f g i − . A map T − ( f ) → D. consists of the same except the maps are indexed over i ≤
0. Note that the canonical map P. → T + ( f ) . is a localization by f and the canonical maps P. → T − ( f ) . → P. are quasi-isomorphisms. Remark . The reverse mapping telescope T − ( f ) has an endomor-phism sh − that shifts one slot to the left. With respect to this A [ t ]-action the collapse T − ( f ) . → P. is A [ t ]-linear, in fact it is essentially arewriting of the characteristic sequence (7) for chain complexes. Sim-ilarly, T + ( f ) has an endomorphism that shifts one slot to the right,making it a chain complex version of the dual characteristic sequence(10) that we used to define the zeta function. We say a complex of A -modules is strictly perfect if it is bounded (i.e.nonzero in only finitely many degrees) and finitely generated projectiveover A at each level. A complex is perfect if it is quasi-isomorphic toa strictly perfect complex. This condition is equivalent to being com-pact in the derived category, so it is preserved by extensions, retracts,kernels, and cokernels. The perfect complexes define a Waldhausensubcategory Ch perf ( A ) := Ch A -proj A -perf ( A ) ⊆ Ch A -proj ( A ) . Next we consider chain complexes with an endomorphism. This isthe same thing as an A [ t ]-chain complex, but a priori, there is morethan one way to characterize when such a complex is perfect over A ,or is S -torsion. The following lemmas eliminate this ambiguity. Lemma 6.2.
The following two conditions are equivalent. We say an A [ t ] -complex P. is A -perfect if either one holds. • There is a zig-zag of A [ t ]-linear quasi-isomorphisms to an A -strictly perfect complex. • There is a zig-zag of A -linear quasi-isomorphisms to an A -strictly perfect complex. Proof.
Clearly the first implies the second. Suppose P. is an A [ t ]-chain complex satisfying the second condition, and denote the action of t ∈ A [ t ] by f : P. → P. . Without loss of generality the zig-zag is a singlequasi-isomorphism g : C. → P. where C. is a strictly perfect A -chaincomplex. Lift f along this equivalence to an A -linear endomorphism f ′ : C. → C. such that the following square commutes up to a chosenchain homotopy. C. f ′ / / g (cid:15) (cid:15) C. g (cid:15) (cid:15) P. f / / P. Let T − ( f ′ ) be the reverse mapping telescope discussed above, as an A [ t ]-chain complex. Let T − ( f ′ ) → C. be the collapse onto the end anddefine T − ( f ′ ) → P. by sending the i th copy of C. to P. by f | i | ◦ g , andextending over the 1-skeleton using the chosen chain homotopy for thesquare above (composed with iterates of f ). Both of these maps are A [ t ]-linear quasi-isomorphisms, giving the desired zig-zag from P. tothe strictly perfect A -chain complex C. . (cid:3) Lemma 6.3. If P. is A -perfect then it is A [ t ] -perfect. -THEORETIC TORSION AND THE ZETA FUNCTION 19 Proof.
In light of the previous lemma, without loss of generality P. is A -strictly perfect and we must show it admits a quasi-isomorphismfrom an A [ t ]-strictly perfect complex. Simply take the reverse mappingtelescope T − ( f ) . → P. ; it is obtained from the bicomplex P. [ t ] t − f −−→ P. [ t ](cf. (7)) and is therefore A [ t ]-strictly perfect. (cid:3) Again, fix a multiplicative central set of monic polynomials S ⊆ A [ t ]. Lemma 6.4.
For an A [ t ] -perfect complex P. , the following five condi-tions are equivalent. We say P. is S -torsion if any of them hold. • There exists p ( t ) ∈ S such that p ( f ) is equivalent to 0 in the A [ t ]-derived category. • There exists p ( t ) ∈ S such that p ( f ) is equivalent to 0 in the A -derived category. • There exists p ( t ) ∈ S such that p ( f ) = 0 on H ∗ ( P. ). • There exists p ( t ) ∈ S such that p ( f ) − H ∗ ( P. ) = 0. • There exists p ( t ) ∈ S such that p ( f ) − P. is acyclic. Proof.
Clearly each one implies the next. So assume that p ( f ) − P. isacyclic. Without loss of generality P. is A [ t ]-strictly perfect. The local-ization p ( f ) − P. can be modeled as the mapping telescope T + ( p ( f )), sothis telescope is acyclic. Therefore the inclusion P. → T + ( p ) . admits achain homotopy to zero, giving an extension to C (0) . → T + ( p ) . , where C (0) . is the mapping cone of the zero map 0 . → P. . The mapping cone C (0) . is also A [ t ]-strictly perfect, so this extension factors through afinite stage of the mapping telescope T n . . This finite stage admits aquasi-isomorphism to P. by collapsing onto the end, and the composite P. → T n . → P. is p ( f ) n . However this factors through the acycliccomplex C (0) . , so it is zero in the A [ t ]-derived category. Therefore p ( t ) n ∈ S acts as 0 on P. in the A [ t ]-derived category. (cid:3) Lemma 6.5. If P. is A [ t ] -perfect and S -torsion then P. is A -perfect.Proof. Let p ( t ) ∈ S be a monic polynomial such that p ( f ) is zero inthe A [ t ]-derived category. Let R. be the two-term complex A [ t ] p −→ A [ t ],whose A [ t ]-module structure extends in a unique way to the structureof a DGA. Note that R. is equivalent as an A -complex to the one-termcomplex A [ t ] /p ∼ = A ⊕ deg p , and therefore R. is perfect over A . Since P is perfect over A [ t ], the extension of scalars P. ⊗ A [ t ] R. is thereforeperfect over A as well. This latter complex is isomorphic to C ( p ) . , themapping cone of p ( f ) : P. → P. . Therefore C ( p ) . is perfect over A . Since P. is A [ t ]-projective and p ( f ) is zero in the A [ t ]-derived cat-egory, p ( f ) : P. → P. is chain null homotopic. Picking a null homo-topy defines an A [ t ]-linear map C ( p ) . → P. that splits the inclusion P. → C ( p ) . . Therefore P. is an A -linear retract of the A -perfect com-plex C ( p ) . , so P. is A -perfect as well. (cid:3) Corollary 6.6. If P. is S -torsion then it is A [ t ] -perfect iff it is A -perfect. Now that the ambiguity has been removed from the definitions, weexplain how these chain complex models are equivalent to the ones ofthe previous sections. Recall that the functor that sends each moduleto the corresponding chain complex supported in degree 0 gives anequivalence K ( A ) = Ω | iS.P ( A ) | ∼ −→ Ω | wS. Ch A -proj A -perf ( A ) | . This is by an application of Waldhausen’s “sphere theorem” [27, thm. 1.71].The main hypothesis can be verified by a method described in Appen-dix C (Proposition C.4). By a standard abuse of notation we let K ( A )refer to either of these spaces.Once we consider endomorphisms and torsion, we have more choicesof model, illustrated in the diagram below.Ch A [ t ]-proj A [ t ]-perf ( A [ t ]) o o Ch A [ t ]-proj A -perf ( A [ t ]) ∼ / / Ch A -proj A -perf ( A [ t ]) ∼ i i End A ∼ o o Ch A [ t ]-proj A [ t ]-perf ( A [ t ]) S ∼ Ch A [ t ]-proj A -perf ( A [ t ]) S ∼ / / Ch A -proj A -perf ( A [ t ]) S ∼ i i End SA ∼ o o The straight maps are all inclusions. The curved map is the reversemapping telescope functor from the proof of Lemma 6.3. The S -superscripts denote the subcategory of S -torsion complexes in the senseof Lemma 6.4. Proposition 6.7.
Every map marked ∼ induces an equivalence on K -theory.Proof. The leftmost equivalence is an equality of categories by Corol-lary 6.6. The arrows in the center are inverses up to equivalence bythe proof of Lemma 6.3. Alternatively, the straight arrows are equiva-lences using the approximation theorem. The remaining inclusions onthe right-hand side are by Waldhausen’s sphere theorem [27, thm. 1.71].In each case the main hypothesis of the theorem is not obvious because -THEORETIC TORSION AND THE ZETA FUNCTION 21 the modules are not finitely generated projective over A [ t ], but thehypothesis does hold (Propositions C.7 and C.8). (cid:3) Corollary 6.8.
The exact functor sending ( P, f ) to the A [ t ] -complex P [ t ] t − f −−→ P [ t ] induces an equivalence K (End SA ) ∼ −→ K (Ch perf ( A [ t ]) S ) . The construction of higher torsion τ and the zeta function ζ can bedefined as in the previous sections, only taking P to be an A -perfectcomplex with an endomorphism, instead of a finitely generated projec-tive A -module with an endomorphism. Theorem 6.9.
The definitions of τ and ζ extend in a natural wayalong the inclusions of categories End SA ⊆ Ch A - proj A - perf ( A [ t ]) S , P ( A [ t ] S ) ⊆ Ch perf ( A [ t ] S ) and the composition K (Ch A - proj A - perf ( A [ t ]) S ) ζ ( t − ) ∗ τ ( t ) −−−−−−→ LK (Ch perf ( A [ t ]) S ) is homotopic to the map t , as maps over K (Ch perf ( A )) → K (Ch perf ( A [ t ]) S ) .Proof. The most significant change in defining the maps is that themap t − f , respectively 1 − f t , is a quasi-isomorphism rather than anisomorphism, because its cofiber is acyclic, not identically zero. Whenproving this for 1 − f t , it is illuminating to refer to f − P as the extensionof scalars P ⊗ A [ t ] A [ t, t − ], and then to observe that inverting p ( f ) and˜ p ( f − ) are the same localization. Then we define the automorphismcategory to consist of chain complexes and self-quasi-isomorphisms.As a result, ζ is not an isomorphism, so it can’t be inverted. Tocorrect for this, instead of defining ζ as (1 − f t ) − , we define it as(1 − f t ) and at the very end apply the flip map to the free loop spaceto accomplish the inversion.It is possible to modify the proof of the identity accommodate thisweaker setup, but it is faster to deduce the identity directly from The-orem 5.1, since all the K -theory spaces involved are equivalent to thoseappearing in Theorem 5.1. (cid:3) The non-linear setting
For a space X , we define a Waldhausen category End X . We con-struct a linearization map on K -theory K (End X ) → K (End Z [ π ] ), where π = π ( X ). For a fixed ring homomorphism Z [ π ] → A and any multiplicativesystem of monic centric polynomials S ⊂ A [ t ], we associate a full sub-category End SX ⊂ End X of S -torsion objects. We then exhibit a lin-earization map K (End SX ) → K (End SA ).7.1. Endomorphisms of retractive spaces.
For each topologicalspace X , let T ( X ) be the category of retractive spaces Y over X .There is a Quillen model category structure on T ( X ) in which the weakequivalences are weak homotopy equivalences and the cofibrations arethe retracts of relative cell complexes. This makes the subcategory R ( X ) ⊂ T ( X ) of cofibrant objects into a Waldhausen category.An object Y ∈ T ( X ) is finite if it is built up from the zero object bya finite number of cell attachments. It is homotopy finite if it is weaklyequivalent to a finite object, and finitely dominated if it is a retract ofa homotopy finite object.Let R fd ( X ) ⊂ R ( X ) be the full subcategory on the finitely dom-inated objects. Then R fd ( X ) together with the above cofibrationsco R fd ( X ) and weak equivalences vR fd ( X ) forms a Waldhausen cate-gory.Let End X = End( R fd ( X )) denote the category of endomorphismsof objects in R fd ( X ). As before, this inherits a Waldhausen structurein which a weak equivalence or cofibration is determined by forgettingthe endomorphisms.7.2. The T. construction. We will define a linearization map back toEnd Z [ π ( X )] , but for this we need a variant of the S. construction due toThomason [27, p. 334]. To each a Waldhausen category ( C, co C, wC )we may associate a simplicial category wT.C . The objects of wT k C aresequences of cofibrations A • := A A · · · A k . A morphism A • → B • consists of maps A i → B i for 0 ≤ i ≤ k suchthat each of the squares A i / / / / (cid:15) (cid:15) A i +1 (cid:15) (cid:15) B i / / / / B i +1 is a homotopy cocartesian square, i.e. the square commutes and themap B i ∪ A i A i +1 → B i +1 is a weak equivalence. The i -th face operatordrops A i from the sequence and the i -th degeneracy operator inserts -THEORETIC TORSION AND THE ZETA FUNCTION 23 the identity map. There is a chain of homotopy equivalences wT.C ≃ ←− wT. + C ≃ −→ wS.C where the middle simplicial category is defined in a way similar to wT.C but where quotient data is included. The left equivalence is given byforgetting quotient data and the right one is given by mapping thesequence A • to the sequence A /A A /A · · · .7.3. Linearization.
Now assume that X is based, path-connected andhas a universal cover e X → X . Let π = π ( X ) be the fundamental groupof X . Under these assumptions we define a linearization map L : End X → End Z [ π ] . For each Y ∈ R fd ( X ), let e Y denote the pullback e Y (cid:15) (cid:15) / / Y (cid:15) (cid:15) e X / / X. Then the reduced singular chain complex C. ( e Y , e X ) is a perfect Z [ π ]-chain complex. The functor L is then the operation that assigns ( Y, f )to ( C. ( e Y , e X ) , f ∗ ).The functor L preserves cofibrations and weak equivalences but doesnot preserve pushouts, so it does not induce a map on the S. con-struction. However L preserves homotopy cocartesian squares, henceit induces a map of simplicial categories wT. End X → wT. End Z [ π ] Taking realization and loops, we obtain the linearization map L : K (End X ) → K (End Z [ π ] ) . Torsion endomorphisms.
Let Z [ π ] → A be any ring homomor-phism and let S ⊂ A [ t ] be a multiplicative subset of monic centric poly-nomials. To define S -torsion for endomorphisms of retractive spaces wesimply apply linearization. Definition 7.1.
An object (
Y, f ) ∈ End X is S -torsion if L ( Y, f ) ∈ End Z [ π ] becomes, after extending scalars to A , an S -torsion endomor-phism of chain complexes in the sense of Lemma 6.4. LetEnd SX ⊂ End X be the full subcategory consisting of the S -torsion objects. It is readily verified that End SX inherits the structure of a Wald-hausen category. Then linearization followed by extension of scalarsinduces a functor L : End SX → End SA , and as above this induces a map on K -theory,(13) L : K (End SX ) → K (End SA ) . Remark . Our definition of End SX differs from that of Levikov [18],who defines S -torsion at the level of suspension spectra rather than atthe chain level.We define the endomorphism torsion τ and zeta function ζ for anyobject ( Y, f ) ∈ End SX by applying linearization (13). These give classesin Ω K ( A [ t ] S ) satisfying the same identity as before. When A is a field,we recover the original form of Milnor’s identity [23, p. 123].8. Families of endomorphisms
In this section we give a few examples of higher classes in endo-morphism K -theory K (End SX ), which by the previous section producehigher classes in K (End SA ) and therefore have higher endomorphismtorsion.8.1. Higher torsion for families.
The essential idea is that to givea map B → K (End X ), where B is some topological space, it is enoughto give a map B → | w. End X | , and such maps correspond to fibrations of the following form. Definition 8.1. A B -family of endomorphisms over X is a retractivespace E over B × X , and an endomorphism f : E → E of retractivespaces, such that the projection p : E → B is a fibration and eachfiber is a finitely dominated retractive space over X . We sometimesabbreviate this by ( p, f ) or just E , but the other data is understood. Proposition 8.2.
Such families, up to the evident notion of weakequivalence, correspond to homotopy classes of maps B → | w. End X | . We omit the proof, which is by classical arguments as in [21]. How-ever we describe the correspondence itself. Given a map to | w. End X | ,we pull back the universal family over | w. End X | to produce a fam-ily over B . This universal family is defined by taking the bar con-struction B ( ι, w End X , ∗ ) where ι : w End X → End X is the functortaking every retractive space Y over X to itself. By the proof of -THEORETIC TORSION AND THE ZETA FUNCTION 25 Quillen’s Theorem B, the projection of this bar construction back to B ( ι, w End X , ∗ ) = | w. End X | is a quasifibration, every fiber of which isa retractive space over X with an endomorphism. In other words, oncewe replace the map to X × | w. End X | by a fibration, it is a | w. End X | -family. Then this is the universal family in the sense that any otherfamily is the pullback of this one along a map B → | w. End X | . Remark . We can also give an explicit inverse to this operation.Given a B -family of endomorphisms, we can regard it as a point inthe classifying space of all B -families up to weak equivalence. Thisclassifying space, as a functor of B , is a contravariant homotopy functorand therefore has a coassembly map, see e.g. [5, §
5] and [20, § B → | w. End X | .Using the universal property of coassembly, we identify this as theinverse to the above correspondence.For each B -family of endomorphisms E → B we call the resultingcompositions Ω K ( A [ t ] S ) B / / K (End SX ) L A / / K (End SA ) τ ❡❡❡❡❡ ζ , , ❨❨❨❨❨ Ω K ( A [ t ] T )the higher endomorphism torsion τ ( E ) and higher zeta function ζ ( E )of the family.8.2. Examples.
We give a few examples of higher endomorphism tor-sion where the base space B = S is the circle, X = ∗ is a point, and A = Q . In this case the interesting part of the endomorphism torsionlies in the group K ( Q ( t )). To give a map S → | w. End S ∗ | it sufficesto select a based space Y with two commuting self-maps f and θ , suchthat θ is a weak equivalence. Example . Let n ≥ θ : S n ∨ S n → S n ∨ S n whose action on n th homology is the invertible matrix(14) Q = (cid:20) −
11 1 (cid:21)
Set f = θ , so that f and θ trivially commute. This determines the S -family of endomorphisms p : E → S , f : E → E. To be more precise, E is the mapping torus of θ , made into a fibrationwith fiber S n ∨ S n . In fact it is a non-trivial homology circle with π ( E ) ∼ = Z (when n ≥ u denote the resulting homotopy class u : S −→ K (End ∗ ) . An unraveling of the construction shows that the composition S u −→ K (End ∗ ) L Q −→ K (End Q ) , where L Q is the linearization map of § θ ∗ : ( e C. ( S n ∨ S n ) , f ∗ ) ≃ −→ ( e C. ( S n ∨ S n ) , f ∗ ) , where e C. ( S n ∨ S n ) is the reduced total singular complex over Q of S n ∨ S n . We then simplify this by the zig-zag of quasi-isomorphisms... (cid:15) (cid:15) ... (cid:15) (cid:15) ... (cid:15) (cid:15) e C n +1 ( S n ∨ S n ) (cid:15) (cid:15) e C n +1 ( S n ∨ S n ) = o o (cid:15) (cid:15) / / (cid:15) (cid:15) e C n ( S n ∨ S n ) (cid:15) (cid:15) e Z n ( S n ∨ S n ) o o (cid:15) (cid:15) / / Z ⊕ Z (cid:15) (cid:15) e C n − ( S n ∨ S n ) (cid:15) (cid:15) o o (cid:15) (cid:15) / / (cid:15) (cid:15) ... ... ...to the chain complex ( Z ⊕ Z )[ n ] with θ ∗ , f ∗ acting by the matrix (14).Alternatively, we can write this chain complex as ( Q [ t ] / ( t − t + 1))[ n ]with θ ∗ , f ∗ acting by multiplication by t . The resulting class θ ∗ ∈ K (End Q ) is therefore the non-trivial unit( − n t ∈ ( Q [ t ] / ( t − t + 1)) × ∼ = K ( Q [ t ] / ( t − t + 1))by § Theorem 8.5.
The endomorphism torsion invariant of the above pair ( p, f ) is non-trivial in K ( Q ( t )) .Example . Another example is given by the matrix R := (cid:20) −
11 0 (cid:21) . -THEORETIC TORSION AND THE ZETA FUNCTION 27 In this case the associated fibration p : E → S with fiber S n ∨ S n is arational homology circle: H ∗ ( E ) ∼ = Z ∗ = 0 , , Z / ∗ = n, R for both the clutch-ing data θ and the fiberwise endomorphism f . The unit is then t ∈ Q [ t ] / ( t + 1), giving rise to a non-trivial element of K ( Q ( t )) which isthe endomorphism torsion of ( p, f ).In this example, the fibration can be chosen as a fiber bundle withmonodromy θ : S n ∨ S n → S n ∨ S n defined by θ ( x, y ) = ( r ( y ) , x ), inwhich r : S n → S n is the reflection r ( y , . . . , y n ) = ( − y , . . . , y n ). Ob-serve that θ is 4-periodic. Example . Consider the trefoil knot K : S ⊂ S . This is a fiberedknot, so one has a smooth fiber bundle p : E → S where E is the knot complement and the fiber V , a Seifert surfacefor K , is a torus with an open disk removed. The geometric mon-odromy θ : V → V is a diffeomorphism which restricts to the identityon ∂V = S . Moreover, θ is 6-periodic [13]. With respect to a suit-able choice of basis for H ( V ; Z ), the homological monodromy matrixis given by (14). Since θ commutes with itself, it induces a fiberwisediffeomorphism f : E → E that covers the identity map of S . Again,the endomorphism torsion τ ( p, f ) ∈ K ( Q ( t )) is non-trivial, by theargument appearing in Example 8.4. Example . There is a variant of the Example 8.6 in which the fi-bration E → S is a smooth fiber bundle over the circle with fiber S n × S n and monodromy map θ : S n × S n → S n × S n , defined by θ ( x, y ) = ( r ( y ) , x ). The endomorphism torsion invariant is again non-trivial; we omit the details.8.3. Higher dimensional examples.
We briefly outline how one canindirectly construct higher dimensional examples with non-trivial en-domorphism torsion in K ∗ ( Q ( t )) in every even degree ≥
4. The detailsof this construction depend on a recent result due to Andrew Salch andgo well beyond the scope of the current work.Let S be the sphere spectrum. Let S [ i ] be a Gaussian sphere , i.e. astructured ( A ∞ ), connective S -algebra such that • The ring π ( S [ i ]) is isomorphic to the Gaussian integers Z [ i ]; • the homology H ∗ ( S [ i ]; Z ) is concentrated in degree zero; • S [ i ] is weakly equivalent to S ∨ S as an S -module.Such an S -algebra exists [26] but we assume this without proof.For a structured ring spectrum R we let M k ( R ) be the derived R -module endomorphisms of a wedge of k -copies of R . Then GL k ( R ) isdefined by as a pullbackGL k ( R ) / / (cid:15) (cid:15) M k ( R ) (cid:15) (cid:15) GL k ( π ( R )) / / M k ( π ( R ))i.e., those endomorphisms which are invertible up to homotopy. ThenGL k ( R ) is a topological monoid and one has stabilization maps GL k ( R ) → GL k +1 ( R ). Let B GL( R ) be the colimit of B GL k ( R ) under stabilization.The algebraic K -theory of R is K ( R ) = K ( π ( R )) × B GL( R ) + (cf. [2, p. 67]). When R = S [ i ] the map K ( S [ i ]) → K ( Z [ i ])is a rational homotopy equivalence [17, lem. 2.4].According to [3], the abelian group K n +1 ( Z [ i ]) ∼ = K n +1 ( Q [ i ]) hasrank one for n ≥
1. Consequently, the abelian group K n +1 ( S [ i ]) hasrank one for n ≥
1. Represent a non-torsion element x ∈ K n +1 ( S [ i ])by a map S n → B GL k ( S [ i ]) + for k sufficiently large, and form thehomotopy pullback Σ / / (cid:15) (cid:15) B GL k ( S [ i ]) (cid:15) (cid:15) S n +1 / / B GL k ( S [ i ]) + . Then Σ is a homology (2 n + 1)-sphere and the top horizontal map isrepresented unstably by a fibration p : E → Σ whose fiber is a wedge ofan even number of spheres. Furthermore “multiplication by i ” inducesa fiberwise endomorphism f : E → E . By construction, the endomor-phism torsion τ ( E ) ∈ [Σ , Ω K ( Q ( t ))] = K n +2 ( Q ( t )) is non-trivial byTheorem B.1. Appendix A. Fundamental theorems of endomorphism K -theory The following result involves ideas of Quillen [24], Grayson [15] andWaldhausen [27]. We make no claim to originality. As before, A is an -THEORETIC TORSION AND THE ZETA FUNCTION 29 associative ring, T, S ⊂ A [ t ] are the multiplicative subsets of centricpolynomials of § e K (End SA ) the homotopy fiber of the forgetfulmap K (End SA ) → K ( A ). Theorem A.1.
There are natural homotopy equivalences (1) Ω K ( A [ t ] S ) ≃ Ω K ( A [ t ]) × K (End SA ) , and (2) Ω K ( A [ t − ] T ) ≃ Ω K ( A ) × e K (End SA ) .Remark A.2 . We aren’t aware of a proof in the literature of the part(1) and so we provide a sketch below. The equivalence is in fact theendomorphism torsion map τ ( t ) on the second factor (Theorem B.1).Part (2) is due to Grayson [15]; a non-linear version was also provedby Levikov [18]. It appears the equivalence in (2) is the zeta function ζ ( t − ) on the second factor, but the proof requires a translation of [15]to Waldhausen categories and goes beyond the scope of this paper.Combining Theorem A.1 with the fundamental theorem of algebraic K -theory [14, p. 236] relates the K -theory of the two localizations: Corollary A.3.
There is a canonical splitting Ω K ( A [ t ] S ) ≃ Ω K ( A [ t − ] T ) × e K (Nil A ) , where Nil A ⊂ End A is the full subcategory of nilpotent endomorphisms. In particular, if A is a regular ring then e K (Nil A ) = 0 and the othertwo terms are equivalent. Proof of (1) . We apply the generic fibration theorem of Waldhausen[27, thm. 1.64.] to the category of perfect A [ t ]-chain complexes. Letthe v -notion of weak equivalence be quasi-isomorphism, and let the w -notion be those maps whose homotopy cofiber is S -torsion in thesense of Lemma 6.4. In particular, a chain complex is w -acyclic iff it is S -torsion.By Corollary 6.8, the K -theory of the w -acyclic objects is identifiedwith K (End SA ). Therefore the sequence in the generic fibration theorembecomes K (End SA ) → K ( A [ t ]) → K ( A [ t ]; w ) , the first map sending ( P, f ) to the two-term A [ t ]-chain complex P [ t ] t − f −−→ P [ t ]. By the additivity theorem this map is null-homotopic, hence weget an equivalenceΩ K ( A [ t ]; w ) ≃ K (End SA ) × Ω K ( A [ t ]) . As A [ t ] S is flat over A [ t ], tensoring is exact, so it induces a map K ( A [ t ]; w ) → K ( A [ t ] S ) . This is well-known to give an equivalence after looping (though not anisomorphism on π ), which finishes the proof. (cid:3) Appendix B. Endomorphism torsion and boundary map
In the previous appendix we recalled a fiber sequence of the form K (End SA ) −→ K ( A [ t ]) → K ( A [ t ] S ) . In this appendix we show the endomorphism torsion splits the bound-ary map. We use this for nonvanishing results for τ (e.g. Proposition3.3), and to show that τ gives the equivalence in part (1) of the funda-mental theorem of endomorphism K -theory (Theorem A.1).This is not the only situation in which a torsion invariant splitsthe boundary map of a K -theory long exact sequence. In general, theboundary map of such a sequence sends every class represented by a w -weak equivalence to its homotopy cofiber, which is w -acyclic. However,in most cases this analysis only applies to path components, and thetorsion map has indeterminacy. In the case of endomorphism torsion,the splitting occurs at the space level and therefore holds for everygroup K i .As in the proof of Theorem A.1, let C be the category of perfect A [ t ]-chain complexes, let v be the quasi-isomorphisms and let w be themaps whose homotopy cofibers are S -torsion in the sense of Lemma 6.4.Let C w ⊂ C be the full subcategory of w -acyclic objects. Waldhausen’sgeneric fibration then fits into a commuting diagram of the formΩ K ( C, w ) ∼ (cid:15) (cid:15) ∂ / / K ( C w , v ) / / K ( C, v ) / / K ( C, w ) (cid:15) (cid:15) Ω K ( A [ t ] S ) K (End SA ) ∼ O O / / K ( A [ t ]) / / K ( A [ t ] S ) . The second vertical map sends (
P, f ) to the two-term complex P [ t ] t − f −−→ P [ t ], while the outside vertical maps extend scalars along A [ t ] → A [ t ] S .We observe that the endomorphism torsion naturally lifts along thisextension of scalars. The lifted torsion map comes from the categoricaloperation that sends ( P, f ) to the one-term complex P [ t ] with self- w -equivalence t − f , instead of the complex P [ t ] S with self-isomorphism t − f . Composing this with coassembly and then projecting to basedloops gives a map τ : K (End SA ) → LK ( C, w ) → Ω K ( C, w ) -THEORETIC TORSION AND THE ZETA FUNCTION 31 that agrees with the endomorphism torsion along the equivalenceΩ K ( C, w ) ∼ −→ Ω K ( A [ t ] S ) . Theorem B.1.
The following commutes in the homotopy category Ω K ( C, w ) ∂ / / K ( C w , v ) K (End SA ) , ∼ O O τ g g ◆◆◆◆◆◆◆◆◆◆ i.e., the endomorphism torsion map is a section to the boundary map.Proof. We first recall from [27] the definition of the boundary map.Let wC denote the subcategory of maps that are both cofibrationsand weak equivalences. Let w.C denote the simplicial category that atsimplicial level k is length k flags of maps in w , B • := B ∼ B ∼ · · · ∼ B k . Let F. ( C, C w ) denote the same construction but equipped addition-ally with choices of quotient B j /B i for all i ≤ j (see [27, p.344] fordetails). This forms a simplicial Waldhausen category. The forgetfulmaps F k ( C, C w ) → w k C are equivalences of categories, hence equiva-lences on the v. -nerve. Together with the equivalence vw.C ≃ vw.C and the swallowing lemma, this gives an equivalence of spaces | v.F. ( C, C w ) | ∼ −→ | v.w.C | ∼ −→ σ | w.C | . Here the map σ takes a square grid of objects ( A ij ) and maps to the flagof diagonal objects ( A ii ) and the maps between them. Equivalently, it isthe most obvious map from the double realization of v.w.C to the singlerealization of w.C , which for each p × q grid of morphisms, subdividesthe corresponding copy of ∆ p × ∆ q and maps each of the simplices tothe evident corresponding simplex in | w.C | .We have a commutative square F. ( C, C w ) (cid:15) (cid:15) ∂ / / S.C w (cid:15) (cid:15) F. ( C, C ) / / S.C where the horizontal maps are induced by the operation B • B • /B . After passing to S. constructions, the square is homotopy cartesian[27, cor. 1.56.] and vS.F. ( C, C ) is contractible. Combining this withthe generic fibration theorem, we have two homotopy pullback squares | w.S.C | o o ∼ | v.S.F. ( C, C w ) | (cid:15) (cid:15) ∂ / / | v.S.S.C w | (cid:15) (cid:15) / / | w.S.S.C w | (cid:15) (cid:15) | v.S.F. ( C, C ) | / / | v.S.S.C | / / | w.S.S.C | . The bottom-left and top-right terms are contractible, giving a four-term homotopy fiber sequence( | w.S.C | ≃ | vS.F. ( C, C w ) | ) ∂ −→ | vS.S.C w | → | vS.S.C | → | wS.S.C | . In particular, the map from | w.S.C | to | v.S.S.C w | may be identifiedwith a (one-fold delooping of) the boundary map Ω | wS.C | → | vS.C w | in the fiber sequence | vS.C w | → | vS.C | → | wS.C | . We need to check that the identification | w.S.C | ≃ Ω | w.S.S.C | fromthe above four-term fiber sequence agrees with the usual one. Thisfollows from the commutativity of the following diagram, because thecommuting square at the very bottom gives the usual map | w.C | → Ω | w.S.C | by [27, Lemma 1.5.2], and the composite along the left-handside agrees in the homotopy category with the identification | w.S.C | ≃| vS.F. ( C, C w ) | from above. For simplicity we have suppressed one copyof S. on every term. v.F. ( C, C w ) / / + + ❱❱❱❱❱❱ ∼ (cid:15) (cid:15) w.S.C w ) ) ❙❙❙❙❙❙❙ v.F. ( C, C ) ∼ (cid:15) (cid:15) / / w.S.Cw.F. ( C, C w ) / / + + ❱❱❱❱❱❱ O O ∼ w.S.C w O O ∼ ) ) ❙❙❙❙❙❙❙ w.F. ( C, C ) / / w.S.Cw.C + + ❲❲❲❲❲❲❲❲❲❲ / / ∗ ) ) ❚❚❚❚❚❚❚❚❚ w.F. ( C, C ) / / w.S.C Finally, to prove that ∂ ◦ τ is the desired equivalence, we extend thedomain of ∂ by adding on the zero map K ( C, w ) → K ( C w , v ), giving -THEORETIC TORSION AND THE ZETA FUNCTION 33 a map out of the free loop space as indicated below. LK ( C, w ) ∂ / / K ( C w , v ) K (End SA ) ∼ O O τ f f ◆◆◆◆◆◆◆◆◆◆ It suffices to prove that this modified diagram commutes. To provethis we rewrite the modified boundary map as the topmost route inthe following diagram. L | w.S.C | o o ∼ L | v.S.F. ( C, C w ) | ∂ / / L | v.S.S.C w | + + Ω | v.S.S.C w | o o | vS.C w | ∼ o o | i.S. End SA | τ h h ◗◗◗◗◗◗◗◗◗◗◗◗ O O ✤✤✤ ∼ ❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞❞ Then we define a dotted map making both triangular regions in thediagram commute up to homotopy. It arises from an exact functorEnd SA → F ( C, C w )with the following description. Given ( P, f ) we regard P [ t ] t − f −−→ P [ t ]as a map of one-term A [ t ]-complexes. We take its reverse mappingcone T − ( t − f ) as in §
6, though in contrast to that section we use theexisting t -action, not the action through sh − . This makes T − ( t − f )a two-term complex of A [ t ]-modules, with an acyclic cofibration sh − whose quotient is the two-term complex P [ t ] t − f −−→ P [ t ]. This describesthe desired exact functor to F ( C, C w ):( P, f ) (cid:16) T − ( t − f ) sh − −−→ T − ( t − f ) , T − ( t − f ) / sh − ∼ = ( P [ t ] t − f −−→ P [ t ]) (cid:17) . It produces a map from End SA to simplicial level 1 of F. ( C, C w ), withthe property that the two faces give the same map to F ( C, C w ), henceon realizations it gives a map to the free loop space.Now we check the two triangular regions commute. The one on theright is fairly straightforward: both routes arise in the same way fromthe exact functor End SA → S C w ∼ = C w taking ( P, f ) to the complex P [ t ] t − f −−→ P [ t ]. For the region on the left,we recall that the weak equivalence is a composite of the two maps L | w.S.C | o o ≃ σ L | w.w.S.C | o o ≃ L | v.S.F. ( C, C w ) | . Unraveling the definition of σ , the composite arises from the naturaltransformation of exact functorsEnd SA × [1] → C sending ( P, f ) to the map of chain complexessh − : T − ( t − f ) → T − ( t − f ) . We collapse these mapping cylinders onto their ends to get a simplermap sending (
P, f ) to the map of chain complexes P [ t ] t − f −−→ P [ t ] . This matches precisely the definition of τ . The collapsing operationmodifies our original map by a homotopy, so we conclude the triangularregion on the left commutes up to homotopy. (cid:3) Appendix C. The sphere theorem for
End A and End SA In this appendix we recall a construction in homological algebra thatallows us to factor each map X → Y of perfect chain complexes as X → X m +1 → X m +2 → . . . → X n ∼ −→ Y where each X q /X q − has homology that is finitely generated projectiveand concentrated in degree q . We then generalize this construction sothat it applies to A -perfect chain complexes over A [ t ], i.e. to chaincomplexes of endomorphisms, with or without S -torsion. This is usedin § K (End A ) and K (End SA )are equivalent to the classical models (Proposition 6.7).Let A be an associative ring. Let f : X → Y be a map of chain com-plexes of A -modules, P a projective A -module, and k : P → H n ( Y, X )any A -linear map. Lemma C.1.
There is a factorization X → X ′ → Y in which H q ( X ′ , X ) is P in degree n and otherwise, and along this identification the in-duced map P → H n ( Y, X ) agrees with k .Proof. Recall that H ∗ ( Y, X ) = H ∗ ( C ( f )) where C ( f ) is the mappingcone. We have C ( f ) n ∼ = Y n ⊕ X n − with boundary map given by theboundary map of Y , the negated boundary of X , and f : X n − → Y n − .Therefore an n -cycle consists of an ( n − x ∈ Z n − ( X ) and an n -chain y ∈ Y n such that ∂y = f ( x ). In other words, a map from a“( n − X and an extension to an “ n -disc” in Y . -THEORETIC TORSION AND THE ZETA FUNCTION 35 Since P is projective, k lifts to a map P → Z n ( C ( f )), giving acommuting diagram P h (cid:15) (cid:15) g / / Y n∂ (cid:15) (cid:15) Z n − ( X ) f / / Z n − ( Y ) . Set X ′ n = X n ⊕ P , with the boundary map on P given by h , and X ′∗ = X ∗ in all other degrees. It is immediate that this is a chaincomplex and there is a factorization of f into maps of chain complexes X → X ′ → Y , the latter map defined on P by g . The relative homologyof ( X ′ , X ) is also clearly P , and the isomorphism is split by P → C ( X → X ′ ) n ∼ = ( X n ⊕ P ) ⊕ X n − p (0 , p, h ( p )) . (The latter term is needed for this map to land in the cycles, andtherefore induce a map to homology.) Therefore the resulting map P → C ( X → X ′ ) n → C ( f ) n sends p to ( g ( p ) , h ( p )) as desired. (cid:3) As a result, by the long exact sequence0 → H n +1 ( Y, X ) → H n +1 ( Y, X ′ ) → P k −→ H n ( Y, X ) → H n ( Y, X ′ ) → , if we replace X by X ′ , the new relative homology H ∗ ( Y, X ′ ) agreeswith the old relative homology H ∗ ( Y, X ) in all degrees except n , whereit is the cokernel of k , and degree n + 1, where it is an extension of H n +1 ( Y, X ) by the kernel of k . In particular, if k is surjective then thisprocedure kills H n ( Y, X ) and only changes H n +1 ( Y, X ) in the process.We recall two more preliminaries. Let H ( A ) the category of A -modules that admit a finite resolution by finitely-generated projectivemodules. Lemma C.2.
A module is in H ( A ) iff it occurs as the sole nonzerohomology group of a perfect complex.Proof. The forward implication is obvious. If our module is the lonehomology group of a strictly perfect complex, and the lowest nonzerogroup of the complex P is beneath the lowest homology group, weinclude a copy of P → P into the bottom of the complex and take thecofiber, giving a new perfect complex that is shorter but has the samehomology. After performing this move finitely many times, we arriveat the desired resolution for our module. (cid:3) Lemma C.3. [29, II.7.7.1] If H has a length ( n + 1) resolution byfinitely generated projective A -modules and → K → P → H → isshort exact with P finitely generated projective, then K has a resolutionof finitely generated projective A -modules of length max(0 , n ) . Now we may verify the main hypothesis of the sphere theorem.Recall that a map of chain complexes X → Y is m -connected if H q ( Y, X ) = 0 for q ≤ m . Proposition C.4.
Given an m -connected map X → Y of A -perfect A -chain complexes, there is a factorization X → X m +1 → X m +2 → . . . → X n ∼ −→ Y where each X q /X q − has homology that is finitely generated projectiveover A and concentrated in degree q .Proof. The mapping cone has lowest homology in degree ( m +1). As themapping cone is perfect, its lowest homology group is finitely generated,therefore we have a surjective map P → H n ( Y, X ) where P is finitelygenerated free. The procedure of Lemma C.1 produces X m +1 suchthat X m +1 /X has homology P concentrated in degree ( m + 1) and X m +1 → Y is ( m + 1)-connected. We repeat until we have exhaustedall of the relative homology groups of the original map. Then C ( f )has only one homology group remaining, so it lies in H ( A ) by LemmaC.2. By Lemma C.3, if we continue the procedure then after finitelymany steps the homology group will be finitely generated projective, inwhich case we apply Lemma C.1 once more with P as the last remaininghomology group. (cid:3) Now we modify this argument to work for endomorphisms, and en-domorphisms with torsion. In each case, an additional trick is needed.
Lemma C.5.
Lemma C.1 remains true if, instead of being projective, P has a length-one resolution by projective modules → P i −→ P j −→ P → . Proof.
We lift the map P j −→ P k −→ H n ( Y, X ) to P → C ( f ) n and applythe procedure of Lemma C.1 to this map. This gives the lower squarebelow. We then add P as shown and define the map b to Y n +1 bynoting that the composite to Y n is zero on homology, therefore mustland in the boundaries, and therefore has a lift because P is projective. -THEORETIC TORSION AND THE ZETA FUNCTION 37 The total modified complex and its map to Y now look like X n +1 ⊕ P ∂ ⊕ i (cid:15) (cid:15) ( f,b ) / / Y n +1 ∂ (cid:15) (cid:15) X n ⊕ P ∂,h ) (cid:15) (cid:15) ( f,g ) / / Y n∂ (cid:15) (cid:15) X n − f / / Y n − This gives a factorization X → X ′′ → Y of A -perfect A [ t ]-chain com-plexes where H ∗ ( X ′′ , X ) is P concentrated in degree n , and the result-ing map to H n ( Y, X ) is k . So we again get the same conclusions as inLemma C.1 for the effect of this operation on relative homology. (cid:3) Remark
C.6 . This argument breaks down if the resolution of P islonger, because one begins to encounter obstructions in the homologygroups of Y . Proposition C.7.
Given an m -connected map X → Y of A -perfect A [ t ] -chain complexes, there is a factorization in A [ t ] -chain complexes X → X m +1 → X m +2 → . . . → X n ∼ −→ Y where each X q /X q − has homology that is finitely generated projectiveover A and concentrated in degree q . (Therefore every term of thefactorization is A -perfect.)Proof. The mapping cone has lowest homology in degree ( m + 1), andis finitely generated over A . Therefore there is a surjective A -linearmap k : P → H m +1 ( Y, X ) where P is a finitely generated projective A -module (with no A [ t ]-action).The key observation is that we can always endow such a P withan A [ t ]-action such that k is A [ t ]-linear. Without loss of generality P is free, and then we define the action one basis element at a time,by applying k , applying the t action in H m +1 ( Y, X ), then taking a liftalong k . We then extend to the rest of P to form an A -linear map α : P → P such that kα = tk . This makes P into an A [ t ]-module suchthat k is A [ t ]-linear.Then by the characteristic sequence, P has a length-one resolutionby projective A [ t ]-modules. By Lemma C.5, we can therefore form thedesired factorization through X m +1 . The rest of the proof now proceedsjust as in Proposition C.4. (cid:3) Proposition C.8.
Given an m -connected map X → Y of A -perfect, S -torsion A [ t ] -chain complexes, there is a factorization in A [ t ] -chaincomplexes X → X m +1 → X m +2 → . . . → X n ∼ −→ Y where each X q /X q − has homology that is S -torsion, finitely generatedprojective over A and concentrated in degree q . (Therefore every termof the factorization is A -perfect and S -torsion.)Proof. Since H m +1 ( Y, X ) is A [ t ]-finitely generated and S -torsion, thereis a single element p ( t ) ∈ S such that p ( t ) acts by zero on H m +1 ( Y, X ).Then we get a surjective A [ t ]-linear map ( A [ t ] /p ( t )) ⊕ i → H m +1 ( Y, X ).Observe that ( A [ t ] /p ( t )) ⊕ i ∼ = A ⊕ i (deg p ) is finitely generated free as an A -module, and S -torsion as an A [ t ]-module, so we may kill it as beforeusing Lemma C.5 without leaving the category of perfect S -torsioncomplexes. We continue this trick for the remaining steps, except thefinal step where the last remaining homology group is finitely generatedprojective, in which case (just as in Proposition C.4) we take P to bethat last remaining homology group. (cid:3) References [1] Stanislaw Betley and Christian Schlichtkrull,
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