A counterexample to vanishing conjectures for negative K -theory
aa r X i v : . [ m a t h . K T ] J un A COUNTEREXAMPLE TO SOME RECENT CONJECTURES
AMNON NEEMAN
Abstract.
In a 2006 article Schlichting conjectured that the negative K– theory ofany abelian category must vanish. This conjecture was generalized in a 2019 articleby Antieau, Gepner and Heller, who hypothesized that the negative K– theory of anycategory with a bounded t– structure must vanish.Both conjectures will be shown to be false. Contents
0. Introduction 11. The t –structure on the category Ac b ( E ) 22. The natural map D b (cid:2) Ac ( E ) ♥ (cid:3) −→ Ac b ( E ) 63. The K –theoretic consequences 114. Vector bundles on nodal curves 135. The counterexample 20References 220. Introduction
Let E be any idempotent-complete exact category. We may form the category Ac b ( E ),whose objects are the acyclic bounded cochain complexes in E . In Lemma 1.2 andRemark 1.3 we prove that Ac b ( E ) always has a bounded t –structure. We eventually showthat, if E = V ect ( Y ) is the category of vector bundles on a projective curve Y with onlysimple nodes as singularities, then there is an injective map K − ( E ) −→ K − (cid:2) Ac b ( E ) (cid:3) .Since there are known examples of nodal curves for which K − (cid:2) V ect ( Y ) (cid:3) = 0, thisprovides a counterexample to Antieau, Gepner and Heller’s [1, Conjecture B].More generally: let E be any idempotent-complete exact category. In Proposition 3.3we produce a homotopy fiber sequence K (cid:2) Ac b ( E ) (cid:3) / / K ( E ⊕ ) / / K ( E ) Mathematics Subject Classification.
Primary 19D35, secondary 18E30, 14F05.
Key words and phrases.
Derived categories, t –structures, homotopy limits.The research was partly supported by the Australian Research Council. where E ⊕ is E with the split exact structure. Thus the vanishing of K n (cid:2) Ac b ( E ) (cid:3) for n < K n ( E ⊕ ) −→ K n ( E ) must be isomorphisms forall n <
0. It is entirely possible that there are many more counterexamples out there;the one computed in this article is the case of projective nodal curves. More precisely:if E = V ect ( Y ) with Y a projective nodal curve, we prove that K − ( E ⊕ ) = 0. But thereare known examples where K − ( E ) = 0.Following Beilinson, Bernstein and Deligne [4, Proposition 3.1.10], if a triangulatedcategory T with a t –structure comes from a model and has suitable “filtered” versions,then there is a natural functor F : D b (cid:2) T ♥ (cid:3) −→ T b , from the bounded derived category ofthe heart of T to the bounded part T b ⊂ T . And what’s important here is that the proofgoes by a way that lifts to models. If we apply this to Ac b ( E ) we deduce an induced mapin K –theory of the form K (cid:2) Ac b ( E ) ♥ (cid:3) −→ K (cid:2) Ac b ( E ) (cid:3) . And to show that the map in K –theory is a homotopy equivalence, it suffices to prove that F : D b (cid:2) Ac b ( E ) ♥ (cid:3) −→ Ac b ( E )is an equivalence of triangulated categories. In Proposition 2.4 we show that the functor F : D b (cid:2) Ac b ( E ) ♥ (cid:3) −→ Ac b ( E ) is an equivalence if and only if the exact category E ishereditary, meaning Ext i ( E, E ′ ) = 0 for all E, E ′ ∈ E and i ≥ F is an equivalence. Therefore K − (cid:2) Ac b ( E ) ♥ (cid:3) = 0, giving acounterexample to Schlichting [15, Conjecture 1 of Section 10], which is also Antieau,Gepner and Heller [1, Conjecture A]. Acknowledgements.
The author would like to thank Ben Antieau, Ching-Li Chai,Bernhard Keller, Henning Krause, Peter Newstead, Sundararaman Ramanan and ChuckWeibel for helpful comments and improvements on earlier incarnations of the manuscript.1.
The t –structure on the category Ac b ( E ) Notation 1.1.
Let E be an idempotent-complete exact category, and let K ( E ) be thecategory whose objects are the cochain complexes of objects in E and whose morphismsare the homotopy equivalence classes of cochain maps. Let Ac ( E ) be the full subcate-gory of acyclic complexes. The full subcategories Ac − ( E ) ⊂ K − ( E ), Ac + ( E ) ⊂ K + ( E )and Ac b ( E ) ⊂ K b ( E ) are the obvious bounded versions . We remind the reader of thedefinition of acyclicity: a cochain complex · · · ∂ i − / / E i − ∂ i − / / E i ∂ i / / E i +1 ∂ i +1 / / · · · possibly bounded, is declared acyclic if there exist admissible short exact sequences0 / / K i α i / / E i β i / / K i +1 / / In the Introduction we followed the notation of Schlichting [15], where Ac b ( E ) is a model category. Inalmost all of the article we will follow the notation of Krause [8], where Ac b ( E ) stands for the associatedtriangulated category. COUNTEREXAMPLE TO SOME RECENT CONJECTURES 3 such that ∂ i = α i +1 ◦ β i . The derived categories D ? ( E ) are defined to be the Verdierquotients K ? ( E ) / Ac ? ( E ), for ? being b , − , + or the empty restriction.Note that we are assuming E idempotent-complete, and [12, Lemma 1.2] proves that Ac ( E ) is a thick subcategory of K ( E ). The fact that Ac − ( E ) ⊂ K − ( E ), Ac + ( E ) ⊂ K + ( E )and Ac b ( E ) ⊂ K b ( E ) are all thick subcategories is older, it may essentially be found inThomason and Trobaugh [18, 1.11.1 (see also Appendix A)]. See also [12, Remark 1.10]for a brief synopsis of the argument in Thomason-Trobaugh.We will usually write E ∗ as a shorthand for the object · · · ∂ i − / / E i − ∂ i − / / E i ∂ i / / E i +1 ∂ i +1 / / · · · in K ( E ). Lemma 1.2.
Let E be an idempotent-complete exact category, and let Ac ( E ) be thesubcategory of acyclics as in Notation 1.1. Define the full subcategories Ac ( E ) ≤ = { E ∗ ∈ Ac ( E ) | E i = 0 for all i > } Ac ( E ) ≥ = { E ∗ ∈ Ac ( E ) | E i = 0 for all i < − } Then the pair (cid:2) Ac ( E ) ≤ , Ac ( E ) ≥ (cid:3) define a t–structure on Ac ( E ) .Proof. The containments Σ Ac ( E ) ≤ ⊂ Ac ( E ) ≤ and Ac ( E ) ≥ ⊂ Σ Ac ( E ) ≥ are obviousfrom the definition.Now suppose we are given a morphism from an object E ∗ ∈ Ac ( E ) ≤ to an object F ∗ ∈ Ac ( E ) ≥ . We may represent it by a cochain map · · · ∂ − / / E − ∂ − / / (cid:15) (cid:15) E − ∂ − / / f (cid:15) (cid:15) E / / g (cid:15) (cid:15) / / (cid:15) (cid:15) · · ·· · · / / / / F − e ∂ − / / F e ∂ / / F e ∂ / / · · · The fact that e ∂ ◦ g = 0 says that g must factor uniquely through the kernel of the map e ∂ , which happens to be the map e ∂ − : F − −→ F . Thus we may find a (unique)morphism θ : E −→ F − with g = e ∂ − ◦ θ . But now we have the equalities e ∂ − ◦ f = g ◦ ∂ − = e ∂ − ◦ θ ◦ ∂ − where the first comes from the commutativity implied by the cochain map E ∗ −→ F ∗ , andthe second is by precomposing g = e ∂ − ◦ θ with ∂ − . And, since e ∂ − is a monomorphism(even an admissible monomorphism), it follows that f = θ ◦ ∂ − . Thus θ provides ahomotopy of the cochain map E ∗ −→ F ∗ with the zero map.Next choose any object E ∗ ∈ Ac ( E ), that is a complex · · · ∂ i − / / E i − ∂ i − / / E i ∂ i / / E i +1 ∂ i +1 / / · · · AMNON NEEMAN such that each morphism ∂ i : E i −→ E i +1 has a factorization E i β i −→ K i +1 α i +1 −→ E i +1 as in Notation 1.1. In particular: we may write ∂ − : E − −→ E as a composite E − β − −→ K α −→ E . But now consider the cochain maps · · · ∂ − / / E − ∂ − / / id (cid:15) (cid:15) E − β − / / id (cid:15) (cid:15) K / / α (cid:15) (cid:15) / / (cid:15) (cid:15) · · ·· · · ∂ − / / E − ∂ − / / (cid:15) (cid:15) E − ∂ − / / β − (cid:15) (cid:15) E ∂ / / id (cid:15) (cid:15) E ∂ / / id (cid:15) (cid:15) · · ·· · · / / / / (cid:15) (cid:15) K α / / id (cid:15) (cid:15) E ∂ / / (cid:15) (cid:15) E ∂ / / (cid:15) (cid:15) · · ·· · · − ∂ − / / E − − β − / / K / / / / / / · · · and we leave it to the reader to check that this is isomorphic in Ac ( E ) to a distinguishedtriangle A ∗ −→ E ∗ −→ B ∗ −→ Σ A ∗ , in which obviously A ∗ ∈ Ac ( E ) ≤ and B ∗ ∈ Ac ( E ) ≥ .This completes the proof that the pair (cid:2) Ac ( E ) ≤ , Ac ( E ) ≥ (cid:3) define a t –structure on Ac ( E ). (cid:3) Remark 1.3.
Given a triangulated category T with a t –structure, it is customary todefine the subcategories T − = ∞ [ n =1 T ≤ n , T − = ∞ [ n =1 T ≥− n , T b = T − ∩ T + . In the special case where T = Ac ( E ) and the t –structure is as in Lemma 1.2, the defini-tions give that[ Ac ( E )] − = Ac − ( E ) , [ Ac ( E )] + = Ac + ( E ) , [ Ac ( E )] b = Ac b ( E ) , with Ac ? ( E ) being as in Notation 1.1. It follows that the t –structure on Ac ( E ) restrictsto t— structures on Ac ? ( E ), with ? being each of − , + and b . And all four categorieshave the same heart. Remark 1.4.
The heart of the t –structure of Lemma 1.2 is, by definition, given by theformula Ac ( E ) ♥ = Ac ( E ) ≤ ∩ Ac ( E ) ≥ , and the formula gives that the objects of Ac ( E ) ♥ are the admissible short exact sequences0 / / E − / / E − / / E / / COUNTEREXAMPLE TO SOME RECENT CONJECTURES 5 in the category E . Since Ac ( E ) ♥ is a full subcategory of K ( E ), the morphisms in Ac ( E ) ♥ are homotopy equivalence classes of cochain maps0 / / E − / / (cid:15) (cid:15) E − / / (cid:15) (cid:15) E / / (cid:15) (cid:15) / / F − / / F − / / F / / Remark 1.5.
The abelian category Ac ( E ) ♥ isn’t new, it may be found in Schlichting [15,Lemma 9 of Section 11]. In Schlichting’s presentation this category doesn’t come as theheart of some t –structure, instead it is described as a subcategory of the category Eff( E ) ⊂ Mod- E , whose objects are the effaceable functors in the category Mod- E = Hom (cid:0) E op , A b (cid:1) of additive functors E op −→ A b . Remark 1.6.
It might help to consider the special case where E is an abelian category.The Yoneda map Y : E −→ mod– E embeds E fully faithfully into the category mod– E ⊂ Hom (cid:0) E op , A b (cid:1) of finitely presented functors E op −→ A b . Recall: a functor F : E op −→ A b is finitely presented if there exists an exact sequence Y ( A ) Y ( f ) / / Y ( B ) / / F / / F in the abelian category Hom (cid:0) E op , A b (cid:1) .Auslander’s work tells us that the functor Y : E −→ mod– E has an exact left adjointΛ : mod– E −→ E . The way to compute Λ( F ) is to choose a finite presentation as above,and define Λ( F ) to be the cokernel of the map f : A −→ B . With eff( E ) defined to be thefull subcategory of mod– E annihilated by the functor Λ, Auslander’s formula [2, page 205]goes on to tell us that E is the Gabriel quotient of mod– E by the Serre subcategory eff( E ),see also Krause [8, Theorem 2.2]. In symbols Auslander’s formula ismod– E eff( E ) = E . Krause [8, Corollary 3.2] goes on to give a derived category version of Auslander’sformula, in the derived category the formula becomes D b (cid:0) mod– E (cid:1) D b eff( E ) (cid:0) mod– E (cid:1) = D b ( E ) . Here D b eff( E ) (cid:0) mod– E (cid:1) is the kernel of the functor Λ : D b (cid:0) mod– E (cid:1) −→ D b ( E ), the mapinduced on derived categories by the exact functor of Λ : mod– E −→ E . Concretely theobjects of D b eff( E ) (cid:0) mod– E (cid:1) are the bounded cochain complexes in mod– E whose cohomol-ogy is in eff( E ).Now the category mod– E has enough projectives, in fact the projective objects ofmod– E are precisely the essential image of the functor Y : E −→ mod– E . Not only AMNON NEEMAN that: every object in mod– E has projective dimension ≤
2. To see this take an object F ∈ mod– E and let Y ( A ) Y ( f ) / / Y ( B ) / / F / / F . If K is the kernel in E of the map f : A −→ B , then thesequence0 / / Y ( K ) / / Y ( A ) Y ( f ) / / Y ( B ) / / F / / (cid:0) E op , A b (cid:1) , and it exhibits a projective resolution of F inthe category mod– E of length ≤
2. Thus every object in D b (cid:0) mod– E (cid:1) is isomorphic to abounded projective resolution, and we obtain an equivalence of triangulated categories K b ( E ) ∼ = D b (cid:0) mod– E (cid:1) . The inverse image of D b eff( E ) (cid:0) mod– E (cid:1) under this equivalence is the category Ac b ( E ) ofNotation 1.1, see Krause [8, top of page 674]. Of course the category D b eff( E ) (cid:0) mod– E (cid:1) has an obvious, standard t –structure with heart eff( E ). Thus what we have really donein Lemma 1.2 is prove that this t –structure on Ac b ( E ) exists for every exact category,there is no need to assume the category E abelian in order to produce the t –structure.And for an abelian category E we have an equivalence of categories Ac ( E ) ♥ ∼ = eff( E ).Thus for abelian categories E , the heart of our new t –structure agrees with Auslander’sold subcategory eff( E ) ⊂ mod– E .2. The natural map D b (cid:2) Ac ( E ) ♥ (cid:3) −→ Ac b ( E )Let T be a triangulated category with a t –structure, and let T ♥ be the heart. Un-der mild hypotheses on T , the inclusion T ♥ ֒ → T b can be naturally factored as T ♥ −→ D b ( T ♥ ) F −→ T b , for a triangulated functor F : D b ( T ♥ ) −→ T b . The reader can find themost general known version of such a result in [13, Theorem 5.1]. But in this articlea more useful version will be the older Beilinson, Bernstein and Deligne [4, Proposi-tion 3.1.10], since the proof presented there comes as a quasifunctor of model categoriesand hence induces a map in K –theory.It becomes natural to wonder when the functor F is an equivalence. The next Lemmais a slight variant of [4, Proposition 3.1.16], and gives a necessary and sufficient condition. Lemma 2.1.
Let T be a triangulated category with a t–structure, let T ♥ be the heartof the t–structure, and let F : D b ( T ♥ ) −→ T b be the natural map. The functor F isan equivalence of categories if and only if every object t ∈ T ≤ ∩ T b admits a triangle a −→ t −→ b with a ∈ T ♥ and b ∈ T ≤− . The reader might wish to compare the necessary and sufficient condition above with Lurie’s notionof
See [9, Definition C.5.3.1 and Proposition C.5.3.2].
COUNTEREXAMPLE TO SOME RECENT CONJECTURES 7
Proof.
Let us start with the necessity: if the functor F is an equivalence then it sufficesto produce the triangle in the category D b ( T ♥ ). The object t ∈ D b ( T ♥ ) ≤ is isomorphicto a cochain complex · · · / / T − / / T − / / T − / / T − / / T / / / / · · · with T i ∈ T ♥ , and the cochain maps · · · / / / / (cid:15) (cid:15) / / (cid:15) (cid:15) / / (cid:15) (cid:15) / / (cid:15) (cid:15) T / / / / (cid:15) (cid:15) · · ·· · · / / T − / / T − / / T − / / T − / / T / / (cid:15) (cid:15) / / (cid:15) (cid:15) · · ·· · · / / T − / / T − / / T − / / T − / / / / / / · · · produce the desired triangle a −→ t −→ b .Now for the sufficiency: we assume the existence of triangles a −→ t −→ b as in theLemma, and need to prove the functor F an equivalence.Let n ≥ A and B be objects of T ♥ , and choose any morphism f : A −→ Σ n B . Form the triangle Σ n − B −→ t −→ A −→ Σ n B . The object t belongsto T ≤ ∩ T b , and by hypothesis there exists a triangle a −→ t −→ b with a ∈ T ♥ and b ∈ T ≤− . Now let H : T −→ T ♥ be the usual homological functor. The exact sequence H ( a ) −→ H ( t ) −→ H ( b ) = 0 tells us that a = H ( a ) −→ H ( t ) is an epimorphism in T ♥ , while the exact sequence H ( t ) −→ H ( A ) −→ H (Σ n B ) = 0 says that H ( t ) −→ H ( A ) = A is also an epimorphism. We conclude that the composite a −→ t −→ A isan epimorphism in T ♥ , and the composite a −→ t −→ A f −→ Σ B obviously vanishes.Since the epimorphism a −→ A can be constructed for every f : A −→ Σ n B , Beilinson,Bernstein and Deligne [4, Proposition 3.1.16] teaches us that F must be an equivalenceof categories. (cid:3) We will soon be applying Lemma 2.1 to the case of F : D b (cid:2) Ac ( E ) ♥ (cid:3) −→ Ac b ( E ).Before proceeding we quickly recall Reminder 2.2.
Let E be an idempotent-complete exact category. Our notion of “acycliccomplexes” is designed in such a way that they go to acyclic complexes under every exactembedding of E as a subcategory of an abelian category. There exists an exact embedding i : E −→ A , with A abelian, and such that(i) The functor i reflects admissible short exact sequences.(ii) A morphism f : x −→ y is an admissible monomorphism in E if and only if i ( f ) isa monomorphism in A .(iii) A morphism f : x −→ y is an admissible epimorphism in E if and only if i ( f ) is aepimorphism in A . AMNON NEEMAN
Thomason and Trobaugh [18, Lemma A.7.15] proves that the Gabriel-Quillen embedding i : E −→ A satisfies not only (i) but also (iii). Dually we obtain an embedding i ′ : E −→ B satisfying (i) and (ii). To achieve (i), (ii) and (iii) we take the embedding of E into A × B .For an embedding i : E −→ A satisfying (i), (ii) and (iii), if a bounded cochaincomplex T ∈ D b ( E ) has the property that i ( T ) is acyclic outside an interval [ a, b ], then T is isomorphic in D b ( E ) to a cochain complex0 / / T a / / T a +1 / / · · · / / T b − / / T b / / a, b ]. Definition 2.3.
An idempotent-complete exact category E is called hereditary if in thecategory D b ( E ) the maps E −→ Σ n F vanish whenever E, F ∈ E and n ≥ . Proposition 2.4.
Let E be an idempotent-complete exact category, let Ac ( E ) be thehomotopy category of acyclic complexes as in Notation 1.1, and let the t–structure on Ac ( E ) be as in Lemma 1.2. Let F : D b (cid:2) Ac ( E ) ♥ (cid:3) −→ Ac b ( E ) be the natural functorfrom the bounded derived category of the heart to the bounded part of the t–structure on Ac ( E ) , the subcategory (cid:2) Ac ( E ) (cid:3) b = Ac b ( E ) .Then the functor F is an equivalence if and only if the category E is hereditary. Remark 2.5.
The term “hereditary” goes back to Cartan and Eilenberg [6, Section I.5],where a ring R is called hereditary if every R –module has projective dimension ≤ ≤
1. In Definition 2.3 we simplyextended the classical term to cover exact categories.
Proof.
Suppose the functor F is an equivalence, let n ≥ f : A −→ Σ n B in D b ( E ). We need to show that f vanishes. Complete f in D b ( E ) to a triangle A f −→ Σ n B −→ T . The object T is such that every exact embedding i : E −→ A takes T to a complex acyclic outside the interval [ − n, − T is isomorphic in D b ( E ) to a cochain complex · · · / / / / T − n / / T − n +1 / / · · · / / T − / / T − / / / / · · · And the triangle Σ n B −→ T −→ Σ A can be represented by cochain maps · · · / / / / B / / (cid:15) (cid:15) / / (cid:15) (cid:15) · · · / / (cid:15) (cid:15) / / / / (cid:15) (cid:15) / / · · ·· · · / / / / T − n (cid:15) (cid:15) / / T − n +1 (cid:15) (cid:15) / / · · · / / T − (cid:15) (cid:15) / / T − (cid:15) (cid:15) / / / / · · ·· · · / / / / / / / / · · · / / / / A / / / / · · · COUNTEREXAMPLE TO SOME RECENT CONJECTURES 9
This being a triangle means that the sequence · · · / / / / B / / T − n / / · · · / / T − / / A / / / / · · · is an object t ∈ Ac ( E ) ≤ ∩ Ac b ( E ). Since we are assuming that F is an equivalence,there exists in Ac ( E ) a triangle a ϕ −→ t −→ b , with a ∈ Ac ( E ) ♥ and b ∈ Ac ( E ) ≤− .The morphism ϕ : a −→ t in the category Ac ( E ) may be represented by a cochainmap · · · / / / / (cid:15) (cid:15) A − / / (cid:15) (cid:15) A − / / (cid:15) (cid:15) A / / (cid:15) (cid:15) / / (cid:15) (cid:15) · · ·· · · / / T − / / T − / / T − / / A / / / / · · · The distinguished triangle a ϕ −→ t −→ b tells us that b is homotopy equivalent toCone( ϕ ), the mapping cone on the cochain map ϕ . Requiring that b ∼ = Cone( ϕ ) shouldbelong to Ac ( E ) ≤− amounts to saying that the morphism A ⊕ T − −→ A must be asplit epimorphism in E .Now the morphism a −→ t is isomorphic in Ac ( E ) to the cochain map · · · / / / / (cid:15) (cid:15) A − / / (cid:15) (cid:15) A − ⊕ T − / / (cid:15) (cid:15) A ⊕ T − / / (cid:15) (cid:15) / / (cid:15) (cid:15) · · ·· · · / / T − / / T − / / T − / / A / / / / · · · and thus, without changing the isomorphism class of the map ϕ : a −→ t in the cat-egory Ac ( E ), we may assume that the cochain map is such that A −→ A is a splitepimorphism. Choose a splitting, meaning choose a morphism g : A −→ A such thatthe composite A −→ A −→ A is the identity. Now form in E the pullback square B − / / (cid:15) (cid:15) A g (cid:15) (cid:15) A − / / A Then the composite · · · / / / / (cid:15) (cid:15) A − / / B − / / (cid:15) (cid:15) A / / g (cid:15) (cid:15) / / (cid:15) (cid:15) · · ·· · · / / / / (cid:15) (cid:15) A − / / (cid:15) (cid:15) A − / / (cid:15) (cid:15) A / / (cid:15) (cid:15) / / (cid:15) (cid:15) · · ·· · · / / T − / / T − / / T − / / A / / / / · · · is a cochain map between acyclic complexes · · · / / / / (cid:15) (cid:15) A − / / (cid:15) (cid:15) B − / / (cid:15) (cid:15) A / / / / (cid:15) (cid:15) · · ·· · · / / T − / / T − / / T − / / A / / / / · · · Coming back to the category D b ( E ): the cochain maps · · · / / / / (cid:15) (cid:15) / / · · · / / / / (cid:15) (cid:15) A − / / (cid:15) (cid:15) B − / / (cid:15) (cid:15) / / (cid:15) (cid:15) · · ·· · · / / / / T − n / / (cid:15) (cid:15) · · · / / T − (cid:15) (cid:15) / / T − (cid:15) (cid:15) / / T − (cid:15) (cid:15) / / / / · · ·· · · / / / / / / · · · / / / / / / A / / / / · · · can be viewed as morphisms A −→ Σ − T −→ A composing to the identity. The triangleΣ − T −→ A f −→ Σ n B −→ T is split, and the map f : A −→ Σ n B that we started outwith must vanish in D b ( E ).Suppose now that in the category D b ( E ) any morphism A −→ Σ B vanishes, whenever A, B ∈ E . Any object in t ∈ Ac ( E ) ≤ ∩ Ac b ( E ) of the form0 / / T − / / T − / / T − / / T / / T −→ Σ T − in the category D b ( E ), and it is classical that thismorphism will vanish in D b ( E ) if and only if there is a cochain map of acyclic complexes0 / / (cid:15) (cid:15) / / C − / / (cid:15) (cid:15) C − / / (cid:15) (cid:15) T / / / / T − / / T − / / T − / / T / / a −→ t with a ∈ Ac ( E ) ♥ whose mapping cone lies in Ac ( E ) ≤− .If t ∈ Ac ( E ) ≤ ∩ Ac b ( E ) is more general, meaning of the form0 / / T − n / / T − n +1 / / · · · / / T − / / T / / / / K / / T − / / T − / / T / / COUNTEREXAMPLE TO SOME RECENT CONJECTURES 11 with K the image of T − −→ T − ; since t is acyclic this image is in E . This producesfor us a cochain map0 / / (cid:15) (cid:15) / / C − / / (cid:15) (cid:15) C − / / (cid:15) (cid:15) T / / / / K / / T − / / T − / / T / / · · · / / (cid:15) (cid:15) / / C − / / (cid:15) (cid:15) C − / / (cid:15) (cid:15) T / / · · · / / T − / / T − / / T − / / T / / ϕ : a −→ t in Ac ( E ), with a ∈ Ac ( E ) ♥ and suchthat the mapping cone of ϕ belongs to Ac ( E ) ≤− . (cid:3) The K –theoretic consequences Reminder 3.1.
Let E be an idempotent-complete exact categry. In Notation 1.1 we re-called the categories Ac ? ( E ) ⊂ K ? ( E ) and the Verdier quotient D ? ( E ) = K ? ( E ) / Ac ? ( E ),where ? is any of b , +, − or the empty restriction. Now a special case is the exactcategory E ⊕ . This means we take any idempotent-complete additive category E , andgive it the exact structure where the admissible exact sequences are the split short exactsequences.Specializing to the case of E ⊕ the general definitions of Notation 1.1, an acyclic complex E ∗ ∈ Ac ( E ⊕ ) is a cochain complex · · · ∂ i − / / E i − ∂ i − / / E i ∂ i / / E i +1 ∂ i +1 / / · · · where there exist split short exact sequences0 / / K i α i / / E i β i / / K i +1 / / ∂ i = α i +1 ◦ β i . This makes E i ∼ = K i ⊕ K i +1 , and the complex E ∗ can bedecomposed as a direct sum of complexes · / / / / K i id / / K i / / / / · · · which vanish except in degrees ( i −
1) and i . Hence all objects in Ac ? ( E ⊕ ) are contractible,and are isomorphic to zero in the homotopy category K ? ( E ⊕ ) = K ? ( E ). This makes theVerdier quotient D ? ( E ⊕ ) = K b ( E ⊕ ) / Ac ? ( E ⊕ ) = K ? ( E ) / K ? ( E ) . Reminder 3.2.
Now it’s time to move on to the K– theoretic consequences, which meanswe need to consider model categories as well as the associated triangulated categories.In the remainder of this section we follow the conventions of Schlichting [15]. Thus M will be a category of models, D : M −→ T will be a functor from M to the category T ofsmall triangulated categories, and K will be a functor from M to spectra. And we willassume that if M ′ −→ M −→ M ′′ is an exact sequence in M then K ( M ′ ) / / K ( M ) / / K ( M ′′ )is a homotopy fibration. Recall: the sequence M ′ −→ M −→ M ′′ is declared to be exactin M if the categories D ( M ′ ), D ( M ) and D ( M ′′ ) are all idempotent-complete, and (1)the functor D ( M ′ ) −→ D ( M ) is fully faithful, (2) the composite D ( M ′ ) −→ D ( M ) −→ D ( M ′′ ) vanishes, and (3) the natural map D ( M ) / D ( M ′ ) −→ D ( M ′′ ) is fully faithful,with D ( M ′′ ) being the idempotent-completion of the essential image of D ( M ) / D ( M ′ ).The next result was presaged in Schlichting [15, Proposition 2 in Section 11]. What’sincorrect about Schlicting’s proof of his old proposition could be rephrased as saying thatthe natural functor D b (cid:2) Ac ( E ) ♥ (cid:3) / / Ac b ( E )need not in general be an equivalence; see Proposition 2.4.We begin with the easy Proposition 3.3.
Let E be an idempotent-complete exact category, and let E ⊕ be thecategory E but with the split exact structure. Then there is a homotopy fibration ofnon-connective K–theory-spectra K ( M ′ ) / / K (cid:0) E ⊕ (cid:1) / / K ( E ) where M ′ ∈ M satisfies D ( M ′ ) = Ac b ( E ) .Proof. Let M be the category of biWaldhausen complicial categories as in Thomasonand Trobaugh [18]. We consider the sequence M ′ −→ M −→ M ′′ in M where, inthe notation of Schlichting [15]—which is in conflict with our notation—one would write M ′ = Ac b ( E ), M = Ch b ( E ) and M ′′ = (cid:0) Ch b ( E , Ac b ( E ) (cid:1) . The conflict of notation is thatin [15] Ac b ( E ) is an object of M , whereas our Ac b ( E ) is what in Schlichting’s notationwould be D (cid:2) Ac b ( E ) (cid:3) . Our notation, which [as we have said] clashes with Schlichting,follows Krause [8]. It is impossible to choose a notation which agrees with every previousarticle in the literature.We remind the reader: Schlichting’s notation means that the objects and morphismsof M, M ′′ are the same, both categories have for objects the bounded complexes ofobjects in E and the morphisms are the cochain maps. The weak equivalences in M arethe homotopy equivalences, whereas the weak equivalences in M ′′ are the cochain mapswhose mapping cones are acyclic. The objects of M ′ are the acyclic complexes, and theweak equivalences are as in M . The natural functor D : M −→ T takes the sequence COUNTEREXAMPLE TO SOME RECENT CONJECTURES 13 M ′ −→ M −→ M ′′ to the sequence of triangulated categories Ac b ( E ) −→ K b ( E ) −→ D b ( E ), where Ac b ( E ) is to be understood in our notation, it is a triangulated category.The categories Ac b ( E ), K b ( E ) and D b ( E ) are all known to be idempotent-complete:in the case of D b ( E ) this is by [3, Theorem 2.8]. For K b ( E ), Reminder 3.1 tells us thatthat K b ( E ) = D b ( E ⊕ ), reducing us to the previous case. And for Ac b ( E ) ⊂ K b ( E ) this isbecause we know Ac b ( E ) to be a thick subcategory of the idempotent-complete triangu-lated category K b ( E ), we already mentioned the thickness of Ac b ( E ) as a subcategory of K b ( E ) in Notation 1.1. Hence the sequence M ′ −→ M −→ M ′′ is exact in M . Thereforethe functor K takes M ′ −→ M −→ M ′′ to a homotopy fibration. In this homotopy fibra-tion we have that K ( M ′′ ) = K ( E ), just because D ( M ′′ ) = D b ( E ). And K ( M ) = K ( E ⊕ ),on the grounds that D ( M ) = K b ( E ) = D b ( E ⊕ ). Thus our homotopy fibration becomes K ( M ′ ) −→ K (cid:0) E ⊕ (cid:1) −→ K ( E ). (cid:3) Remark 3.4.
Proposition 3.3 was straightforward, but in combination with Lemma 1.2it becomes remarkable. The homotopy fiber on the map K (cid:0) E ⊕ (cid:1) −→ K ( E ) is identifiedwith K ( M ′ ), and D ( M ′ ) ∼ = Ac b ( E ) has a bounded t –structure.Antieau, Gepner and Heller [1, Conjecture B on page 244], if true, would imply that K − n ( M ′ ) = 0 for all n >
0, and we would deduce that the natural map K − n (cid:0) E ⊕ (cid:1) −→ K − n ( E ) would have to be an isomorphism for all n >
0. But this will be shown to be false,see Example 5.2 below. As it happens in Example 5.2 the exact category E will be heredi-tary, and Proposition 2.4 informs us that the natural map F : D b (cid:2) Ac ( E ) ♥ (cid:3) −→ Ac b ( E ) isan equivalence of categories. From Beilinson, Bernstein and Deligne [4, proof of Proposi-tion 3.1.10] we know that the map F can be realized as D ( f ) for some suitable morphismin M , and hence induces an isomorphism in K –theory. It follows that K − n (cid:2) Ac ( E ) ♥ (cid:3) isalso nonzero for some n >
0, contradicting [1, Conjecture A], which is a restatement ofan older conjecture due to Schlichting [15, Conjecture 1 of Section 10].There exists an abelian category Ac ( E ) ♥ with non-vanishing negative K –theory.4. Vector bundles on nodal curves
We begin by recalling classical facts about smooth projective curves.
Reminder 4.1.
Fix an algebraically closed field k , and let X be a smooth, projectivecurve over k . We allow X to have more than one connected component. A vector bundleon X will mean a locally free sheaf, locally of finite rank. Of course: the rank maydepend on the connected component we’re at.Each vector bundle V on X gives rise to three continuous functionsrank : X −→ N , degree : X −→ Z , slope : X −→ Q . These continuous functions assign a number to each connected component of X ; for therank this number is a positive integer, the degree may be any integer, and the slope isa rational number. The rank is obvious. The degree of a line bundle, on a connectedcomponent X i ⊂ X , is the usual degree—the number of zeros minus the number of poles of a rational section. For a vector bundle V of rank n the degree of V is defined to bethe degree of the line bundle ∧ n V , and of course it will depend on the component. Andthe slope—whose early history will be recalled in Remark 4.2—is defined by the formulaslope ( V ) = degree ( V )rank ( V ) , and we repeat that this fomula yields a rational number for each component of X .We remind the reader that, if f : V −→ V ′ is an injective map of vector bundles ofequal rank, then degree( V ) ≤ degree( V ′ ) with equality if and only if f is an isomorphism.Since for us this fact is key we quickly recall the argument: assume X connected, andlet n = rank( V ) = rank( V ′ ). The determinant ∧ n f : ∧ n V −→ ∧ n V ′ is an injective mapbetween line bundles, and may be viewed as a regular, nonzero section of the line bundle L = ( ∧ n V ′ ) ⊗ ( ∧ n V ) − . Being a regular section of L , the determinant ∧ n f has no poles.The degree of L can be computed as the number of zeros of ∧ n f and must be non-negative—therefore degree( V ′ ) − degree( V ) = degree( L ) ≥
0. Equality is equivalent to ∧ n f having no zeros, which happens if and only if ∧ n f : ∧ n V −→ ∧ n V ′ is an isomorphism.But this is equivalent to f : V −→ V ′ being an isomorphism.And for us the most important consequence is: any injective endomorphism of a vectorbundle V must be an isomorphism. Remark 4.2.
Since I’ve been asked about the history: it’s traditional to assume X irreducible, which we will do in this Remark. Identifying both H ( X ) and H ( X ) with Z , we have that rank( V ) = ch ( V ) and degree( V ) = ch ( V ) are the zeroth and first Cherncharacters of V and go back a long way. The quotient µ ( V ) = degree( V ) / rank( V ) wasfirst explicitly introduced in print in 1969 by Narasimhan and Ramanan [11, beginning ofSection 2]. The name “slope” for the rational number µ ( V ) came later—Ramanan tellsme that it was coined by Quillen, but the first occurrence I can find in print is in the1977 article by Shatz [17, Section 2]. Of course in some sense it all goes back Mumford’s1962 ICM talk [10, Definition on page 529], which discusses the formation of the modulispace of stable vector bundles. Discussion 4.3.
Now it’s time to move on to singular curves—but for simplicity theonly singularities we allow are simple nodes. Let Y be a nodal projective curve, andlet π : X −→ Y be the normalization. Then X is a smooth projective curve as inReminder 4.1. A vector bundle V on Y , which is a locally free sheaf on Y locally of finiterank, pulls back to a vector bundle π ∗ V on X . And by Reminder 4.1 the vector bundle π ∗ V has associated to it rank, degree and slope functions.Let { p , p . . . , p n } be the singular points of Y . Then each p i has two distinct inverseimages in X ; let us call them p ′ i and p ′′ i . Thus for each i we have a commutative diagram The reader wishing to generalize to allow other singular curves can proceed along the lines of Serre [16,Chapter IV, Section 1].
COUNTEREXAMPLE TO SOME RECENT CONJECTURES 15 of schemes Spec( k ) α i / / β i (cid:15) (cid:15) γ i % % ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ X π (cid:15) (cid:15) X π / / Y where the image of α i is p ′ i and the image of β i is p ′′ i . The composite γ i = πα i = πβ i hasimage p i = π ( p ′ i ) = π ( p ′′ i ). For any coherent sheaf W on X we have units of adjunction η ( α i ) : W −→ α i ∗ α ∗ i W and η ( β i ) : W −→ β i ∗ β ∗ i W . Now let V be a coherent sheaf on Y .There is the unit of adjunction η ( π ) : V −→ π ∗ π ∗ V , and we may combine with the unitsof adjunction above to obtain a commutative square γ i ∗ γ ∗ i V = π ∗ α i ∗ α ∗ i π ∗ V = π ∗ β i ∗ β ∗ i π ∗ V π ∗ π ∗ V η ( α i ) o o π ∗ π ∗ V η ( β i ) O O V η ( π ) o o η ( π ) O O And it is a classical fact that, for V a vector bundle on Y , the sequence0 / / V η ( π ) / / π ∗ π ∗ V ⊕ ni =1 (cid:2) η ( α i ) − η ( β i ) (cid:3) / / n M i =1 γ i ∗ γ ∗ i V / / Y . Moreover: this sequence can be used to constructvector bundles on Y . A vector bundle V on Y is uniquely determined by the vectorbundle W = π ∗ V on X , together with the isomorphisms α ∗ i W ∼ = β ∗ i W that arise from thecanonical isomorphism α ∗ i π ∗ V ∼ = β ∗ i π ∗ V . Concretely: given any vector bundle W on X and, for each i , an isomorphism α ∗ i W ∼ = β ∗ i W , we define V to be the kernel of the map ofsheaves π ∗ W ⊕ ni =1 (cid:2) η ( α i ) − η ( β i ) (cid:3) / / n M i =1 (cid:2) γ i ∗ α ∗ i W ∼ = γ i ∗ β ∗ i W (cid:3) where the isomorphism γ i ∗ α ∗ i W ∼ = γ i ∗ β ∗ i W is by applying the functor γ i ∗ to the chosenisomorphism α ∗ i W ∼ = β ∗ i W . By construction V is a coherent sheaf on Y . And checkingthat V is a vector bundle on Y , with the natural map π ∗ V −→ W an isomorphism, islocal in Y in the flat topology. Hence we may do it separately at each singular point p i ∈ Y , and simplify the argument by first completing at p i . This we leave to the reader. Moreover: morphisms of vector bundles f : V −→ V ′ on Y are uniquely determined bythe “descent data” above. The morphism f gives rise to a map of short exact sequences0 / / V η ( π ) / / f (cid:15) (cid:15) π ∗ π ∗ V π ∗ π ∗ f (cid:15) (cid:15) ⊕ ni =1 (cid:2) η ( α i ) − η ( β i ) (cid:3) / / n M i =1 (cid:2) γ i ∗ α ∗ i π ∗ V = γ i ∗ β ∗ i π ∗ V (cid:3) / / ⊕ ni =1 (cid:2) γ i ∗ α ∗ i π ∗ f = γ i ∗ β ∗ i π ∗ f (cid:3) (cid:15) (cid:15) / / V ′ η ( π ) / / π ∗ π ∗ V ′ ⊕ ni =1 (cid:2) η ( α i ) − η ( β i ) (cid:3) / / n M i =1 (cid:2) γ i ∗ α ∗ i π ∗ V ′ = γ i ∗ β ∗ i π ∗ V ′ (cid:3) / / Lemma 4.4.
Let k be an algebraically closed field, let Y be a projective curve over k withonly simple nodes, and let V be a vector bundle on Y . Then any self-map f : V −→ V leads to a canonical decomposition V = V ′ ⊕ V ′′ such that (i) f decomposes as f = f ′ ⊕ f ′′ for maps f ′ : V ′ −→ V ′ and f ′′ : V ′′ −→ V ′′ . (ii) The map f ′ is an isomorphism while the map f ′′ is nilpotent. Remark 4.5.
The beginning of the proof below, which will treat the case where Y issmooth, has similarities with the proof of Fitting’s Lemma, see Jacobson [7, pp. 113-114].Since the category of coherent sheaves on Y isn’t Artinian the arguments aren’t identical;nevertheless the reader might wish to compare the two proofs. Proof.
We first treat the case where Y is smooth. The coherent sheaves Ker( f n ) are anincreasing sequence of subsheaves of the coherent sheaf V , hence must stabilize. Thereexists an integer N ≫ n ≥ N , the inclusion Ker( f n ) ⊂ Ker( f n +1 ) isan isomorphism.Now: for each integer n > / / Ker( f n ) ψ n / / Ker( f n ) / / Ker( f n ) ∩ Im( f n ) / / n ≥ N the map ψ n is an isomorphism. We conclude that, as long as n ≥ N , wemust have Ker( f n ) ∩ Im( f n ) = 0.Next note that Ker( f n ) and Im( f n ) are coherent subsheaves of V . They must betorsion-free, and torsion-free coherent sheaves on a smooth curve are vector bundles.Choose any n ≥ N and consider the natural composable morphismsIm( f n ) / / V / / Im( f n ) COUNTEREXAMPLE TO SOME RECENT CONJECTURES 17
The kernel of the composite is zero, since the map V −→ Im( f n ) has kernel Ker( f n )which intersects Im( f n ) ⊂ V trivially. Thus the composite is an injective endomorphismof the vector bundle Im( f n ) and must be an isomorphism. We conclude that Im( f n ) is adirect summand of V , more precisely we learn that the inclusion Im( f n ) −→ V providesa splitting of the short exact sequence0 / / Ker( f n ) / / V / / Im( f n ) / / . This gives us our canonical decomposition V = Im( f n ) ⊕ Ker( f n ). Obviously the map f : V −→ V takes Im( f n ) ⊂ V to itself and takes Ker( f n ) ⊂ V to itself, that is f : V −→ V decomposes as f ′ ⊕ f ′′ for a unique f ′ : Im( f n ) −→ Im( f n ) and a unique f ′′ : Ker( f n ) −→ Ker( f n ). Obviously f ′′ is nilpotent, more precisely ( f ′′ ) n = 0. And the kernel of themap f ′ is Ker( f ) ∩ Im( f n ) ⊂ Ker( f n ) ∩ Im( f n ) = 0 , which shows that the map f ′ : Im( f n ) −→ Im( f n ) is an injective endomorphism of thevector bundle Im( f n ) and hence an isomorphism.Now for the case where Y is allowed to have simple nodes as singularities. Let f : V −→ V be an endomorphism of the vector bundle V , and let π : X −→ Y be thenormalization of Y . Then π ∗ f : π ∗ V −→ π ∗ V is an endomorphism of the vector bundle π ∗ V on X , and by the above there exists a decomposition of π ∗ V as π ∗ V = W ′ ⊕ W ′′ suchthat(iii) There exists an integer n ≫ W ′ = Im( π ∗ f n ) and W ′′ = Ker( π ∗ f n ).(iv) The map π ∗ f : W ′ ⊕ W ′′ −→ W ′ ⊕ W ′′ is equal to ϕ ′ ⊕ ϕ ′′ for some morphisms ϕ ′ : W ′ −→ W ′ and ϕ ′′ : W ′′ −→ W ′′ .(v) The map ϕ ′ is an isomorphism while the map ϕ ′′ satisfies ( ϕ ′′ ) n = 0.With α i : Spec( k ) −→ X and β i : Spec( k ) −→ X as in Discussion 4.3, we have induceddecompositions α ∗ i π ∗ V = α ∗ i W ′ ⊕ α ∗ i W ′′ and β ∗ i π ∗ V = β ∗ i W ′ ⊕ β ∗ i W ′′ such that(vi) α ∗ i π ∗ f = α ∗ i ϕ ′ ⊕ α ∗ i ϕ ′′ and β ∗ i π ∗ f = β ∗ i ϕ ′ ⊕ β ∗ i ϕ ′′ .Of course we have canonical isomorphisms ρ i : α ∗ i π ∗ V −→ β ∗ i π ∗ V , and these ismorphisms ρ i must be such that the squares below commute α ∗ i π ∗ V α ∗ i π ∗ f / / ρ i (cid:15) (cid:15) α ∗ i π ∗ V ρ i (cid:15) (cid:15) β ∗ i π ∗ V β ∗ i π ∗ f / / β ∗ i π ∗ V Raising the horizontal maps to the n th power, the squares α ∗ i π ∗ V α ∗ i π ∗ f n / / ρ i (cid:15) (cid:15) α ∗ i π ∗ V ρ i (cid:15) (cid:15) β ∗ i π ∗ V β ∗ i π ∗ f n / / β ∗ i π ∗ V must also commute. But these squares can be rewritten as α ∗ i W ′ ⊕ α ∗ i W ′′ (cid:2) α ∗ i ( ϕ ′ ) n (cid:3) ⊕ / / ρ i (cid:15) (cid:15) α ∗ i W ′ ⊕ α ∗ i W ′′ ρ i (cid:15) (cid:15) β ∗ i W ′ ⊕ β ∗ i W ′′ (cid:2) β ∗ i ( ϕ ′ ) n (cid:3) ⊕ / / β ∗ i W ′ ⊕ β ∗ i W ′′ The fact that ϕ ′ is an isomorphism means that so are α ∗ i ( ϕ ′ ) n and β ∗ i ( ϕ ′ ) n . And thecommutativity of the square forces the map ρ i to take the kernel of the top horizontal mapto the kernel of the bottom horizontal map, and the image of the top horizontal map tothe image of the bottom horizontal map. That is: the isomorphism ρ i : α ∗ i W ′ ⊕ α ∗ i W ′′ −→ β ∗ i W ′ ⊕ β ∗ i W ′′ must split as the direct sum ρ ′ i ⊕ ρ ′′ i , for isomorphisms ρ ′ i : α ∗ i W ′ −→ β ∗ i W ′ and ρ ′′ i : α ∗ i W ′′ −→ β ∗ i W ′′ .And this allows us to descend to Y ; the maps ϕ ′ : W ′ −→ W ′ and ϕ ′′ : W ′′ −→ W ′′ , together with the descent data given by the isomorphisms ρ ′ i and ρ ′′ i , allow us touniquely define vector bundles V ′ , V ′′ on Y , as well as endomorphisms f ′ : V ′ −→ V ′ and f ′′ : V ′′ −→ V ′′ , so that π ∗ f ′ : π ∗ V ′ −→ π ∗ V ′ agrees with ϕ ′ : W ′ −→ W ′ and π ∗ f ′′ : π ∗ V ′′ −→ π ∗ V ′′ agrees with ϕ ′′ : W ′′ −→ W ′′ . The uniqueness forces f ′ ⊕ f ′′ toagree with f : V −→ V . And the fact that f ′ is an isomorphism and ( f ′′ ) n = 0 can bechecked after pulling back to X . (cid:3) Remark 4.6.
We should perhaps spell out what we meant in Lemma 4.4, when we saidthat the decomposition of V as V = V ′ ⊕ V ′′ is “canonical”.The fact that f : V −→ V decomposes as f = f ′ ⊕ f ′′ , with f ′ : V ′ −→ V ′ anisomorphism and f ′′ : V ′′ −→ V ′′ nilpotent, makes the decomposition unique. Choose an n ≫ f ′′ ) n = 0; then f n : V −→ V has kernel V ′′ and image V ′ . Thus we canwrite the formulas V ′ = ∞ \ n =1 Im( f n ) and V ′′ = ∞ [ n =1 Ker( f n ) , which show that these subbundles are canonically unique—what isn’t immediate fromthe formulas is that these are vector bundles and that their direct sum is V .Suppose we are given a commutative square of maps of vector bundles on Y V f / / ρ (cid:15) (cid:15) V ρ (cid:15) (cid:15) W g / / W Then obviously ρ ∞ \ n =1 Im( f n ) ! ⊂ ∞ \ n =1 Im( g n ) and ρ ∞ [ n =1 Ker( f n ) ! ⊂ ∞ [ n =1 Ker( g n ) COUNTEREXAMPLE TO SOME RECENT CONJECTURES 19
This means that, in the decomposition V = V ′ ⊕ V ′′ that comes from f and the decompo-sition W = W ′ ⊕ W ′′ that comes from g , we must have the compatibility that ρ : V −→ W must split as ρ = ρ ′ ⊕ ρ ′′ for a unique choice of ρ ′ : V ′ −→ W ′ and ρ ′′ : V ′′ −→ W ′′ . Lemma 4.7.
Let k be an algebraically closed field, let Y is a projective curve over k with only simple nodes, and let V ∗ be a cochain complex of vector bundle on Y . Thenany cochain map f ∗ : V ∗ −→ V ∗ leads to a canonical decomposition V ∗ = V ∗ ⊕ V ∗ suchthat (i) f ∗ decomposes as f ∗ = f ∗ ⊕ f ∗ for maps f ∗ : V ∗ −→ V ∗ and f ∗ : V ∗ −→ V ∗ . (ii) The map f ∗ is an isomorphism while the map f ∗ is locally nilpotent. By locallynilpotent we mean that, for any integer i ∈ Z , there exists an integer n i (which maydepend on i ) for which f i : V i −→ V i satisfies ( f i ) n i = 0 .Moreover: if f is null homotopic then the complex V ∗ must be contractible.Proof. The existence of the canonical decomposition is by Remark 4.6; in each degree i the map f i : V i −→ V i leads to a decomposition V i = V i ⊕ V i , and the differential ∂ i : V i −→ V i +1 must split as a direct sum ∂ i = ∂ i ⊕ ∂ i for suitable ∂ i : V i −→ V i +11 and ∂ i : V i −→ V i +12 .It remains to prove the “moreover” statement. Suppose therefore that f ∗ is nullhomotopic. Then the isomorphism f ∗ : V ∗ −→ V ∗ can be factored as the composite V ∗ i / / V ∗ f / / V ∗ p / / V ∗ where V ∗ i −→ V ∗ p −→ V ∗ are the canonical inclusion and projection from the direct sum.Thus the fact that f is null homotopic forces the isomorphism f ∗ : V ∗ −→ V ∗ to alsobe. (cid:3) Proposition 4.8.
Let k be an algebraically closed field, let Y be a projective curve over k with only simple nodes, and let K (cid:2) V ect ( Y ) (cid:3) be the homotopy category of cochaincomplexes of vector bundles on Y . Then every idempotent in K (cid:2) V ect ( Y ) (cid:3) splits.Proof. Choose an idempotent in K (cid:2) V ect ( Y ) (cid:3) , and let it be represented by the cochainmap e ∗ : V ∗ −→ V ∗ . The assumption that e ∗ is idempotent in K (cid:2) V ect ( Y ) (cid:3) means that e ∗ and ( e ∗ ) are homotopic.By Lemma 4.7 we may decompose V ∗ , along the map e ∗ : V ∗ −→ V ∗ , as V ∗ = V ∗ ⊕ V ∗ is such a way that(i) The map e ∗ may be written as e ∗ ⊕ e ∗ , for cochain maps e ∗ : V ∗ −→ V ∗ and e ∗ : V ∗ −→ V ∗ .(ii) The map e ∗ is an isomorphism while the map e ∗ is locally nilpotent.The fact that ( e ∗ ) − e ∗ is null homotopic forces the direct summands ( e ∗ ) − e ∗ and( e ∗ ) − e ∗ to be null homotopic. Since e ∗ is an isomorphism and e ∗ ( e ∗ −
1) is nullhomotopic we deduce that e ∗ is homotopic to the identity. It remains to show that e ∗ is null homotopic. Replacing e ∗ by e ∗ , we are reduced toshowing(iii) Suppose e ∗ : V ∗ −→ V ∗ is a cochain map, and assume that e ∗ is locally nilpotentand that e ∗ − ( e ∗ ) is null homotopic. Then e ∗ is null homotopic.Observe the formal equalities e ∗ = ∞ X i =1 (cid:2) ( e ∗ ) i − ( e ∗ ) i +1 (cid:3) = (cid:2) e ∗ − ( e ∗ ) (cid:3) · " ∞ X i =0 ( e ∗ ) i The infinite sums make sense since the local nilpotence guarantees that in each degree thesums are finite. Thus we have produced a factorization of the cochain map e ∗ : V ∗ −→ V ∗ as the composite of two cochain maps V ∗ P ∞ i =0 ( e ∗ ) i / / V ∗ e ∗ − ( e ∗ ) / / V ∗ where the second is null homotopic. Hence e ∗ is null homotopic. (cid:3) Remark 4.9.
The proof of Proposition 4.8 generalizes, it works to show that idempotentssplit not only in the unbounded homotopy category K (cid:2) V ect ( X ) (cid:3) , but also in the boundedsubcategories K b (cid:2) V ect ( X ) (cid:3) , K − (cid:2) V ect ( X ) (cid:3) and K + (cid:2) V ect ( X ) (cid:3) . But for the categories K b , K − and K + the fact that idempotents split is not new: the reader can constructproofs using the methods of [5, Proposition 3.1]. See the proof of [5, Proposition 3.4] foran outline. For a later proof that is more K– theoretical see Balmer and Schlichting [3,Section 2]. More precisely: the idempotent-completeness of K − ( E ) = D − ( E ⊕ ) and of K + ( E ) = D + ( E ⊕ ) may be found in [3, Lemma 2.4], while the idempotent-completenessof K b ( E ) = D b ( E ⊕ ) follows from [3, Theorem 2.8].5. The counterexample
Throughout this section k will be a fixed algebraically closed field. If Y is a projectivecurve over k with only simple nodes as singularities, then V ect ( Y ) will be the categoryof vector bundles over Y . And now we return to the K– theoretic implications of whatwe have proved. Remark 5.1.
The K– theoretic import of Proposition 4.8 is that K − (cid:2) V ect ( Y ) ⊕ (cid:3) = 0.This follows from Schlichting [15, Corollary 6 of Section 9]. We briefly remind the reader: COUNTEREXAMPLE TO SOME RECENT CONJECTURES 21 in the category M of model categories we have a commutative square Ch b (cid:2) V ect ( Y ) (cid:3) / / (cid:15) (cid:15) Ch − (cid:2) V ect ( Y ) (cid:3) (cid:15) (cid:15) Ch + (cid:2) V ect ( Y ) (cid:3) / / Ch (cid:2) V ect ( Y ) (cid:3) The functor D : M −→ T takes this to the commutative square K b (cid:2) V ect ( Y ) (cid:3) / / (cid:15) (cid:15) K − (cid:2) V ect ( Y ) (cid:3) (cid:15) (cid:15) K + (cid:2) V ect ( Y ) (cid:3) / / K (cid:2) V ect ( Y ) (cid:3) and recalling that, by Reminder 3.1, we have K ? ( E ) = D ? ( E ⊕ ) for ? any of b , +, − orthe empty restriction, this puts us squarely in the situation of Schlichting [15, Corollary6 of Section 9]. The connective part of the functor K takes the commutative square in M to the homotopy cartesian square K ≥ (cid:16) Ch b (cid:2) V ect ( Y ) (cid:3)(cid:17) / / (cid:15) (cid:15) (cid:15) (cid:15) / / K ≥ (cid:16) Ch (cid:2) V ect ( Y ) (cid:3)(cid:17) making K ≥ (cid:16) Ch (cid:2) V ect ( Y ) (cid:3)(cid:17) a delooping of K ≥ (cid:16) Ch b (cid:2) V ect ( Y ) (cid:3)(cid:17) ∼ = K ≥ (cid:2) V ect ( Y ) ⊕ (cid:3) .And K − (cid:2) V ect ( Y ) ⊕ (cid:3) can therefore be computed as K of the idempotent completionof D (cid:16) Ch (cid:2) V ect ( Y ) (cid:3)(cid:17) = K (cid:2) V ect ( Y ) (cid:3) . Proposition 4.8 tells us that the triangulatedcategory K (cid:2) V ect ( Y ) (cid:3) is idempotent-complete, and since it has vanishing K we deducethat K − (cid:2) V ect ( Y ) ⊕ (cid:3) = 0. Example 5.2.
As in Remark 5.1 we let Y be a projective curve over k with simple nodesas singularities. Now recall the homotopy fibration of Proposition 3.3. With E = V ect ( Y )we have an exact sequence K − ( E ⊕ ) −→ K − ( E ) −→ K − ( M ′ ), where D ( M ′ ) = Ac b ( E )is the homotopy category of bounded acyclic cochain complexes.Remark 5.1 gives the vanishing of K − ( E ⊕ ), and the sequence of Proposition 3.3 be-comes 0 −→ K − ( E ) −→ K − ( M ′ ). This shows that K − ( M ′ ) contains K − ( E ) as asubmodule. But there are known examples of nodal curves with non-vanishing K − ; seeWeibel [21, Exercise III.4.4]. Hence there are examples of model categories M ′ ∈ M ,with non-vanishing negative K –theory, and such that D ( M ′ ) = Ac b ( E ) has a bounded t –structure.Note: as presented in Weibel’s book the nodal curves U for which K − (cid:2) V ect ( U ) (cid:3) = 0are affine. But the Mayer-Vietoris exact sequence, for assembling K (cid:2) V ect ( Y ) (cid:3) from Zariski open covers, permits us to pass to the compactification of these affine curves,which can be chosen to have only simple nodes as singularities and also have non-vanishing K − (cid:2) V ect ( Y ) (cid:3) . The Mayer-Vietoris sequence was proved to be exact byWeibel [20, Main Theorem] for reduced quasiprojective varieties with isolated singu-larities, and in general by Thomason and Trobaugh [18, Theorem 8.1]. The non-vanishing of K − (cid:2) V ect ( Y ) (cid:3) can also be deduced from the Mayer-Vietorissequence for the conductor square; see Pedrini and Weibel [14, Theorem A.3]. Remark 5.3.
In Remark 3.4 we promised the reader that, for the E = V ect ( Y ) ofExample 5.2, the category E will be proved hereditary. It is time to deliver on thepromise.If V , V ′ are vector bundles on Y then there is a spectral sequence, converging toExt p + q ( V , V ′ ), whose E term has entries H p (cid:2) Y, E xt q ( V , V ′ ) (cid:3) . Now as V is a locally freesheaf the local Ext sheaves E xt q ( V , V ′ ) vanish when q >
0. And H p (cid:2) Y, E xt ( V , V ′ ) (cid:3) vanishes if p >
1, just because Y is a curve. References [1] Benjamin Antieau, David Gepner, and Jeremiah Heller, K -theoretic obstructions to bounded t -structures , Invent. Math. (2019), no. 1, 241–300.[2] Maurice Auslander, Coherent functors , Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965),Springer, New York, 1966, pp. 189–231.[3] Paul Balmer and Marco Schlichting,
Idempotent completion of triangulated categories , J. Algebra (2001), no. 2, 819–834.[4] Alexander A. Be˘ılinson, Joseph Bernstein, and Pierre Deligne,
Analyse et topologie sur les ´espacessinguliers , Ast´erisque, vol. 100, Soc. Math. France, 1982 (French).[5] Marcel B¨okstedt and Amnon Neeman,
Homotopy limits in triangulated categories , Compositio Math. (1993), 209–234.[6] Henri Cartan and Samuel Eilenberg, Homological algebra , Princeton University Press, Princeton, N.J., 1956. MR 0077480[7] Nathan Jacobson,
Basic algebra. II , second ed., W. H. Freeman and Company, New York, 1989.[8] Henning Krause,
Deriving Auslander’s formula , Doc. Math. (2015), 669–688.[9] Jacob Lurie, Spectral Algebraic Geometry , Prepint, available from the author’s web page.[10] David Mumford,
Projective invariants of projective structures and applications , Proc. Internat.Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, pp. 526–530. For the discerning, careful reader who checks the reference to [18]: the only schemes Y considered inthe current article have been quasiprojective, they have ample line bundles, this makes them satisfy theresolution property, and the natural map D b (cid:2) V ect ( Y ) (cid:3) −→ D perf ( Y ) is an equivalence of categories.Thomason and Trobaugh [18] allows pathological schemes which do not satisfy the resolution prop-erty, and for such schemes the right K –theory to work with, the one for which the Mayer-Vietorissequence is exact, is the K –theory that comes from perfect complexes and not the one correspondingto (cid:16) Ch b (cid:2) V ect ( Y ) (cid:3) , Ac b (cid:2) V ect ( Y ) (cid:3)(cid:17) . It isn’t known how pathological a scheme Y has to be for this tobe an issue. Non-separated schemes can fail to satisfy the resolution property, but the reader can checkTotaro [19, Question 1, page 3 of the Introduction]: for all we know every scheme with an affine diagonalmight have the resolution property. COUNTEREXAMPLE TO SOME RECENT CONJECTURES 23 [11] M. S. Narasimhan and S. Ramanan,
Moduli of vector bundles on a compact Riemann surface , Ann.of Math. (2) (1969), 14–51.[12] Amnon Neeman, The derived category of an exact category , J. Algebra (1990), 388–394.[13] ,
Some new axioms for triangulated categories , J. Algebra (1991), 221–255.[14] Claudio Pedrini and Charles Weibel,
Divisibility in the Chow group of zero-cycles on a singularsurface , no. 226, 1994, K -theory (Strasbourg, 1992), pp. 10–11, 371–409.[15] Marco Schlichting, Negative K -theory of derived categories , Math. Z. (2006), no. 1, 97–134.[16] Jean-Pierre Serre, Groupes alg´ebriques et corps de classes , Publications de l’institut de math´ematiquede l’universit´e de Nancago, VII. Hermann, Paris, 1959.[17] Stephen S. Shatz,
The decomposition and specialization of algebraic families of vector bundles , Com-positio Math. (1977), no. 2, 163–187.[18] Robert W. Thomason and Thomas F. Trobaugh, Higher algebraic K–theory of schemes and ofderived categories , The Grothendieck Festschrift ( a collection of papers to honor Grothendieck’s60’th birthday), vol. 3, Birkh¨auser, 1990, pp. 247–435.[19] Burt Totaro,
The resolution property for schemes and stacks , J. Reine Angew. Math. (2004),1–22.[20] Charles A. Weibel,
A Brown-Gersten spectral sequence for the K -theory of varieties with isolatedsingularities , Adv. in Math. (1989), no. 2, 192–203.[21] , The K -book , Graduate Studies in Mathematics, vol. 145, American Mathematical Society,Providence, RI, 2013, An introduction to algebraic K -theory. Centre for Mathematics and its Applications, Mathematical Sciences Institute, Building145, The Australian National University, Canberra, ACT 2601, AUSTRALIA
E-mail address ::