aa r X i v : . [ m a t h . K T ] A ug A survey of Gersten’s conjecture
Satoshi Mochizuki
Abstract
This article is the extended notes of my survey talk of Gersten’s conjecture given at the workshop“Bousfield classes form a set: a workshop in a memory of Tetsusuke Ohkawa” at Nagoya University inAugust 2015. In the last section, I give an explanation of my recent work of motivic Gernsten’s conjecture.
Introduction
This article is the extended notes of my survey talk of Gersten’s conjecture in [Ger73] given at the workshop“Bousfield classes form a set: a workshop in a memory of Tetsusuke Ohkawa” at Nagoya University inAugust 2015. (The slide movie of my talk at the workshop is [Moc15].) In this article, we provide anexplanation of a proof of Gersten’s conjecture in [Moc16a] by emphasizing the conceptual idea behind theproof rather than its technical aspects. In the last section, I give an explanation of my recent work of motivicGernsten’s conjecture in [Moc16b]. acknowledgement
I wish to express my deep gratitude to all organizers of the workshop and speciallyprofessor Norihiko Minami for giving me the opportunity to present my work.
We start by recalling the following result. For n =
0, 1, the results are classical and for n =
2, it was givenby Spencer Bloch in [Blo74] by utilzing the second universal Chern class map in [Gro68].
Proposition 1.1.
For a smooth variety X over a field, we have the canonical isomorphisms CH n ( X ) ≃ H n Zar ( X , K n ) (1) for n = , , where K n is the Zariski sheafication of the K-presheaf U K n ( U ) . On the other hand, there are the following spectral sequences. To give a precise statement, we introducesome notations. For a noetherian scheme X , we write M X for the category of coherent sheaves on X . Thereis a filtration 0 ⊂ ··· ⊂ M X ⊂ M X ⊂ M X = M X (2)by the Serre subcategories M iX of those coherent sheaves whose support has codimension ≥ i . Proposition 1.2. ( Quillen spectral sequence [Qui73], [Bal09].)
For a noetherian scheme X, the fil-tration ( ) induces the strongly convergent spectral sequenceE p , q ( X ) = M x ∈ X p K − p − q ( k ( x )) ⇒ G − p − q ( X ) , (3) where X p is the set of points codimension p in X. Moreover if X is regular separated, then we havethe canonical isomorphism E p , − p ( X ) ≃ CH p ( X ) . (4) ( Brown-Gersten-Thomason spectral sequence [BG73], [TT90].)
For a noetherian scheme of finiteKrull dimension X, we have the canonical equivalence K ( X ) ≃ H • Zar ( X ; K ( − )) . (5) In particular, there exists the strongly convergent spectral sequence ′ E p , q ( X ) = H p Zar ( X ; e K q ) ⇒ K q − p ( X ) (6) where e K q is the Zariski sheafication of the presheaf U p q K ( U ) of non-connective K-theory. o regard the isomorphisms ( ) as the isomorphisms of E -terms of the spectral sequences above, Gerstengave the following observation in [Ger73]. For simplicity, for a commutative noetherian ring A with 1, wewrite M A and M pA for M Spec A and M p Spec A respectively. Proposition 1.3.
The following conditions are equivalent:1. For any regular separated noetherian scheme X, we have the canonical isomorphism between E -terms of Quillen and Brown-Gersten-Thomason spectral sequencesE p , q ( X ) ≃ ′ E p , − q ( X ) . In particular the isomorphisms ( ) hold for X and any non-negative integer n.2. For any commutative regular local ring R of dimension , E -terms of Quillen spectral sequence ( ) for Spec
R yields an exact sequence → K n ( R ) → K n ( frac ( R )) → M ht p = K n − ( k ( p )) → M ht p = K n − ( k ( p )) → ··· . (7)
3. For any commutative regular local ring R and natural number ≤ p ≤ dim R, the canonical inclusion M pR ֒ → M p − R induces the zero map on K-theoryK ( M pR ) → K ( M p − R ) where K ( M iR ) denotes the K-theory of the abelian category M iR . Here is Gersten’s conjecture:
Conjecture . The conditions above are true for any commutative regular localring.
Historical Note
The conjecture has been proved in the following cases:
The case of general dimension
1. If A is of equi-characteristic, then Gersten’s conjecture for A is true.We refer to [Qui73] for special cases, and the general cases [Pan03] can be deduced from limit argu-ment and Popescu’s general N´eron desingularization [Pop86]. (For the case of commutative discretevaluation rings, it was first proved by Sherman [She78]).2. If A is smooth over some commutative discrete valuation ring S and satisfies some condition, thenGersten’s conjecture for A is true. [Blo86].3. If A is smooth over some commutative discrete valuation ring S and if we accept Gersten’s conjecturefor S , then Gersten’s conjecture for A is true. [GL87]. See also [RS90]. The case of that A is a commutative discrete valuation ring
1. For the cases of n = , n = A is true.We refer to [Ger73] for the cases of that its residue field is a finite field, the proof of general cases[She82] is improved from that of special cases by using Swan’s result [Swa63], the universal propertyof algebraic K -theory [Hil81] and limit argument. The case for Grothendieck groups
Gersten’s conjecture for Grothendieck groups has several equivalent forms. Namely we can show the fol-lowing three conditions are equivalent. See [CF68], [Lev85], [Dut93] and [Dut95].1. (Gersten’s conjecture for Grothendieck groups).
For any commutative regular local ring R andnatural number ≤ p ≤ dim R, the canonical inclusion M pR ֒ → M p − R induces the zero map onGrothendieck groups K ( M pR ) → K ( M p − R ) . . (Generator conjecture). For any commutative regular local ring R and any natural number ≤ p ≤ dim R, the Grothendieck group K ( M pR ) is generated by cyclic modules R / ( f , ··· , f p ) where thesequence f , ··· , f p forms an R-regular sequence. (Claborn and Fossum conjecture). For any commutative regular local ring R, the Chow homologygroup CH k ( Spec R ) is trivial for any k < dim R. For historical notes of the generator conjecture, please see [Moc13a].
The case for singular varieties
Gersten’s conjecture for non-regular rings is in general false in the literal sense of the word and severalappropriate modified versions are studied by many authors. See [DHM85], [Smo87], [Lev88], [Bal09],[HM10] and [Moc13a]. See also [Mor15] and [KM16].
Other cohomology theories
1. For torsion coefficient K -theory, Gersten’s conjecture for a commutative discrete valuation ring istrue. [Gil86], [GL00].2. For Gersten’s conjecture for Milnor K -theory, see [Ker09], [Dah15].3. For Gersten’s conjecture for Witt groups, see [Par82], [OP99], [BW02] and [BGPW03].4. For an analogue of Gersten’s conjecture for bivariant K -theory, see [Wal00].5. In the proof of geometric case of Gersten’s conjecture by Quillen in [Qui73], he introduced a strength-ning of Noether normalization theorem. There are several variants of Noether normalization theoremin [Oja80], [GS88], [Gab94] and [Wal98] and by utilizing them, there exists Gersten’s conjecturetype theorem for universal exactness (see [Gra85]) for Cousin complexes (see [Har66]) of certaincohomology theories. For axiomatic approaches of these topics, see [BO74], [C-THK97]. See also[Gab93], [C-T95], [Lev08] and [Lev13].6. For an analogue of Gersten’s conjecture for Hochschild coniveau spectral sequences, see [BW16].7. For an analogue of Gersten’s conjecture for infinitesimal theory, see [DHY15].8. For Gersten’s complexes for homotopy invariant Zariski sheaves with transfers, see [Voe00], [MVW06,Lecture 24]. See also [SZ03]. For injectivity result for pseudo pretheory, see [FS02]. Logical connections with other conjectures
1. Some conjectures imply that Gersten’s conjecture for a commutative discrete valuation ring is true.[She89].2. Parshin conjecture in [Bei84] implies Gersten’s conjecture of motivic cohomology for a localization ofsmooth varieties over a Dedekind ring. (see [Gei04].) It is also known that Tate-Beilinson conjecture(see [Tat65], [Bei87] and [Tat94]) implies Parshin conjecture. (see [Gei98].)
Counterexample for non-commutative discrete valuation rings (Due to Kazuya Kato)
In Gersten’s conjecture, the assumption of commutativity is essential. Let D be a skew field finite over Q p , A its integer ring and a its prime element. As the inner automorphism of a induces non-trivial automorphismon its residue field, we have x ∈ A × with y = axa − x − is non vanishing in its residue field, a fortiori in K ( A ) = ( A × ) ab . On the other hand y is a commutator in D × . Hence it turns out that the canonical map K ( A ) → K ( D ) = ( D × ) ab is not injective. My idea of how to prove Gersten’s conjecture has come from weight argument of Adams operations in[GS87] and [GS99]. In my viewpoint, difficulty of solving Gersten’s conjecture consists of ring theoreticside and homotopy theoretic side. We will explain ideas about how to overcome each difficulty. ing theoretic side Combining the results in [GS87] and [TT90], for a commutative regular local ring R , there exists Adamsoperations { j k } k ≥ on K ( M pR ) and we have the equality j k ([ R / ( f , ··· , f p )] = k p [ R / ( f , ··· , f p )] (8)where a sequence f , ··· , f p is an R -regular sequence. Thus roughly saying, the generator conjecture saysthat for each p , K ( M pR ) is spanned by objects of weight p and Gersten’s conjecture could follow fromweight argument of Adams operations. We illustrate how to prove that the generator conjecture impliesGersten’s conjecture for K without using Adams operations. Proof.
Let a sequence f , ··· , f p be an R -regular sequence. Then there exists the short exact sequence0 → R / ( f , ··· , f p − ) f p → R / ( f , ··· , f p − ) → R / ( f , ··· , f p ) → M p − R . Thus the class [ R / ( f , ··· , f p )] in K ( M pR ) goes to [ R / ( f , ··· , f p )] = [ R / ( f , ··· , f p − )] − [ R / ( f , ··· , f p − )] = K ( M p − R ) . Strategy 1.
We will establish and prove a higher analogue of the generator conjecture.Inspired from the works [Iwa59], [Ser59], [Bou64], [Die86] and [Gra92], we establish a classification theoryof modules by utilizing cubes in [Moc13a], [Moc13b] and [MY14]. In this article, we will implicitly usethese theory and simplify the arguments in [Moc13a] and [Moc16a].
Homotopy theoretic side
Roghly saying, we will try to compare the following two functors on K -theory. We denote the category ofbounded complexes on M p − R by Ch b ( M p − R ) . M pR → Ch b ( M p − R ) , R / ( f , ··· , f p ) R / ( f , ··· , f p ) ↓ f p R / ( f , ··· , f p ) ∼ qis R / ( f , ··· , f p ) R / ( f , ··· , f p ) ↓ id R / ( f , ··· , f p ) ∼ qis K -theory by the additivity theorem. A problemis that the functors above are not 1 -funtorial !! We need to a good notion of K -theory for higher categorytheory or need to discuss more subtle argument for such exotic functors. Strategy 2.
We give a modified definition of algebraic K -theory in a particular situation and establish a technique ofrectifying lax functors to 1-functors and by utilizing this definition and these techniques, we will treat suchexotic functors inside the classical Waldhausen K -theory. Let A be a commutative noetherian ring with 1 and let I be an ideal of A with codimension Y = V ( I ) ≥ p in Spec A . Let M IA be a full subcategory of M pA consisting of those modules M supported on Y = V ( I ) and M IA , red be a full subcategory of M IA consisting of those modules M such that IM are trivial. We calla module in M IA , red a reduced module with respect to I . Let P A be the category of finitely generatedprojective A -modules. For a commutative regular local ring R , let J be an ideal generated by R -regularsequence f , ··· , f p such that f i is an prime element for any 1 ≤ i ≤ p .First notice the following isomorphisms: ( M pR ) ∼ → I colim codim Spec R V ( I )= p Spec R / I ֒ → regular Spec R K ( M IR ) , (9) K ( P R / J ) ∼ → II K ( M JR , red ) ∼ → III K ( M JR ) . (10)Since R is Cohen-Macaulay, the ordered set of all ideals of R that contains an R -regular sequence of length p with usual inclusion is directed. Thus M pR is the filtered limit colim I M IR where I runs through any idealgenerated by any R -regular sequence of length p . Thus the isomorphism I follows from cocontinuity of K -theory. The isomorphism II follows from the resolution theorem and regularity of R . Finally the isomor-phism III follows from the d´evissage theorem. Let f S = { f s } s ∈ S be an R -regular sequence such that f s is aprime element of R for any s ∈ S . We call such a sequence f S a prime regular sequence . For a commutativenoetherian ring A with 1 and for an ideal I of A , let M IA ( ) and M IA , red ( ) be the full subcategory of M IA and M IA , red respectively consisting of those A -modules M with projdim A M ≤
1. We fix an element s ∈ S .Since the inclusion functor P R / f S R ֒ → M S − R factors through M R / f S r { s } R ( ) , the problem reduce to thefollowing: For any prime regular sequence f S = { f s } s ∈ S of R and any element s ∈ S, the inclusion functor P R / f S R ֒ → M R / f S r { s } R ( ) induces the zero map K ( P R / f S R ) → K ( M R / f S r { s } R ( )) on K-theory. For simplicity we set B = R / f S r { s } R and g = f s . We let Ch b ( M B ( )) denote the category of boundedcomplexes on M B ( ) . (We use homological index notation.) Let C be the full subcategory of Ch b ( M B ( )) consisting of those complexes x such that x i = i =
0, 1 and x and x are free B -modules and thebounded map d x : x → x is injective and H x : = Coker ( x d x → x ) is annihilated by g . We can show that C is an idempotent exact category such that the inclusion functor h : C ֒ → Ch b ( M B ( )) is exact and reflectsexactness and the functor H : C → P B / gB is exact. Thus we obtain the commutative diagram K ( C ) K ( h ) / / K ( H ) (cid:15) (cid:15) K ( Ch b ( M B ( )) ;qis ) K ( P B / gB ) / / K ( M B ( )) ≀ I O O where qis is the class of all quasi-isomorphisms in Ch b ( M B ( )) and the map I which is induced from theinclusion functor M B ( ) ֒ → Ch b ( M B ( )) is a homotopy equivalence by Gillet-Waldhausen theorem. Wewill prove that1. The map K ( H ) is a split epimorphism in the stable category of spectra (see §
4) and2. The map K ( h ) is the zero map in the stable category of spectra. (See § . corresponds with Strategy 1 and assertion 2 . corresponds with Strategy 2 in the previous section. In this section, we will give a brief proof of assertion that K ( H ) is a split epimorphism in the stable categoryof spectra.Let D be the full subcategory of Ch b ( M B ( )) consisting of those complexes x such that x i = i = d x : x → x is injective and H x : = Coker ( x d x → x ) is in M gBB , red ( ) . We can showthat D is an exact category such that the inclusion functor D ֒ → Ch b ( M B ( )) is exact and reflects exactnessand we can also show that the functor H : D → M gBB , red ( ) is exact. Thus we obtain the commutative squarebelow K ( C ) / / K ( H ) (cid:15) (cid:15) K ( D ) K ( H ) (cid:15) (cid:15) K ( P B / gB ) / / K ( M gBB , red ( )) . here the horizontal maps are induced from the inclusion functors. Since the functor H : D → M gBB , red ( ) admits a section which is defined by sending an object x in M gBB , red ( ) to the complex [ → x ] in D ,the right vertical map in the diagram above is a split epimorphism. Moreover the inclusion functors P B / gB ֒ → M gBB , red ( ) and C ֒ → D induce equivalences of triangulated categories on bounded derived cat-egories respectively. (Compare [Moc13a, 2.21] and [Moc16a, 2.1.1].) Thus the horizontal maps in thediagram above are homotopy equivalences and the left vertical map is also a split epimorphism in the stablecategory of spectra. In this section we will give an outline of the proof of assertion that K ( h ) is the zero map in the stablecategory of spectra.Let B be the full subcategory of Ch b M B ( ) consiting of those complexes x such that x i = i = i =
1. Let s i : B → M B ( ) ( i =
0, 1) be an exact functor defined by sending an object x in B to x i in M B ( ) .By the additivity theorem, the map s × s : iS · B → iS · M B ( ) × iS · M B ( ) is a homotopy equivalence. Let j : B → Ch b ( M B ( )) be the inculsion functor.We wish to define two exact ’functors’ m , m : C → B which satisfy the following conditions:1. We have the equality s × s m = s × s m . (11)2. There are natural transformations h → j m and 0 → j m such that all componets are quasi-isomorphisms.Then we have the equalities K ( h ) = K ( j m ) = K ( j ) K ( s × s ) − K ( s × s m ) = K ( j ) K ( s × s ) − K ( s × s m ) = K ( j m ) = . To define the ’functors’ m i ( i =
1, 2), we analyze morphisms in C . C Definition 5.1. (Compare [Moc16a, 1.1.3.].) For a pair of non-negative integers ( n , m ) , we write ( n , m ) B for the complex of the form B ⊕ n ⊕ B ⊕ m ↓ (cid:18) gE n E m (cid:19) B ⊕ n ⊕ B ⊕ m in C where E k is the k × k unit matrix. Lemma 5.2. ( ) ( Compare [Moc16a, 1.2.10., 1.2.13.].)
Let n be a positive integer. For any endomorphism a : ( n , ) B → ( n , ) B , the following conditions are equivalent. ( i ) a is an isomorphism. ( ii ) a is a quasi-isomorphism. ( ) ( Compare [Moc13a, 2.17.].)
An object in C is projective. In particular, C is a semi-simple exactcategory. ( ) ( Compare [Moc16a, 1.2.15.].)
For any object x in C , there exists a pair of non-negative integers ( n , m ) such that x is isomorphic to ( n , m ) B .Proof. ( ) We assume condition ( ii ) . In the commutative diagram below B ⊕ n gE n / / a (cid:15) (cid:15) B ⊕ n / / a (cid:15) (cid:15) H (( n , ) B ) H ( a ) (cid:15) (cid:15) B ⊕ n gE n / / B ⊕ n / / H (( n , ) B ) , first we will prove that a is an isomorphism. Then a is also an isomorphism by the five lemma. By takingdeterminant of a , we shall assume that n =
1. Then assertion follows from Nakayama’s lemma. ( ) Let t : y ։ z be an admissible epimorphism in C and let f : x → z be a morphism in C . Then since H ( x ) is a projective B / gB -module, there exist a homomorphism of B / gB -modules s : H ( x ) → H ( y ) such that ( t ) s = H ( f ) . Since x is a complex of free B -modules, there exists a morphism of complexes s ′ : x → y such that H ( s ′ ) = s and ts ′ is chain homotopic to f by [Wei94, Comparison theorem 2.2.6.]. Namely thereis a map h : x → z such that ( f − ts ′ ) = d z h and ( f − ts ′ ) = hd x . x d x (cid:15) (cid:15) ( f − ts ′ ) / / z d z (cid:15) (cid:15) x h > > ⑥⑥⑥⑥⑥⑥⑥⑥ ( f − ts ′ ) / / z . Since x is projective, there is a map u : x → y such that t u = h . We set s : = s ′ + ud x and s : = s ′ + d y u .Then we can check that s is a morphism of complexes of B -modules and f = ts . ( ) By considering x ⊗ B B (cid:20) g (cid:21) , we notice that x and x are same rank. Thus we shall assume x = x = B ⊕ m .First we assume that x is acyclic. Then the boundary map d x : x → x is invertible and x ↓ d x x d x →→ id B ⊕ m ↓ id B ⊕ m B ⊕ m gives an isomorphism between x and ( , m ) B . Thus we obtain the result in this case.Next assume that H ( x ) =
0. Then sicne H ( x ) is a finitely generated projective B / gB -module, there isa positive integer n and an isomorphism s : ( B / gB ) ⊕ n ∼ → H ( x ) . Then by [Wei94, Comparison theorem2.2.6.], there exists morphisms of complexes in C , ( n , ) B a → x and x b → ( n , ) B such that H ( a ) = s andH ( b ) = s − . Thus by ( ) , ba is an isomorphism. By replacing a with a ( ba ) − , we shall assume that ba = id. Hence there exists a complex y in C and a split exact sequence: ( n , ) B a x ։ y . (12)Since boundary maps of objects in C are injective, the functor H from C to the category of finitly generatedprojective B / gB -modules is exact. By taking H to the sequence ( ) , it turns out that y is acyclic and bythe first paragraph, we shall assume that y is of the form ( , m ) B . Hence x is isomorphic to ( n , m ) B . Remark C ) . (Compare [Moc16a, 1.2.16.].) We can denote a morphism j : ( n , m ) B → ( n ′ , m ′ ) B of C by B ⊕ n ⊕ B ⊕ m ↓ (cid:18) gE n E m (cid:19) B ⊕ n ⊕ B ⊕ m j →→ j B ⊕ n ′ ⊕ B ⊕ m ′ ↓ (cid:18) gE n ′ E m ′ (cid:19) B ⊕ n ′ ⊕ B ⊕ m ′ with j = (cid:18) j ( n ′ , n ) j ( n ′ , m ) g j ( m ′ , n ) j ( m ′ , m ) (cid:19) and j = (cid:18) j ( n ′ , n ) g j ( n ′ , m ) j ( m ′ , n ) j ( m ′ , m ) (cid:19) where j ( i , j ) are i × j matrices whose coef-ficients are in B . In this case we write (cid:18) j ( n ′ , n ) j ( n ′ , m ) j ( m ′ , n ) j ( m ′ , m ) (cid:19) (13)for j . In this matrix presentation of morphisms, the composition of morphisms between objects ( n , m ) B j → ( n ′ , m ′ ) B y → ( n ′′ , m ′′ ) B in C is described by (cid:18) y ( n ′′ , n ′ ) y ( n ′′ , m ′ ) y ( m ′′ , n ′ ) y ( m ′′ , m ′ ) (cid:19)(cid:18) j ( n ′ , n ) j ( n ′ , m ) j ( m ′ , n ) j ( m ′ , m ) (cid:19) = (cid:18) y ( n ′′ , n ′ ) j ( n ′ , n ) + g y ( n ′′ , m ′ ) j ( m ′ , n ) y ( n ′′ , n ′ ) j ( n ′ , m ) + y ( n ′′ , m ′ ) j ( m ′ , m ) y ( m ′′ , n ′ ) j ( n ′ , n ) + y ( m ′′ , m ′ ) j ( m ′ , n ) g y ( m ′′ , n ′ ) j ( n ′ , m ) + y ( m ′′ , m ′ ) j ( m ′ , m ) (cid:19) . (14)Thus the category C is categorical equivalent to the category whose objects are oredered pair of non-negativeintegers ( n , m ) and whose morphisms from an object ( n , m ) to ( n ′ , m ′ ) are 2 × i × j matrices j ( i , j ) whose coefficients are in B and compositions are given by the formula (14). Wesometimes identify these two categories. .2 Modified algebraic K -theory A candidate of a pair of m i : C → B ( i =
1, 2) are following. We define m ′ , m ′ : C → B to be associationsby sending an object ( n , m ) B in C to ( n , ) B and ( , n ) B respecitvely and a morphism j = (cid:18) j ( n ′ , n ) j ( n ′ , m ) j ( m ′ , n ) j ( m ′ , m ) (cid:19) : ( n , m ) B → ( n ′ , m ′ ) B in C to B ⊕ n ↓ gE n B ⊕ n j ( n ′ , n ) →→ j ( n ′ , n ) B ⊕ n ′ ↓ gE n ′ B ⊕ n ′ and B ⊕ n ↓ E n B ⊕ n j ( n ′ , n ) →→ j ( n ′ , n ) B ⊕ n ′ ↓ E n ′ B ⊕ n ′ respectively. Notice that they are not 1-functors.We need to make revisions in the previous idea. We introduce a modified version of algebraic K -theory of C . Definition 5.4 (Triangular morphisms) . (Compare [Moc16a, 2.3.5.].) We say that a morphism j : ( n , m ) B → ( n ′ , m ′ ) B in C of the form ( ) is an upper triangular if j ( m ′ , n ) is the zero morphism, and say that j is a lower triangular if j ( n ′ , m ) is the zero morphism. We denote the class of all upper triangular isomorphismsin C by i △ . Next we define S ▽· C to be a simplicial subcategory of S · C consisting of those objects x suchthat x ( i ≤ j ) → x ( i ′ ≤ j ′ ) is a lower triangular morphism for each i ≤ i ′ , j ≤ j ′ . Proposition 5.5. ( ) ( Compare [Moc16a, 2.3.5].)
The inclusion functor iS ▽· C → iS · C is a homotopy equivalence. ( ) ( Compare [Moc16a, 2.3.6].)
The inclusion functor i △ S ▽· C → iS ▽· C is a homotopy equivalence. ( ) ( Compare [Moc16a, 2.3.7].)
The associations m ′ , m ′ : C → Ch b ( M B ( )) induces simplicial maps m , m : i △ S ▽· C → iS · B respectively. ( ) ( Compare [Moc16a, 2.3.7].) m is homotopic to m .Proof. ( ) Since C is semi-simple by Lemma 5.2 ( ) , the inclusion functor k : iS ▽· C → iS · C is an equiva-lence of categories for each degree. Therefore the inclusion functor k induces a weak homotopy equivalence NiS ▽· C → NiS · C . ( ) First for non-negative integer n , let i n C be the full subcategory of C [ n ] the functor category from thetotally ordered set [ n ] = { < < ··· < n } to C consisting of those objects x : [ n ] → C such that x ( i ≤ i + ) is an isomorphism in C for any 0 ≤ i ≤ n −
1. Next for integers n ≥ n − ≥ k ≥
0, let i n C ( k ) bethe full subcategory of i n C consisting of those objects x : [ n ] → C such that x ( i ≤ i + ) is in i △ for any k ≤ i ≤ n −
1. In particular i n C ( ) = i △ n C and by convention, we set i n C ( n ) = i n C . There is a sequence ofinclusion functors; i △ n C = i n C ( ) j ֒ → i n C ( ) j ֒ → ··· j n − ֒ → i n C ( n ) = i n C . For each 0 ≤ k ≤ n −
1, we will define q k : i n C ( k + ) → i n C ( k ) to be an exact functor as follows. First for anyobject z in i n C ( k + ) , we shall assume that all z ( i ) are the same object, namely z ( ) = z ( ) = ··· = z ( n ) . Thenwe define a z to be an isomorphism of z ( i ) in C by setting a z : = UT ( z ( k ≤ k + )) . Here for the definitionof the upper triangulation UT of z ( k ≤ k + ) , see Definition 5.8 below. Next for an object x : [ n ] → C anda morphism x q → y in i n C ( k + ) , we define q k ( x ) : [ n ] → C and q k ( q ) : q k ( x ) → q k ( y ) to be an object and amorphism in i n C ( k ) respectively by setting q k ( x )( i ) : = x ( i ) , (15) q k ( x )( i ≤ i + ) : = a − x x ( k − ≤ k ) if i = k − x ( k ≤ k + ) a x if i = kx ( i ≤ i + ) otherwise , (16) q k ( q )( i ) : = ( a − y q ( k ) a x if i = k q ( i ) otherwise . (17)Obviously q k j k = id. We define g k : j k q k ∼ → id to be a natural equivalence by setting for any object x in i n C ( k + ) g k ( x )( i ) : = ( a x if i = k id x i otherwise . (18)Let s ▽· : = Ob S ▽ be a variant of s = Ob S -construction. Notice that there is a natural identification s ▽· i n C ( l ) = i n S ▽ C ( l ) for any 0 ≤ l ≤ n . We will show that g induces a simplicial homotopy between the maps ▽· j k q k and s ▽· id. The proof of this fact is similar to [Wal85, Lemma 1.4.1]. The key point of well-definedness of the simplicial homotopy is that each component of g is lower triangular. Therefore theinclusion i n S ▽· C ( k ) → i n S ▽· C ( k + ) is a homotopy equivalence. Hence by realization lemma [Seg74, Ap-pendix A] or [Wal78, 5.1], NiS ▽· C ( k ) → NiS ▽· C ( k + ) is also a homotopy equivalence for any 0 ≤ k ≤ n − ( ) Notice that for a pair of composable morphisms in C , ( n , m ) B j → ( n ′ , m ′ ) B y → ( n ′′ , m ′′ ) B , (19) ( i ) if both j and y are upper triangular or both j and y are lower triangular, then we have the equality m ′ i ( yj ) = m ′ i ( y ) m ′ i ( j ) for i =
1, 2, ( ii ) if the sequence ( ) is exact in C , then the sequence m ′ i (( n , m ) B ) m ′ i ( j ) → m ′ i (( n ′ , m ′ ) B ) m ′ i ( y ) → m ′ i (( n ′′ , m ′′ ) B ) is exact in B for i =
1, 2 by Lemma 5.9 below and ( iii ) if j is an isomorphism in C , then m ′ i ( j ) is an isomorphism in B for i =
0, 1 by Lemma 5.7 below.Thus the associations m ′ and m ′ induce the simplicial functors m , m : i △ S ▽· C → iS · B . ( ) Inspection shows the equalitiy ( ) . Hence m is homotopic to m by the additivity theorem. Definition 5.6 (Upside-down involution) . (Compare [Moc16a, 1.2.17.].) Let x = [ x d x → x ] be an objectin C . Since d x is a monomorphism and x and x have the same rank, d x is invertible in B (cid:20) g (cid:21) and g ( d x ) − : x → x is a morphism of B -modules. We define UD: C → C to be a functor by sending anobject [ x d x → x ] to [ x g ( d x ) − → x ] and a morphism j : x → y to x ↓ g ( d x ) − x j →→ j y ↓ g ( d y ) − y . Namely we have the equations: UD (( n , m ) B ) = ( m , n ) B , UD (cid:18)(cid:18) j ( n ′ , n ) j ( n ′ , m ) j ( m ′ , n ) j ( m ′ , m ) (cid:19) : ( n , m ) B → ( n ′ , m ′ ) B (cid:19) = (cid:18) j ( m ′ , m ) j ( m ′ , n ) j ( n ′ , m ) j ( n ′ , n ) (cid:19) . Obviously UD is an involution and an exact functor. We call UD the upside-down involution . Lemma 5.7. ( Compare [Moc16a, 1.2.18.].)
Let j = (cid:18) j ( n , n ) j ( n , m ) j ( m , n ) j ( m , m ) (cid:19) : ( n , m ) B → ( n , m ) B be an isomor-phism in C . Then j ( n , n ) and j ( m , m ) are invertible.Proof. For j ( n , n ) , assertion follows from Lemma 5.2 ( ) . For j ( m , m ) , we apply the same lemma to UD ( j ) . Definition 5.8 (Upper triangulation) . (Compare [Moc16a, 2.2.5.].) Let j = (cid:18) j ( n , n ) j ( n , m ) j ( m , n ) j ( m , m ) (cid:19) : ( n , m ) B → ( n , m ) B be an isomorphism in C . By Lemma 5.7, j ( m , m ) is an isomorphism. We define UT ( j ) : ( n , m ) B → ( n , m ) B to be a lower triangular isomorphism by the formula UT ( j ) : = E n − j − ( m , m ) j ( m , n ) E m ! . Then we have anequality j UT ( j ) = j ( n , n ) − g j ( n , m ) j − ( m , m ) j ( m , n ) j ( n , m ) j ( m , m ) ! . (20)We call UT ( j ) the upper triangulation of j . Notice that if j is upper triangular, then UT ( j ) = id ( n , m ) B . emma 5.9 (Exact sequences in C ) . ( Compare [Moc16a, 1.2.19.].)
Let ( n , ) B a → ( n ′ , ) B b → ( n ′′ , ) B (21) be a sequence of morphisms in C such that ba = . If the induced sequence of projective B / gB-modules H (( n , ) B ) H ( a ) → H (( n ′ , ) B ) H ( b ) → H (( n ′′ , ) B ) (22) is exact, then the sequence ( ) is also ( split ) exact.Proof. Since the sequence ( ) is an exact sequence of projective B / gB -modules, it is a split exact se-quence and hence we have the equality n ′ = n + n ′′ and there exists a homomorphism of B / gB -modules g : H (( n ′′ , ) B ) → H (( n ′ , ) B ) such that H ( b ) g = id H (( n ′′ , ) B ) . Then by [Wei94, Comparison theorem2.2.6.], there is a morphism of complexes of B -modules g : ( n ′′ , ) B → ( n ′ , ) B such that H ( g ) = g . Since bg is an isomorphism by Lemma 5.2 ( ) , by replacing g with g ( bg ) − , we shall assume that bg = id ( n ′′ , ) B .Therefore there is a commutave diagram ( n , ) B a / / d (cid:15) (cid:15) ✤✤✤ ( n ′ , ) B b / / ( n ′′ , ) B ( n , ) B / / a ′ / / ( n ′ , ) B b / / / / ( n ′′ , ) B such that the bottom line is exact. Here the dotted arrow d is induced from the universality of Ker b .By applying the functor H to the diagram above and by the five lemma, it turns out that H ( d ) is anisomorphism of projective B / gB -modules and hence d is also an isomorphism by Lemma 5.2 ( ) . Wecomplete the proof. We denote the simplicial morphism i △ S ▽· C → qis S · Ch b ( M B ) induced from the inclusion functor h : C ֒ → Ch b ( M B ) by the same letter h . For simplicial functors h , j m , j m , i △ S ▽· C → qis S · Ch b ( M B ) , thereis a canonical natural transformation j m → h → j m . A candidate of h → j m is the following.For any object ( n , m ) B in C , we write d ( n , m ) B : h (( n , m ) B ) → j m ′ (( n , m ) B ) for the canonical projection B ⊕ n ⊕ B ⊕ m ↓ (cid:18) gE n E m (cid:19) B ⊕ n ⊕ B ⊕ m (cid:16) E n (cid:17) →→ (cid:16) E n (cid:17) B ⊕ n ↓ gE n B ⊕ n . This is not a simplicial natural transformation. But it has thefollowing nice properties. For any morphism j = (cid:18) j ( n ′ , n ) j ( n ′ , m ) j ( m ′ , n ) j ( m ′ , m ) (cid:19) : ( n , m ) B → ( n ′ , m ′ ) B in C , ( i ) d ( n , m ) B is a chain homotopy equivalence, ( ii ) if j is lower triangular, we have the equality j m ′ ( j ) d ( n , m ) B = d ( n ′ , m ′ ) B h ( j ) , ( iii ) if j is upper triangular, there is a unique chain homotopy between j m ′ ( j ) d ( n , m ) B and d ( n ′ , m ′ ) B h ( j ) .Namely since we have the equality j m ′ ( j ) d ( n , m ) B − d ( n ′ , m ′ ) B h ( j ) = (cid:18) − j ( n ′ , m ) (cid:19) , the map (cid:0) − j ( n ′ , m ) (cid:1) : B ⊕ n ⊕ B ⊕ m → B ⊕ n ′ gives a chain homotopy between j m ′ ( j ) d ( n , m ) B and d ( n ′ , m ′ ) B h ( j ) . B ⊕ n ⊕ B ⊕ m ( − j ( n ′ , m ) ) / / gE n E m ! (cid:15) (cid:15) B ⊕ n ′ gE n (cid:15) (cid:15) B ⊕ n ⊕ B ⊕ m ( − j ( n ′ , m ) ) : : ✉✉✉✉✉✉✉✉✉✉ ( − g j ( n ′ , m ) ) / / B ⊕ n ′ . hus we establish a theory of homotopy natural transformations. Then it turns out that d induces a sim-plicial homotopy natural transformation h ⇒ simp j m . (For the definition of simplicial homotopy naturaltransformations, see Definition 5.13 below.) Therefore by the theory below, it turns out that there exists azig-zag sequence of simplicial natural transformations which connects h and j m . Thus finally we obtainthe fact that h is homotopic to 0. We complete the proof of zero map theorem.The rest of this subsection, we will establish a theory of homotopy natural transformations and justify theargument above. Conventions.
For simplicity, we set E = Ch b ( M B ( )) . The functor C : E → E is given by sending a chain complex x in E to Cx : = Cone id x the canonical mapping cone of the identity morphism of x . Namely the degree n part of Cx is ( Cx ) n = x n − ⊕ x n and the degree n boundary morphism d Cxn : ( Cx ) n → ( Cx ) n − is given by d Cxn = (cid:18) − d xn − − id x n − d xn (cid:19) . For any complex x , we define i x : x → Cx and r x : CCx → Cx to be chain morphismsby setting ( i x ) n = (cid:18) x n (cid:19) and ( r x ) n = (cid:18) x n − id x n −
00 0 0 id x n (cid:19) .Let f , g : x → y be a pair of chain morphisms f , g : x → y in E . Recall that a chain homotopy from f to g is a family of morphisms { h n : x n → y n + } n ∈ Z in M B ( ) indexed by the set of integers such that it satisfiesthe equality d yn + h n + h n − d xn = f n − g n (23)for any integer n . Then we define H : Cx → y to be a chain morphism by setting H n : = (cid:0) − h n − f n − g n (cid:1) for any integer n . We can show that the equality f − g = H i x . (24)Conversely if we give a chain map H : Cx → y which satisfies the equality ( ) above, it provides a chainhomotopy from f to g . We denote this situation by H : f ⇒ C g and we say that H is a C-homotopy from fto g . We can also show that for any complex x in E , r x is a C -homotopy from id Cx to 0.Let [ f : x → x ′ ] and [ g : y → y ′ ] be a pair of objects in E [ ] the morphisms category of E . A ( C- ) homotopycommutative square ( from [ f : x → x ′ ] to [ g : y → y ′ ] ) is a triple ( a , b , H ) consisting of chain morphisms a : x → y , b : x ′ → y ′ and H : Cx → y ′ in E such that H i x = ga − b f . Namely H is a C -homotopy from ga to b f .Let [ f : x → x ′ ] , [ g : y → y ′ ] and [ h : z → z ′ ] be a triple of objects in E [ ] and let ( a , b , H ) and ( a ′ , b ′ , H ′ ) be homotopy commutative squares from [ f : x → x ′ ] to [ g : y → y ′ ] and from [ g : y → y ′ ] to [ h : z → z ′ ] respectively. Then we define ( a ′ , b ′ , H ′ )( a , b , H ) to be a homotopy commutative square from [ f : x → x ′ ] to [ h : z → z ′ ] by setting ( a ′ , b ′ , H ′ )( a , b , H ) : = ( a ′ a , b ′ b , H ′ ⋆ H ) (25)where H ′ ⋆ H is a C -homotopy from ha ′ a to b ′ b f given by the formula H ′ ⋆ H : = b ′ H + H ′ Ca . (26)We define E [ ] h to be a category whose objects are morphisms in E and whose morphisms are homotopycommutative squares and compositions of morphisms are give by the formula ( ) and we define E [ ] → E [ ] h to be a functor by sending an object [ f : x → x ′ ] to [ f : x → x ′ ] and a morphism ( a , b ) : [ f : x → x ′ ] → [ g : y → y ′ ] to ( a , b , ) : [ f : x → x ′ ] → [ g : y → y ′ ] . By this functor, we regard E [ ] as a subcategory of E [ ] h .We define Y : E [ ] h → E to be a functor by sending an object [ f : x → y ] to Y ( f ) : = y ⊕ Cx and a homotopycommutative square ( a , b , H ) : [ f : x → y ] → [ f ′ : x ′ → y ′ ] to Y ( a , b , H ) : = (cid:18) b − H Ca (cid:19) .We write s and t for the functors E [ ] h → E which sending an object [ f : x → y ] to x and y respectively. Wedefine j : s → Y and j : t → Y to be natural transformations by setting j f : = (cid:18) f − i x (cid:19) and j f : = (cid:18) id y (cid:19) respectively for any object [ f : x → y ] in E [ ] h . Definition 5.10 (Homotopy natural transformations) . Let I be a category and let f , g : I → E be apair of functors. A homotopy natural transformation ( from f to g ) is consisting of a family of morphims { q i : f i → g i } i ∈ Ob I indexed by the class of objects of I and a family of C -homotopies { q a : g a q a ⇒ C q j f a } a : i → j ∈ Mor I indexed by the class of morphisms of I such that for any object i of I , q id i = or any pair of composable morphisms i a → j b → k in I , q ba = q b ⋆ q a (= g b q a + q b C f a ) . We denote thissituation by q : f ⇒ g . For a usual natural transformation k : f → g , we regard it as a homotopy naturaltransformation by setting k a = a : i → j in I .Let h and k be another functors from I to E and let a : f → g and g : h → k be natural transformations and b : g ⇒ h a homotopy natural transformation. We define ba : f ⇒ h and gb : g ⇒ k to be homotopy naturaltransformations by setting for any object i in I , ( ba ) i = b i a i and ( gb ) i = g i b i and for any morphism a : i → j in I , ( ba ) a : = b a C ( a i ) and ( gb ) a = g j b a . Example 5.11 (Homotopy natural transformations) . We define e : s ⇒ t and p : Y ⇒ t to be homotopynatural transformations between functors E [ ] h → E by setting for any object [ f : x → y ] in E [ ] h , e f : = f : x → y and p f : = (cid:0) id y (cid:1) : Y ( f ) = y ⊕ Cx → y and for a homotopy commutative square ( a , b , H ) : [ f : x → y ] → [ f ′ : x ′ → y ′ ] , e ( a , b , H ) : = H : f ′ a ⇒ C b f and p ( a , b , H ) : = (cid:0) − Hr x (cid:1) : b (cid:0) id y (cid:1) ⇒ C (cid:0) id y ′ (cid:1)(cid:18) b − H Ca (cid:19) .Then we have the commutative diagram of homotopy natural transformations. s j / / e (cid:31) (cid:31) ❃❃❃❃❃❃❃ Y p (cid:15) (cid:15) t j o o id t (cid:127) (cid:127) (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0) t . Here we can show that for any object [ f : x → y ] in E [ ] h , p f and j f are chain homotopy equivalences. Inparticular if f is a chain homotopy equivalence, then j f is also a chain homotopy equivalence. Definition 5.12 (Mapping cylinder functor on Nat h ( E I ) ) . Let I be a small category. We will defineNat h ( E I ) the category of homotopy natural transformations ( between the functors from I to E ) as follows.An object in Nat h ( E I ) is a triple ( f , g , q ) consisting of functors f , g : I → E and a homotopy naturaltransformation q : f ⇒ g . A morphism ( a , b ) : ( f , g , q ) → ( f ′ , g ′ , q ′ ) is a pair of natural transformations a : f → f ′ and b : g → g ′ such that q ′ a = b q . Compositions of morphisms is given by componentwisecompositions of natural transformations.We will define functors S , T , Y : Nat h ( E I ) → E I and natural transformations J : S → Y and J : T → Y as follows. For an object ( f , g , q ) and a morphism ( a , b ) : ( f , g , q ) → ( f ′ , g ′ , q ′ ) in Nat h ( E I ) and an object i and a morphism a : i → j in I , we set S ( f , g , q ) : = f , S ( a , b ) : = a , T ( f , g , q ) : = g , T ( a , b ) : = b , Y ( f , g , q ) i (= Y ( q ) i ) : = Y ( q i )(= g i ⊕ C ( f i )) , Y ( f , g , q ) a (= Y ( q ) a ) : = Y ( f a , g a , q a ) (cid:18) = (cid:18) g a − q a C f a (cid:19)(cid:19) , Y ( a , b ) i : = (cid:18) b i C ( a i ) (cid:19) , J ( f , g q ) i : = j q i , J ( f , g q ) i : = j q i . In particular for an object ( f , g , q ) in Nat h ( E I ) if for any object i of I , q i is a chain homotpy equivalence,then there exists a zig-zag diagram which connects f to g , f J → Y ( q ) J ← g such that for any object i , J i and J i are chain homotopy equivalences. Definition 5.13 (Simplicial homotopy natural transformations) . Let J be a simplicial object in the categoryof small categories which we call shortly a simplicial small category and let f , g : J → Ch b ( S · M B ( )) besimplicial functors. Recall that a simplicial natural transformaion ( from f to g ) is a family of naturaltransformations { r n : f n → g n } n ≥ indexed by non-negative integers such that for any morphism j : [ n ] → [ m ] , we have the equality r n f j = g j r m .A simplicial homotopy natural transformation ( from f to g ) is a family of homotopy natural transformations { q n : f n ⇒ g n } n ≥ indexed by non-negative integers such that for any morphism j : [ n ] → [ m ] , we have theequality q n f j = g j q m . We denote this situation by q : f ⇒ simp g .For a simplicial homotopy natural transformation q : f ⇒ simp g , we will define Y ( q ) : J → Ch b ( S · M B ( )) and J : f → Y ( q ) and J : g → Y ( q ) to be a simplicial functor and simplicial natural transformations re-spectively as follows. For any [ n ] and any morphism j : [ m ] → [ n ] , we set Y ( q ) n : = Y ( q n ) , J n : = J q n , J n : = J q n and Y ( q ) j : = Y ( f j , g j ) . In particular if for any non-negative integer n , any object j of J n , q n j is a chain homotpy equivalence, then there exists a zig-zag sequence which connects f to g , f J → Y ( q ) J ← g such that for any non-negative integer n and any object j , J n j and J n j are chain homotopyequivalences. Motivic Gersten’s conjecture
In this last section, I briefly explain my recent work of motivic Gernsten’s conjecture in [Moc16b].
We denote the category of small dg-categories over Z the ring of integers by dgCat and let M ot adddg and M ot locdg be symmetric monoidal stable model categories of additve noncommutative motives and localizingmotives over Z respectively. (See [CT12, § M by Ho ( M ) . There are functors from ExCat the category of small exact categories to dgCat which sendinga small exact category E to its bounded dg-derived category D b dg ( E ) (see [Kel06, § U add : dgCat → Ho ( M ot adddg ) and U loc : dgCat → Ho ( M ot locdg ) . We denote the compositions ofthese functors ExCat → Ho ( M ot adddg ) and ExCat → Ho ( M ot locdg ) by M add and M loc respectively. Then wecan ask the following question: Conjecture . For any commutative regular local ring R and for any integers 1 ≤ p ≤ dim R , the inclusion functor M p − R ֒ → M pR induces the zero morphism M ( M p − R ) → M ( M pR ) in Ho ( M ot ) where ∈ { add , loc } .Since in the proof of zero morphism theorem in §
5, what we just using are basically the resolution theoremand the additivity theorem, by mimicking the proof of zero morphism theorem, we obtain the followingresult:
Proposition 6.2 (Motivic zero morphism theorem) . Let B be a commutative noetherian ring with and letg be an element in B such that g is a non-zero divisor and contained in the Jacobson radical of B. Assumethat every finitely generated projective B-modules are free. Then the inclusion functor P B / gB ֒ → M B ( ) induces the zero morphism M ( P B / gB ) → M ( M B ( )) in M ot where ∈ { add , loc } .Remark . Strictly saying, in the proof above, we shall moreover utilize the following two facts.1. For any pair of exact functors f , g : E → Ch b ( F ) from an exact category E to the category of boundedchain complexes on an exact category F , if there exists a zig-zag sequence of natural equivalenceswhich connects f and g and whose components are quasi-isomorphisms, then f and g induce the samemorphisms M ( f ) = M ( g ) in M ot where ∈ { add , loc } .2. Let D be the simplicial category and let e be the final object in the category of small categories andlet p : D → e be the projection functor. We denote the derivator associated with the model category M ot locdg by Mot locdg . For a small dg category A , there is an object U loc ( S · A ) in Mot locdg ( D ) where S · is the Segal-Waldhausen S -construction. We have the canonical isomorphism in Ho ( M ot locdg ) = Mot locdg ( e ) p ! U loc ( S · A ) = U loc ( A )[ ] . (See [Tab08, Notation 11.11, Theorem 11.12].) A problem to obtain the motivic Gersten’s conjecture from Proposition 6.2 is that in the proof in § ( ) III .) Thus if we imitate the proof, then weneed to modify statement of the conjecture and we need to construct a new stable model category of non-commutative motives which we denote by M ot nilpdg and which we construct by localizing M ot locdg .We briefly recall the construction of the stable model category of localizing non-commutative motives M ot locdg from [Tab08]. First notice that the category dgCat carries a cofibrantly generated model structurewhose weak equivalences are the derived Morita equivalences. [Tab05, Th´eor`eme 5.3]. We fix on dgCat afibratnt resolution functor R , a cofibrant resolution functor Q and a left framing G ∗ (see [Hov99, Definition5.2.7 and Theorem 5.2.8].) and we also fix a small full subcategory dgCat f ֒ → dgCat such that it containsall finite dg cells and any objects in dgCat f are ( Z -)flat and homotopically finitely presented (see [TV07,Definition 2.1 (3)].) and dgCat f is closed uder the operations Q , QR and G ∗ and ⊗ . The construction belowdoes not depend upon a choice of dgCat f up to Dwyer-Kan equivalences. et s \ dgCat f and s \ dgCat f • be the category of simplicial presheaves and that of pointed simplicial preshaveson dgCat f respectively. We have the projective model structures on s \ dgCat f and s \ dgCat f • where the weakequivalences and the fibrations are the termwise simplicial weak equivalences and termwise Kan fibrationsrespectively. (see [Hir03, Theorem 11.6.1].)We denote the class of derived Morita equivalences in dgCat f by S and we also write S + for the im-age of S by the composition of the Yoneda embedding h : dgCat f → s \ dgCat and the canonical functor ( − ) + : s \ dgCat f → s \ dgCat f • .Let P be the canonical map /0 → h ( /0 ) in s \ dgCat f and we write P + for the image of P by the functor ( − ) + .We write L S , P s \ dgCat f • for the left Bosufield localization of s \ dgCat f • by the set S + ∪ { P + } . The Yonedaembedding functor induces a functor R h : Ho ( dgCat ) → Ho ( L S , P s \ dgCat f • ) which associates any dg category A to the pointed simplicial presheaves on dgCat f : R h ( A ) : B Hom ( G ∗ ( Q B ) , R ( A )) + . Let E be the class of morphisms in L S , P s \ dgCat f • of shapeCone [ R h ( A ) → R h ( B )] → R h ( C ) associated to each exact sequence of dg categories A → B → C , with B in dgCat f where Cone means homotopy cofiber. We write M ot ulocdg for the left Bosufield localizationof L S , P s \ dgCat f • by E and call it the model category of unstable localizing non-commutative motives .Finally we write M ot locdg for the stable symmetric monoidal model category of symmetric S ⊗ M ot ulocdg (see [Hov01, § model category of localizing non-commutative motives .Next we construct the stable model category M ot nilpdg . First recall that we say that a non-empty full sub-category Y of a Quillen exact category X is a topologizing subcategory of X if Y is closed under finitedirect sums and closed under admissible sub- and quotient objects. The naming of the term ‘topologizing’comes from noncommutative geometry of abelian categories by Rosenberg. (See [Ros08, Lecture 2 1.1].)We say that a full subcategory Y of an exact category X is a Serre subcategory if it is an extensional closedtopologizing subcategory of X . For any full subcategory Z of X , we write S √ Z for intersection of allSerre subcategories which contain Z and call it the Serre radical of Z ( in X ). Definition 6.4 (Nilpotent immersion) . Let A be a noetherian abelian category and let B a topologizingsubcategory. We say that B satisfies the d´evissage condition ( in A ) or say that the inclusion B ֒ → A is a nilpotent immersion if one of the following equivalent conditions holds:1. For any object x in A , there exists a finite filtration of monomorphisms x = x x x ··· x n = i < n , x i / x i + is isomorphic to an object in B .2. We have the equality A = S √ B . (For the proof of the equivalence of the conditions above, see [Her97, 3.1], [Gar09, 2.2].) Definition 6.5.
We write N for the class of morphisms in M ot ulocdg of shape R h ( D b dg ( B )) → R h ( D b dg ( A )) asscoated with each noetherian abelian category A and each nilpotent immersion B ֒ → A .We write M ot unilpdg the left Bousfield localization of M ot ulocdg by N and call it the model category of unstablenilpotent invariant non-commutative motives .Finally we write M ot nilpdg for the stable model category of symmetric S ⊗ M ot unilpdg and call itthe stable model category of nilpotent invariant non-commutative motives . We denote the compositons ofthe following functorsExCat D b dg → dgCat → Ho ( dgCat ) R h → Ho ( L S , P s \ dgCat f • ) → Ho ( M ot unilpdg ) S ¥ → Ho ( M ot nilpdg ) by M nilp . e obtain the following theorem. Theorem 6.6 (Motivic Gersten’s conjecture) . For any commutative regular local ring R and for any integers ≤ p ≤ dim R, the inclusion functor M p − R ֒ → M pR induces the zero morphismM nilp ( M p − R ) → M nilp ( M pR ) in Ho ( M ot nilpdg ) . References [Bal09] P. Balmer,
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