aa r X i v : . [ m a t h . K T ] A p r Gersten’s conjecture
Satoshi Mochizuki
Abstract
The purpose of this article is to prove that Gersten’s conjecture for a commutativeregular local ring is true. As its applications, we will prove the vanishing conjecture forcertain Chow groups, generator conjecture for certain K -groups and Bloch’s formula forabsolute case. Contents
The purpose of this note is to prove the following theorem.
Theorem 0.1 (Gersten’s conjecture) . For any commutative regular local ring R , Gersten’s conjecture is true.That is for any naturalnumbers n , p , the canonical inclusion M p +1 ( R ) ֒ → M p ( R ) induces the zero map on K -groups K n ( M p +1 ( R )) → K n ( M p ( R )) , where M i ( R ) is the category of finitely generated R -modules M with Codim
Spec R Supp M ≧ i . Gersten’s conjecture is proposed in [Ger73]. More precise historical back grounds of this con-jecture are explained in [Moc07]. In §
1, we will prove the main theorem and in §
2, we will alsodiscuss applications of this conjecture.
Acknowledgement
The author thankful to Shuji Saito for encouraging him, to Fabrice Or-gogozo for stimulating argument about Corollary 2.4, to Takeshi Saito for making him to getto the reduction argument in Lemma 1.3, and to Kazuhiko Kurano for teaching him condition(iii) in Proposition 2.2. 1
Proof of the main theorem
From now on, let R be a commutative regular local ring. Proof of the main theorem is dividedseries of lemmas. First we will improve Quillen’s reduction argument in the proof of Gersten’sconjecture in [Qui73]. Lemma 1.1 (Quillen induction) . To prove the main theorem, we shall only check the following assertion:For any non-negative integers n , p , and any regular sequence f , . . . , f p +1 in R , the canonicalmap induced from the inclusion map P ( R/ ( f , . . . , f p +1 )) ֒ → M p ( R ) , K n ( P ( R/ ( f , . . . , f p +1 ))) → K n ( M p ( R )) is zero.Proof. In the proof of Theorem 5.11 in [Qui73], we have the following formula K n ( M p +1 ( R )) = colim t : regularelement K n ( M p ( R/tR )) . Since R is UFD by [AB59], for any regular element t in R , we can write t = p e p e . . . p e r r where p i are prime elements. By d´evissage theorem in [Qui73], we have the following formula K n ( M p ( R/tR )) ∼ → K n ( M p ( R/p p . . . p r R )) . Claim
We have the following formula K n ( M p ( R/p p . . . p r R )) ∼ → r M i =1 K n ( M p ( R/p i R )) . Proof of
Claim . We put X = Spec R/p p . . . p r R and X i = Spec R/p i R . For any closed set Y ⊂ X ? , we put Perf Y ( X ? ) the category of strictly perfect complexes which are acyclic on X ? − Y . We also put Perf p ( X ? ) := ∪ Y ⊂ X ? Codim Y ≧ p Perf Y ( X ? ). Then we have the following identities K n ( M p ( X )) ∼ → I colim Y ⊂ X Codim Y ≧ p K ′ n ( Y ) ∼ → II colim Y ⊂ X Codim Y ≧ p K n ( X on Y ) ∼ → colim Y ⊂ X Codim Y ≧ p r M i =1 K n ( X i on X i ∩ Y ) ∼ → III r M i =1 colim Y ⊂ X i Codim Y ≧ p K n ( X i on Y ) ∼ → II r M i =1 colim Y ⊂ X i Codim Y ≧ p K ′ n ( Y ) ∼ → I p M i =1 K n ( M p ( X i ))where the isomorphisms I are proved by continuity [Qui73], [TT90], the isomorphisms II areproved by the Poincar´e duality and comparing the following fibration sequences [Qui73] and[TT90] K ′ ( Y ) → K ′ ( X ? ) → K ′ ( X ? − Y ) ,K ( X ? on Y ) → K ( X ? ) → K ( X ? − Y )for any closed set Y ⊂ X and to prove the isomorphism III, we are using the fact that all X i are equidimensional. 2herefore to prove Gersten’s conjecture we shall only check that for any prime element f ,the inclusion map M p ( R/f R ) → M p ( R ) induces the zero map K n ( M p ( R/f R )) → K n ( M p ( R )) . Since
R/f R is regular, inductive argument implies that to prove Gersten’s conjecture we shallonly check that for any regular sequence f , . . . , f p +1 such that ( f , . . . , f p +1 ) is prime ideal,the inclusion map M ( R/ ( f , . . . , f p +1 )) → M p ( R ) induces the zero map K n ( M ( R/ ( f , . . . , f p +1 )) → K n ( M p ( R )) . Since R/ ( f , . . . , f p +1 ) is regular, we have K n ( P ( R/ ( f , . . . , f p +1 )) ∼ → K n ( M ( R/ ( f , . . . , f p +1 ))by resolution theorem in [Qui73]. Hence we get the result.Now Lemma 1.1 implies the following assertion by famous Gersten-Sherman argument in[Ger73], [She82] p.240, which is an application of the universal property for algebraic K -theoryassociated with semisimple exact categories [She92] Corollary 5.2. From now on let F bethe category of finite pointed connected CW-complexes and frequently using the notations in[Moc07]. Lemma 1.2 (Gersten-Sherman reduction argument) . To prove the main theorem, we shall only check the following assertion:For any X ∈ F , any non-negative integer p , and any regular sequence f , . . . , f p +1 in R , thecanonical map induced from the inclusion map P ( R/ ( f , . . . , f p +1 )) ֒ → M p ( R ) , ˜ R ( π ( X ) , P ( R/ ( f , . . . , f p +1 ))) → ˜ R ( π ( X ) , M p ( R )) ˜Sh → [ X, ( K ( M p ( R ))) ] ∗ is zero.Proof. We have the following commutative diagram for each X ∈ F :˜ R ( π ( X ) , P ( R/ ( f , . . . , f p +1 ))) / / ˜Sh (cid:15) (cid:15) ˜ R ( π ( X ) , M p ( R )) ˜Sh (cid:15) (cid:15) [ X, ( K ( P ( R/ ( f , . . . , f p +1 )))) ] ∗ / / [ X, ( K ( M p ( R ))) ] ∗ It is well-known that K ( M p ( R )) is a H -space, P ( R/ ( f , . . . , f p +1 )) is semi-simple and by theuniversal property [She92] Corollary 5.2, we learn that we shall only prove the composition˜ R ( π ( X ) , P ( R/ ( f , . . . , f p +1 ))) → ˜ R ( π ( X ) , M p ( R )) ˜Sh → [ X, ( K ( M p ( R ))) ] ∗ is the zero map for any X ∈ F .Next we will define equivalence relations between morphisms in M p ( R ) as follows:For any R -modules M , N in M p ( R ), and morphisms f, g : M → N , we will declare f ∼ g .Then M p ( R ) is an exact category with equivalence relations satisfying the cogluing axiom inthe sense of [Moc07]. So we can define the Grothendieck group of lax G -representations in M p ( R ). (For the precise definition, see [Moc07] Definition 3.9). Lemma 1.3 (Retraction principle) . To prove main theorem, we shall only check the following assertion:In the notation
Lemma 1.2 , the canonical map induced from the inclusion map P ( R/ ( f , . . . , f p +1 )) ֒ → M p ( R ) , ( G, P ( R/ ( f , . . . , f p +1 ))) → R lax ( G, M p ( R )) is zero.Proof. We have the following commutative diagram for each X ∈ F :˜ R ( π ( X ) , P ( R/ ( f , . . . , f p +1 ))) / / ˜ R ( π ( X ) , M p ( R )) ˜Sh / / (cid:15) (cid:15) [ X, ( K e ( M p ( R ))) ] ∗ I (cid:15) (cid:15) ˜ R lax0 ( π ( X ) , M p ( R )) ˜Sh lax / / [ X, ( K lax , e ( M p ( R ))) ] ∗ where the morphism I is a injection by retraction theorem 3.13 in [Moc07]. Hence we get theresult.The following argument is one of a variant of weight argument of the Adams operations. (See[Moc07] § Lemma 1.4 (Weight changing argument) . The assertion in
Lemma 1.3 is true. Therefore Gersten’s conjecture is true.Proof.
We put B = R/ ( f , . . . , f p ). Let G be a group and ( X, ρ X ) be a representation in P ( B/f p +1 B ). Since B/f p +1 B is local, X is isomorphic to ( B/f p +1 B ) ⊕ m for some m as a B/f p +1 B -module. Then there is a short exact sequence0 → B ⊕ m f p +1 → B ⊕ m π → X → . For each g ∈ G , we have a lifting of ρ X ( g ), that is, a R -module homomorphism ˜ ρ ( g ) : B ⊕ m → B ⊕ m such that ˜ ρ ( g ) mod f p +1 = ρ X ( g ). Since [ B ⊕ m f p +1 → B ⊕ m ] is a minimal resolution of X as a B -module, (For the definition of a minimal resolution, see [Ser00] p.84.) we can easilylearn that ˜ ρ ( g ) is an isomorphism as a B -modules by Nakayama’s lemma. Therefore ˜ ρ ( g ) isan isomorphism as a R -modules. Obviously assignment ˜ ρ : G → Aut( B ⊕ m ) defines a laxrepresentation ( B ⊕ m , ˜ ρ ) in M p ( R ) and we have a short exact sequence( B ⊕ m , ˜ ρ ) f p +1 → ( B ⊕ m , ˜ ρ ) π → ( X, ρ X )in LAX (G , M p ( R )) s . Notice that proving f p +1 is a strict deformation, we need the assumptionthat R is commutative!! So we have an identity[( X, ρ X )] = [( B ⊕ m , ˜ ρ )] − [( B ⊕ m , ˜ ρ )] = 0in R lax0 ( G, M p ( R )). Hence we get the result. In this section, we will discuss applications of
Theorem 0.1 . First we get the following absoluteversion of Bloch’s formula. 4 orollary 2.1.
For a regular noetherian scheme X , there is a canonical isomorphism H p ( X, K p ) ∼ → A p ( X ) where K p is the Zariski sheaf on X associated to the presheaf U K n ( U ) and A p ( X ) is definedby the following formula A p ( X ) := Coker( a x ∈ X p − k ( x ) × ord x → a x ∈ X p Z ) . Here X i is the set of points of codimension i in X .Proof. Combining Propositions 5.8 and 5.14 and Remark 5.17 in [Qui73] and
Theorem 0.1 ,we can easily obtain the result.Next we will cite the following well-known statement.
Proposition 2.2. (c.f. [Lev85] P.452, Proposition 1.1, [Moc07] Proposition 1.2)
Let A be a commutative regularlocal ring. Then the following statements are equivalent. (i) The maps K ( M p ( A )) → K ( M p − ( A )) are zero for p = 1 , · · · , dim A . (ii) K ( M p ( A )) is generated by cyclic modules A/ ( f , · · · , f p ) where f , · · · , f p forms a regularsequence for p = 1 , · · · , dim A . (iii) A p (Spec A ) = 0 for any p < dim A . Therefore we get the following results.
Corollary 2.3 (Vanishing conjecture) . For any commutative regular local ring R and any p < dim R , we have A p (Spec A ) = 0 . Corollary 2.4 (Generator conjecture) . For any commutative regular local ring R , K ( M p ( R )) is generated by cyclic modules R/ ( f , · · · , f p ) where f , · · · , f p forms a regular sequence for p = 1 , · · · , dim R . References [AB59] M. Auslander and D. Buchsbaum,
Unique factorization in regular local rings , Proc.Nat. Acad. Sci. USA., (1959), p.733-734.[Ger73] S. Gersten, Some exact sequences in the higher K-theory of rings , In Higher K-theories,Springer Lect. Notes Math. (1973), p.211-243.[Lev85] M. Levine, A K -theoretic approach to multiplicities , Math. Ann. (1985), p.451-458.[Moc07] S. Mochizuki, Gersten conjecture for commutative discrete valuation rings
Higher algebraic K-theory I , In Higher K-theories, Springer Lect. NotesMath. (1973), p.85-147.[Ser00] J. P. Serre,
Local algebra , Springer monographs in Mathematics (2000)5She82] C. Sherman,