aa r X i v : . [ m a t h . K T ] A ug K-THEORY OF n -COHERENT RINGS EUGENIA ELLIS AND RAFAEL PARRA
Abstract.
Let R be a strong n -coherent ring such that each finitely n -presented R -module has finite projective dimension. We consider FP n ( R ) the full sub-category of R -Mod of finitely n -presented modules. We prove that K i ( R ) = K i ( FP n ( R )) for every i ≥ i ( R ). Introduction
Let R be an associative ring with unit. We consider the following full subcate-gories of the category of left R -modules denoted by R -Mod: FP ( R ) = { M ∈ R -Mod : M is finitely generated }FP ( R ) = { M ∈ R -Mod : M is finitely presented }FP n ( R ) = { M ∈ R -Mod : M is finitely n -presented }FP ∞ ( R ) = { M ∈ R -Mod : M has a resolution by finitely generated free modules } Proj( R ) = { M ∈ R -Mod : M is finitely generated and projective } . Observe there is a chain of inclusionsProj( R ) ⊆ FP ∞ ( R ) ⊆ FP n ( R ) ⊆ . . . ⊆ FP ( R ) ⊆ FP ( R ) ⊆ R -Mod . If R is Noetherian then FP ( R ) = FP ∞ ( R ) is an abelian category. If R is coherentthen FP ( R ) = FP ∞ ( R ) is an abelian category. In this work we consider rings R such that FP n ( R ) = FP ∞ ( R ) which are called strong n -coherent rings.We prove in Theorem 2.5 that FP n ( R ) also coincides with the category of n -coherent modules. We say that R is n -regular if the projective dimension of eachfinitely n -presented module is finite. We investigate which techniques used in [21]for regular and coherent rings can be applied to n -regular and strong n -coherentrings. In this case we prove in Theorem 3.2 such that K i ( R ) ≃ K i ( FP n ( R )) ∀ i ≥ . The main tool for this result is the Quillen’s Resolution Theorem given in [18]applied to Proj( R ) ֒ → FP n ( R ). Theorem 1.1 (Resolution Theorem) . [8, Th 2.1] Let M be an exact category andlet P be a full subcategory closed under extension in M . Suppose in addition that: (1) If → M → P → P ′ → is exact in M with P , P ′ in P , then M is in P . (2) For every M in M there is a finite P -resolution of M → P k → . . . → P d −→ P d −→ M → The first author was partially supported by ANII and CSIC. Both authors were partiallysupported by PEDECIBA and by the grant ANII FCE-3-2018-1-148588.
Then the inclusion P ֒ → M induces isomorphisms K m ( P ) ≃ −→ K m ( M ) for all m ≥ . We can also apply this theorem to the inclusion Nil(Proj( R )) ֒ → Nil( FP n ( R ))and we obtain the Proposition 4.2 which give us an expression of Nil i ( R ). If n = 1Swan in [21] prove that the new expression obtained is cero, then K i ( R [ t ]) = K i ( R ) and K i ( R [ t, t − ]) = K i ( R ) ⊕ K i − ( R ) for all i ≥ FP ( R ) ֒ → Nil( FP ( R )) be-cause FP ( R ) is abelian when R is coherent. Let us recall the Quillen’s DevissageTheorem: Theorem 1.2 (Devissage Theorem) . [8, Th 2.1] If A is an abelian category and B is a non-empty full subcategory closed under subobjects, quotient objects, and finiteproducts in A and if every object A in A has a finite filtration A ⊂ A ⊂ . . . ⊂ A k = A with A i /A i − in B for each i , then the inclusion B ֒ → A induces isomorphisms K m ( B ) ≃ −→ K m ( A ) for all m ≥ . Can we apply Devissage Theorem if n >
1? We can do it if FP n ( R ) is abelian butin Proposition 5.1 we prove this only happen when R is coherent. The propertywhich arise in FP n ( R ) when R is strong n -coherent is thickness, R is strong n -coherent if and only if FP n ( R ) is a thick category.In Section 2 we state the definition of strong n -coherent and n -regular ring.We compare this regularity with the n -Von Neumann regularity. We show thatProj( R ) = FP n ( R ) when R is n -Von Neumann regular but this do not happennecessarily when R is strong n -coherent and n -regular ring. In Section 3 we provethat Proj( R ) ֒ → FP n ( R ) induces isomorphisms K i ( R ) ≃ K i ( FP n ( R )) for all i ≥ R is strong n -coherent and n -regular. In Section 4 we generalize [21, Lemma6.3] in order to apply Resolution Theorem to Nil(Proj( R )) ֒ → Nil( FP n ( R )). Thisshows that Nil i ( R ) coincides with Nil ni ( R ) which is the cokernel of the map K i ( FP n ( R )) → K i (Nil( FP n ( R )))induced by inclusion. In Section 5 we prove that we can use Devissage Theoremto prove Nil n ( R ) = 0 if and only if R is coherent, because FP n ( R ) is abelian onlywhen R is coherent. In the last section we present an application of Theorem 3.2in the particular cases when R is an arithmetic or a valuation ring.The projective dimension of an R -module M is denoted by pd ( M ), the weakdimension of M is denoted by wd ( M ) and w.dim ( R ) is the weak dimension of R .The expression R -module means left R -module.2. n -coherent modules and rings Let n ≥ R -module M is finitely n -presented ifthere exists a sequence F n → F n − → F n − → . . . → F → M → -THEORY OF n -COHERENT RINGS 3 where F i is a finitely generated and free (or projective) module, for every 0 ≤ i ≤ n. Following [4], we denote by FP n ( R ) to the full subcategory of finitely n -presentedmodules. Note that FP ( R ) is the category of finitely generated modules and FP ( R ) is the category of finitely presented modules. Consider the λ -dimension of M λ R ( M ) = sup { n ≥ M is finitely n -presented } . If M is not finitely generated we set λ R ( M ) = −
1. The category formed by mod-ules that posses a resolution by finitely generated free (or projective) modules is FP ∞ ( R ). We say that M is finitely ∞ -presented if M ∈ FP ∞ ( R ). We inmediatelyobserve the following chain of inclusions: FP ∞ ( R ) = \ n ≥ FP n ( R ) ⊆ . . . ⊆ FP n +1 ( R ) ⊆ FP n ( R ) ⊆ . . . FP ( R ) ⊆ FP ( R ) . The finitely ∞ -presented modules can be found in the literature as pseudo coherentmodules, see [23, Example 7.1.4, Ch 2, § λ -dimension and thefinitely n -presented modules are related as follows. Lemma 2.1. [4, Remark 1.5]
For every n ≥ . (1) M ∈ FP n ( R ) if and only if λ R ( M ) ≥ n. (2) M ∈ FP n ( R ) \FP n +1 ( R ) if and only if λ R ( M ) = n. (3) M ∈ FP ∞ ( R ) if and only if λ R ( M ) = ∞ . The λ -dimension of a ring R , denoted by λ ( R ), was formulated by Vasconcelosin [22] in order to study the power series and polynomial rings. The λ -dimension of R is the greatest integer n (or ∞ if none such exits) such that λ R ( M ) ≥ n implies λ R ( M ) = ∞ . Recall that R is Noetherian ring if and only if λ ( R ) = 0 and R iscoherent if and only if λ ( R ) ≤ Definition 2.2 ( n -coherent module ) . Let n ≥ R -module M iscalled n -coherent if: • M belongs to the category FP n ( R ) . • For each submodule
N ֒ → M such that N is ( n − N isalso n -presented.Denote by n -Coh( R ) to the collection of all n -coherent R -modules. Note that if n = 1 the definition is the same as coherent module. Remark . A ring R is n -coherent if it is n -coherent as a R -module with theregular action, (i.e. if each ( n − R is n -presented). We saythat R is strong n -coherent if each n -presented R -module is ( n + 1)-presented. Theidea of n -coherent rings has been stated in the literature by D. L. Costa [5] but thisterminology is not the same as in [7]. A strong n -coherent ring is equivalent to a n -coherent ring for n = 1, but it is an open question for n ≥
2. A coherent ring is a1-coherent ring (1-strong coherent ring) and 0-strong coherent ring is a Noetherianring.In our definition, R is a strong n -coherent ring if and only if λ ( R ) ≤ n . Denoteby n -SCoh the class of all strong n -coherent rings. We observe the following chainof inclusions: 0-SCoh ⊂ ⊂ ⊂ . . . ⊂ n -SCoh ⊂ . . . EUGENIA ELLIS AND RAFAEL PARRA
Example . [4, Example 1.3] Let k be a field. Consider the following ring R = k [ x , x , x , . . . ]( x i x j ) i,j ≥ . In [4] it is shown that FP ( R ) = FP ∞ ( R ), ( x ) ∈ FP ( R ) \FP ( R ) and R/ ( x ) ∈FP ( R ) \FP ( R ). We obtain strict inclusions FP ∞ ( R ) = FP ( R ) ⊂ FP ( R ) ⊂ FP ( R )which show that R is strong 2-coherent but is it not coherent.It is well know that R is a coherent ring if and only if the category of coher-ent modules (1-Coh( R )) coincides with the category of finitely presented modules( FP ( R )). Let n ≥ n -Coh( R ) the full subcategory of n -coherentmodules. Theorem 2.5.
Let n ≥ . If R is a strong n -coherent ring then FP n ( R ) = n - Coh(R) .Proof.
Suppose M ∈ FP n ( R ). Then from [24, Proposition 1.1] we have a shortexact sequence with K ∈ FP n − ( R ).0 → K → R k → M → R k is a n -coherent module by [7, Remark 3.5] andtherefore M ∈ n -Coh(R) by [7, Theorem 2.3]. (cid:3) Corollary 2.6. [21, Corollary 2.7] If R is a coherent ring then FP ( R ) = 1 - Coh(R) . The category of FP ∞ ( R ) is thick, see [4, Theorem 1.8]. That means it is closedunder taking direct summands and for every short exact sequence0 → A → B → C → A , B , C in FP ∞ ( R ) so is the third. Because of [4,Theorem 2.4], R is strong n -coherent if and only if FP n ( R ) = FP ∞ ( R ). Proposition 2.7. [4, Corolary 2.6]
Let n ≥ . R is a strong n -coherent ring if,and only if, the category FP n ( R ) is closed under taking kernels of epimorphims. Corollary 2.8.
Let n ≥ . If R is a strong n -coherent ring then the category of n -coherent modules in R -Mod is closed under direct summands, extensions and takingcokernels of monomorphisms and kernels of epimorphism. A result of Gersten [8, Theorem 2.3] shows that if R is a regular coherent ringthen the inclusion of the finitely generated projective modules in the category of FP ( R ) induces isomorphisms K i ( FP ( R )) ≃ K i ( R ) for all i ≥
0. Recall that aring is said to be regular if every finitely generated ideal of R has finite projectivedimension. If R is coherent, Quentel in [17] shows that R is regular if and onlyif every finitely presented module has finite projective dimension, see [9, Theorem6.2.1]. Motivated by this fact, we introduce the following definition: Definition 2.9 ( n -regular ring ) . Let n ≥ R is called n -regular if each finitely n -presented R -module has finite projective dimension. -THEORY OF n -COHERENT RINGS 5 A 1-regular ring is a regular ring. In the next section we extend the mentionedresult of Gersten to strong n -coherent rings which are n -regular, see Theorem 3.2.The idea of strong n -coherent and n -regularity has been studied before for commu-tative rings. Following [5], let n, d ≥ n, d )-Ring ifevery n -presented module has projective dimension at most d . This rings are strong sup { n, d } -coherent. In [5, Example 6.5] there is an example of a non-coherent ring R of weak dimension one such that is a (2 , n -Von Neumann regular if it is an ( n, Proposition 2.10. [16, Theorem 2.1]
A commutative ring R is n -Von Neumannregular if and only if R is a strong n -coherent and n -regular ring such that everyfinitely generated proper ideal of R has non-zero annihilator. The rings we are interested in are those with the first two conditions of Propo-sition 2.10. A ring R such that every finitely generated proper ideal of R has anon-zero annihilator is called CH -Ring. Another equivalent definition is as follows:every finitely generated submodule of a projective R -module P is a direct summandof P [2, Theorem 5.4].Let n ≥
1. By [24, Theorem 3.9], every n -presented R -module is flat if andonly if R is n -Von Neumann regular. In particular every module of FP n ( R ) isprojective. Then if R is n -Von Neumann regular we obtainProj( R ) = FP n ( R ) . Under the hypothesis of n -Von-Neumann regularity there is no sense to study therelation between the K-theory of R and FP n ( R ).Let k be a field and E be a k -vector space with infinite rank. Set B = k ⋉ E the trivial extension of k by E . Let A be a Noetherian ring of global dimension1, and R = A × B the direct product of A and B . By [16, Theorem 3.4] R is a(2 , R ) = FP ( R ) . K-theory of FP n ( R )In this section we show FP n ( R ) is an exact category in the sense of [19, Definition3.1.1]. We also prove that if R is a strong n -coherent and n -regular ring then K i ( FP n ( R )) ≃ K i ( R ) ∀ i ≥ . Proposition 3.1. FP n ( R ) is an exact category.Proof. The category FP n ( R ) is a full additive subcategory of R -Mod which is anabelian category. We have to prove that FP n ( R ) is closed under extension andit has a small skeleton. The first assertion follows from [4, Proposition 1.7]. Thecategory of finitely generated R -modules FP ( R ) has a small skeleton S the setof quotient modules of { R k : k ∈ N } . The set S n = FP n ( R ) ∩ S is an skeleton of FP n ( R ). (cid:3) EUGENIA ELLIS AND RAFAEL PARRA
Theorem 3.2. If R is a strong n -coherent and n -regular ring then K i ( R ) ≃ K i ( n - Coh(R)) ≃ K i ( FP n ( R )) ∀ i ≥ . Proof.
We are going to use the Resolution Theorem with M = FP n ( R ) and P =Proj( R ). Note that Proj( R ) ⊆ FP ∞ ( R ) ⊆ FP n ( R ). Consider the following exactsequence0 → P → M → P → P , P ∈ Proj( R ) and M ∈ FP n ( R ).Because P is projective we obtain M ≃ P ⊕ P then M belongs to Proj( R ). Weconclude that Proj( R ) is closed under extension in FP n ( R ).Let0 → M → P → P → P , P ∈ Proj( R ) and M ∈ FP n ( R ).we have P ≃ M ⊕ P and M is also projective because P is projective. As FP ∞ ( R )is thick then M belongs to FP ∞ ( R ) and in particular is finitely generated. Weconclude M ∈ Proj( R ).Let M ∈ FP n ( R ) then there exists a resolution of free modules F n u n −−→ F n − → . . . → F → F → M → R is strong n -coherent we can take a free resolution as long as we want . . . → F n +1 → F n → F n − → . . . → F → F → M → M has a finite projective dimension, then there exists d → ker u d → F d u d −→ . . . → F → F → M → u d projective. Asker u d = Img u d +1 ≃ F d +1 / ker u d +1 we obtain ker u d is finitely generated. (cid:3) Corollary 3.3. [8, Theorem 2.3] If R is a coherent and regular ring then K i ( R ) ≃ K i ( FP ( R )) ∀ i ≥ .
4. Nil( FP n )Let C be an exact category, Nil( C ) is the category whose objects are the pairs( A, α ) where A ∈ C and α : A → A is a nilpotent endomorphism. A morphism( A, α ) → ( B, β ) in Nil( C ) is a morphism f : A → B in C such that f ◦ α = β ◦ f .The category Nil( C ) is an exact category and there exist exact functors C →
Nil( C ) A ( A,
0) Nil( C ) → C ( A, α ) A. Taking C = Proj( R ) and N = Nil(Proj( R )), note that K i ( R ) is a direct summandof K i ( N ) and define Nil i ( R ) the cokernel of K i ( R ) → K i ( N ). We obtain K i ( N ) = K i ( R ) ⊕ Nil i ( R ) i ≥ . We consider C = FP n ( R ). Let N ∗ n = Nil( FP n ( R )) and Nil ni ( R ) the cokernel of K i ( FP n ( R )) → K i ( N ∗ n ). We obtain K i ( N ∗ n ) = K i ( FP n ( R )) ⊕ Nil ni ( R ) i ≥ . -THEORY OF n -COHERENT RINGS 7 Lemma 4.1. (1) [21, Lemma 6.3]
For every ( M, α ) ∈ N ∗ there exist a ( P, β ) ∈N and a epimorphism ϕ : P ։ M such that the following diagram iscommutative P ϕ / / / / β (cid:15) (cid:15) M α (cid:15) (cid:15) P ϕ / / / / M (2) If R is coherent and regular then every ( M, α ) ∈ N ∗ has a finite N -resolution. (3) If R is strong n -coherent and n -regular then every ( M, α ) ∈ N ∗ n has a finite N -resolution.Proof. (1) Consider n ∈ N such that α n +1 = 0. Let Q be a projective andfinitely generated module with f : Q ։ M . Let P = Q n +1 and define β : P → P β ( x , x , . . . x n ) = (0 , x , . . . x n − ) ϕ : P ։ M ϕ ( x , x , . . . x n ) = f ( x ) + α ( f ( x )) + α ( f ( x )) + . . . α n ( f ( x n )) . (2) This case is a particular case of (3) with n = 1.(3) Given ( M, α ) ∈ N ∗ n ⊆ N ∗ , by (1) there exist ( P , β ) ∈ N and a epi-morphism in N ∗ n such that ϕ : ( P , β ) ։ ( M, α ) . Because R is strong n -coherent then FP n ( R ) is closed by taking kernel of epimorphisms. Then(ker( ϕ ) , β | ker( ϕ ) ) belongs to N ∗ n ⊆ N ∗ and we can apply (1) again. Thereexist ( P , β ) ∈ N and a epimorphism in N ∗ n such that ϕ : ( P , β ) ։ (ker( ϕ ) , β | ker( ϕ ) ). By this way we construct a resolution . . . ( P i , β i ) ϕ i −→ ( P i − , β i − ) ϕ i − −−−→ . . . ( P , β ) ϕ −→ ( M, α ) → R is n -regular, every M ∈ FP n ( R ) has finite projective dimension,then there exist k ∈ N such that P k = 0. We obtain a finite N -resolutionof ( M, α ). (cid:3) Proposition 4.2. If R is a strong n -coherent and n -regular ring then Nil ni ( R ) ≃ Nil i ( R ) . Proof.
We can regard N ∗ n as the full subcategory of R [ t ]-modules consisting ofmodules M which are n -coherent over R and are such that the action of t is anilpotent endomorphism t : M → M m t · m. We obtain
N ⊆ N ∗ n ⊆ R [ t ]-Mod. The category N ∗ n is exact and N is a fullsubcategory of N ∗ n which is closed under kernels of epimorphisms and extensions.As R is strong n -coherent and n -regular we can use the Lemma 4.1 (3) to obtainthat every M ∈ N ∗ n has a finite N -resolution. Because of the Resolution Theoremwe obtain that the inclusion N ֒ → N ∗ n induces isomorphisms K i ( N ) ≃ K i ( N ∗ n ) . In Theorem 3.2 we prove such that the inclusion Proj( R ) ֒ → FP n ( R ) inducesisomorphisms K i (Proj( R )) ≃ K i ( FP n ( R )) . EUGENIA ELLIS AND RAFAEL PARRA
Then we have the following diagram0 / / Nil i ( R ) (cid:15) (cid:15) / / K i ( N ) / / ≃ (cid:15) (cid:15) K i (Proj( R )) ≃ (cid:15) (cid:15) / / / / Nil ni ( R ) / / K i ( N ∗ n ) / / K i ( FP n ( R )) / / ni ( R ) ≃ Nil i ( R ) . (cid:3) Remark . The previous result with n = 1 is [21, Proposition 6.2].5. Categorical properties of FP n ( R )The Fundamental Theorem in K-theory relates the K-theory of R [ t ] and R [ t, t − ]with the K-theory of a ring R . Let R be a ring then if i ≥ K i ( R [ t ]) ≃ K i ( R ) ⊕ Nil i − ( R ) K i ( R [ t, t − ]) ≃ K i ( R ) ⊕ K i − ( R ) ⊕ Nil i − ( R ) ⊕ Nil i − ( R ) . We want to know when Nil i ( R ) = 0 in order to obtain an expression which relatesthe K-groups of R [ t, t − ] or R [ t ] only with the K-groups of R .If R is coherent it is proved in [21, Lema 6.4] that Nil i ( R ) = 0 using DevissageTheorem 1.2. We are going to check that FP ( R ) ֒ → N ∗ satisfyies the hypothesis.The category N ∗ is abelian because FP ( R ) is abelian when R is coherent. Wealso have FP ( R ) is closed under subobjetcs, quotient object and finite productin N ∗ . For each object [ M, α ] ∈ N ∗ we can filter [ M, α ] by [ M i , α | M i ] where M i = α n − i ( M ) and n the lowest natural number such that α n ( M ) = 0. Thequotient [ M i , α i ] / [ M i − , α i − ] is [ M i /M i − , α ]. As α ( M i ) = M i − then α = 0. Forthis reason [ M i , α i ] / [ M i − , α i − ] ∈ FP ( R ). If R is a regular coherent ring, byProposition 4.2 we have Nil i ( R ) = Nil i ( R ), as we seen above Nil i ( R ) = 0 and thenNil i ( R ) = 0. It means that K i ( R [ t ]) = K i ( R ) and K i ( R [ t, t − ]) = K i ( R ) ⊕ K i − ( R ) for all i ≥ R is a strong n -coherent and n -regular ring then by Proposition 4.2 we haveNil ni ( R ) ≃ Nil i ( R ). We can not follow the steps of the case n = 1 because FP n ( R )is not necessarily abelian. We prove in Proposition 5.1 that this only happen when R is coherent. Although FP n ( R ) is not abelian if R is a non-coherent and strong n -coherent ring with n ≥
2, we have that FP n ( R ) is a thick category. We expectthat Nil ni ( R ) are easier to handle than Nil i ( R ).Recall from [12] that a full subcategory C of R -Mod is wide if it is abelian andclosed under extensions. Let C the wide subcategory generated by R . Observe C contains all finitely presented modules then FP ( R ) ⊆ C . By [12, Lema 1.6] a ring R is coherent if and only if C = FP ( R ). Proposition 5.1.
Let n ≥ . The category FP n ( R ) is abelian if, and only if R isa coherent ring.Proof. If FP n ( R ) is abelian then C ⊆ FP n ( R ) ⊆ FP ( R ). We obtain C = FP ( R ) then R is coherent. Reciprocally, if R is coherent then FP ( R ) is an abeliancategory. By [4, Theorem 2.4] we obtain that FP ( R ) = FP ∞ ( R ) = FP n ( R )therefore FP n ( R ) is an abelian category. (cid:3) -THEORY OF n -COHERENT RINGS 9 Arithmetic and valutation rings.
In this section R is a commutative ring. For coherent rings, the regularity condi-tion is related to the weak dimension of R -modules. Examples of regular coherentrings includes Von Neumann regular and semihereditary rings. Coherent rings withfinite weak dimension are regular, however the converse is not necessarily true.Let us discuss the properties of R depending only on its weak dimension. If w.dim ( R ) = 0 then R is Von Neumann regular which also is coherent. If w.dim ( R ) =1 the coherence is not guaranteed, see [14, Example 2.3], but in this case R iscoherent if and only if R is semihereditary (see [10, Proposition 2.2]). Finally w.dim ( R ) ≤ R is an arithmetic reduced ring.Recall a ring R is arithmetic if the ideals of R M are totally order by inclusion forall maximal ideals M of R . It is proved in [6, Theorem 2.1] that arithmetic ringsare strong 3-coherent and that reduced arithmetic rings are strong 2-coherent.Arithmetical condition is not equivalent to strong n -coherence. The Example2.4 is strong 2-coherent but is not arithmetical neither coherent, see [1, Example3.13]. Proposition 6.1.
Let R be an arithmetic ring. The ring R is coherent if and only if the annihilator of every element isfinitely generated. In particular, if w.dim ( R ) < ∞ then R is a supercoher-ent regular reduced ring. If R is 3-regular: K i ( R ) ≃ K i (3 - Coh(R)) ≃ K i ( FP ( R )) ∀ i ≥ . If R is a reduced ring then: K i ( R ) ≃ K i (2 - Coh(R)) ≃ K i ( FP ( R )) ∀ i ≥ . If R has a Krull dimension and is 2-regular then: [ Spec ( R ) , Z ] = K ( R ) ≃ K (2 - Coh(R)) ≃ K ( FP ( R )) . Proof.
1. By [15, 1.4 Fact, Ch XII, § Arithmetic Rings ] an arithmetic ring iscoherent if and only if the annihilator of every element is finitely generated.An arithmetical coherent ring with w.dim ( R ) < ∞ is a semiheditary ring.2. It follows from Theorem 3.2 and [6, Theorem 2.1].3. If R is an arithmetic reduced ring then w.dim ( R ) ≤ pd ( M ) ≤ M ∈ FP ( R ) [3, Lemma 8].4. By Pierce’s Theorem, [23, Theorem 2.2.2], [ Spec ( R ) , Z ] = K ( R ) for every R with Krull dimension 0. By [6, Corollary 2.7] R is a strong 2-coherentring. (cid:3) Recall R is arithmetic if R is locally a valuation ring. A ring R is a valuationring if the set of ideals of R is totally order by inclusion. Proposition 6.2.
Let R be a valuation ring, M its maximal ideal and Z the subsetof its zero divisors. If Z = 0 then R is a coherent ring. If Z = 0 , Z = M and R is -regular then: K i ( R ) ≃ K i (2 - Coh(R)) ≃ K i ( FP ( R )) ∀ i ≥ . FP ∞ ( R ) = FP ( R ) FP ( R ) . If Z = 0 , Z = M and R is -regular then: K i ( R ) ≃ K i (2 - Coh(R)) ≃ K i ( FP ( R )) ∀ i ≥ . FP ∞ ( R ) = FP ( R ) ⊆ FP ( R ) .Proof. Straightforward from Theorem 3.2 and [6, Theorem 2.11]. (cid:3)
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