About proregular sequences and an application to prisms
aa r X i v : . [ m a t h . A C ] S e p ABOUT PROREGULAR SEQUENCES AND AN APPLICATION TO PRISMS
PETER SCHENZELA
BSTRACT . Let x = x , . . . , x k denote an ordered sequence of elements of a commutative ring R . Let M bean R -module. We recall the two notions that x is M -proregular given by Greenlees and May (see [5]) andLipman (see [1]) and show that both notions are equivalent. As a main result we prove a cohomologicalcharacterization for x to be M -proregular in terms of ˇCech homology. This implies also that x is M -weaklyproregular if it is M -proregular. A local-global principle for proregularity and weakly proregularity is proved.This is used for a result about prisms as introduced by Bhatt and Scholze (see [3]). I NTRODUCTION
Let R denote a commutative ring and let M be an R -modue. Let x = x , . . . , x k denote an orderedsequence of elements of R . In their paper Greenlees and May (see [5]) defined x to be M -proregular iffor all i =
1, . . . , k and an integer n ≥ m ≥ n such that the multiplication map ( x , . . . , x i − ) m M : M x mi / ( x , . . . , x i − ) m M x m − ni −→ ( x , . . . , x i − ) n M : M x ni / ( x , . . . , x i − ) n M is zero. In their paper (see [1]) Lipman et all called x to be M -proregular if for all i =
1, . . . , k and aninteger n ≥ m ≥ n such that the multiplication map ( x m , . . . , x mi − ) M : M x mi / ( x m , . . . , x mi − ) M x m − ni −→ ( x n , . . . , x ni − ) M : M x ni / ( x n , . . . , x ni − ) M is zero (see also the Definition 1.1). The advantage of the second notion of proregularity is its relationto Koszul complexes. Moreover, both definitions are equivalent (see 1.2). The notion of proregularsequences is needed by Greenlees and May for their study of the left derived functors of the completion.This was continued by Lipman et all (see [1] and [2]). Moreover, it turned out that the notion that x isweakly proregular (see 1.6) is more appropriate for the study of ˇCech homology and cohomology. A firstsystematic study of weakly proregular sequences is done in [8]. See also the monograph [10] for a moresystematic investigation and its relation to completions. As a first main result we prove a homologicalcharacterization of M -proregular sequences. Theorem 0.1.
Let x = x , . . . , x k denote an ordered sequence of a commutative ring R. For an R-module M thefollowing conditions are equivalent. (i) The sequence x is M-proregular. (ii) ˇ H x i ( Γ x i − ( Hom R ( M , I )) = for i =
1, . . . , k and any injective R-module I.
For the proof see 2.1. Here ˇ H x i ( · ) denotes the ˇCech homology with respect to x i (see 1.5); and Γ x i − ( · ) is the torsion functor with respect to the ideal generated by x i − = x , . . . , x i − . As an applicationthere is a behavior similar to that of a regular sequence. For a sequence x = x , . . . , x k and any k -tupel ( n , . . . , n k ) ∈ ( N + ) k we write x ( n ) = x n , . . . , x n k k . Corollary 0.2.
With the notation of 0.1 the following conditions are equivalent. (i))
The sequence x is M-proregular. (ii)
There is an n ∈ ( N + ) k such that the sequence x ( n ) is M-proregular. (iii) The sequence x ( n ) is M-proregular for all n ∈ ( N + ) k . Mathematics Subject Classification.
Primary: 13C12; Secondary: 13C11, 13D07.
Key words and phrases. torsion, injective module, non-Noetherian commutative ring, proregular sequences.
If the sequence x is M -regular it is also M -proregular. For a Noetherian R -module M any sequence is M -proregular. Therefore, the notion of proregularity is important in the non-Noetherian situation. Anapplication of this is the following result inspired by the work of Bhatt and Scholze (see [3]). For thenotation we refer to Chapter 4. Corollary 0.3.
Let ( R , I ) denote a prism. Suppose that I is of bounded p-torsion. Then it follows. (a) ( I , p ) is proregular, i.e. for an integer n there is an m ≥ n such that I m : R p m / I m p m − n −→ I n : R p n / I n is the zero map. (b) Γ I ( I ) / Γ ( I , p ) ( I ) is p-divisible for any injective R-module I, i.e. Γ I ( I ) = p Γ I ( I ) + Γ ( I , p ) ( I ) .Moreover, the conditions (a) and (b) are equivalent. In Section 1 we summarize the notation and prove some basic results. Section 2 is devoted the homo-logical approach of proregular sequences and their consequences. A few more results about M -weaklyproregular sequences are shown in Section 3. This provides a relative version of some results of [10].In Section 4 we investigate the local-global behavior of proregular and weakly proregular sequencesand prove the Corollary 0.3. For results of Commutative Algebra we follow Matsumura’s book [6].For homological preliminaries and definitions about ˇCech homology and cohomology we refer to themonograph [10]. 1. D EFINITIONS AND P RELIMINARIES
In the following let R denote a commutative ring. Let x = x , . . . , x k denote an ordered sequence ofelements of R . For a positive integer n we put x ( n ) = x n , . . . , x nk . Let M denote an R -module. Note that x nk M ⊆ x ( n ) M ⊆ x n M for all n ≥ xR = Rad x ( n ) R for all n ≥
1. The R -module M is of bounded x -torsion for anelement x ∈ R if the increasing sequence { M x n } n ≥ stabilizes, i.e. there is an integer c such that0 : M x n = M x c for all n ≥ c . Note that M is of bounded x -torsion for an M -regular element x ∈ R . If N is a submodule of M and M is of bounded x -torsion, then N is of bounded x -torsion too. Moreover,if M is a Noetherian module, then M is of bounded torsion for any element. Let m ≥ n denote twopositive integers. The multiplication map by x m − n on M induces a map0 : M x m x m − n −→ M x n for all m ≥ n .Therefore M is of bounded x -torsion if and only if the previous inverse system is pro-zero. This leadsGreenlees and May to the definition of an M -proregular sequence (see [5, Definition 1.8]). We followhere the notion of proregularity as given in [1] and show that it is equivalent to the notion of Greenleesand May (see [5]). Definition 1.1.
Let M denote an R -module and let x = x , . . . , x k denote a sequence of elements of R .Then x is called M-proregular if for i =
1, . . . , k and any positive integer n there is an integer m ≥ n suchthat the multiplication map ( x m , . . . , x mi − ) M : M x mi / ( x m , . . . , x mi − ) M x m − ni −→ ( x n , . . . , x ni − ) M : M x ni / ( x n , . . . , x ni − ) M ( ) is zero. This is equivalent to saying that for i =
1, . . . , k and any positive integer n there is an integer m ≥ n such that ( x m , . . . , x mi − ) M : M x mi ⊆ ( x n , . . . , x ni − ) M : M x m − ni . ( ) Note that an element x ∈ R is M -proregular if and only if M is of bounded x -torsion.A first result about the behavior of M -proregular sequences is the following. Proposition 1.2.
Let x = x , . . . , x k denote an ordered sequence of elements of R. Let M denote an R-module.Then the following conditions are equivalent. (i) The sequence x is M-proregular.
ROREGULAR SEQUENCES 3 (ii)
For i =
1, . . . , k and any positive integer n there is an integer m ≥ n such that ( x , . . . , x i − ) m M : M x mi ⊆ ( x , . . . , x i − ) n M : M x m − ni .(iii) For i =
1, . . . , k and any integer n there is an integer m ≥ n such that the multiplication map ( x , . . . , x i − ) m M : M x mi / ( x , . . . , x i − ) m M x m − ni −→ ( x , . . . , x i − ) n M : M x ni / ( x , . . . , x i − ) n Mis zero.Proof.
We fix l and put x = x , . . . , x l − and y = x l . The equivalence of (ii) and (iii) is trivially true. Thenwe prove the implication (i) = ⇒ (iii). By the assumption and the definition 1.1 for a given n there is an m ≥ n such that x ( m ) M : M y m ⊆ x ( n ) M : M y m − n ⊆ x n M : M y m − n . Because of x ml M : M y ml ⊆ x ( m ) M : M y ml this implies that x ml M : M y ml ⊆ x ( m ) M : M y ml ⊆ x ( n ) M : M y ml − n ⊆ x n M : M y ml − n as required. In order to prove (ii) = ⇒ (i) fix n and choose an integer m ≥ nl such that x m M : M y m ⊆ x nl : M y m − nl . Then we get the inclusions x ( m ) M : M y m ⊆ x m M : M y m ⊆ x nl M : M y m − nl ⊆ x ( n ) M : M y m − nl ⊆ x ( n ) M : M y m − n which finishes the proof. (cid:3) The following example by J. Lipman (see [2]) shows that a proregular sequence is not permutable.
Example 1.3.
Define R = ∏ n ≥ Z /2 n Z . Let x = ( + n ) n ≥ and 1 = ( + n ) n ≥ two elements of R . Then the sequence { x } is R -regular, in particular R -proregular. Moreover, R is not of bounded x -torsion. Therefore { x , 1 } is not a proregular sequence.Clearly an M -regular sequence x is also M -proregular. An interesting extension is the followingresult. Proposition 1.4.
Let M be an R-module. Let x = x , . . . , x k denote an M-regular sequence. Suppose thatM / xM is of bounded y-torsion for an element y ∈ R. Then x , y = x , . . . , x k , y is an M-proregular sequence.Proof. Now x i is regular on M / ( x , . . . , x i − ) n M for all i =
1, . . . , k and all n ≥ x = x , . . . , x k is an M -regular sequence (see e.g. [6, Theorem 16.2]). That is, condition (ii) in 1.2 is satisfied for all i =
1, . . . , k . In order to finish the proof we have to show that M / x n M is of bounded y -torsion for all n ≥
1. Namely, if x n M : M y c = x n M : M y d for all d ≥ c choose m = n + c and therefore x m M : M y m ⊆ x n M : M y m = x n M : M y m − n . It remains to show the previous claim. Since x is an M -regular sequence x n M / x n + M ∼ = ⊕ b n M / xM with b n = ( k + n − n ) (see e.g. [6]). Then the short exact sequence0 → x n M / x n + M → M / x n + M → M / x n M → Γ y the exact sequence 0 → Γ y ( x n M / x n + M ) → Γ y ( M / x n + M ) → Γ y ( M / x n M ) .This proves – by induction on n – that M / x n M is of bounded y -torsion for all n ≥
1. Note that asubmodule of Γ y ( M / x n M ) is of bounded y -torsion too. (cid:3) For our further investigations we need a few more notation and definitions.
Notation 1.5. (A) Let a ⊂ R denote an ideal of R . For an R -module X let Γ a ( X ) = { x ∈ X | a n x = n ≥ } the a -torsion submodule of X . Its left derived functors H i a ( X ) , i ≥
0, are the localcohomology modules of X with respect to a (see e.g. [4] or [10] for more details).(B) Let x = x , . . . , x k denote a sequence of elements of R . Let ˇ C x denote the ˇCech complex with re-spect to R . For an R -module X we write ˇ C x ( X ) = ˇ C x ⊗ R X . We denote the cohomology of ˇ C x ( X ) byˇ H ix ( X ) , i ≥ x = x , . . . , x k a sequence of elements and a = xR . Then it follows easily that Γ a ( X ) = ˇ H x ( X ) .The more general isomorphisms H i a ( X ) ∼ = ˇ H ix ( X ) for all i ≥ x is a weakly proregular sequence (see [10] for more details about weakly proregular sequences).(D) For a sequence of elements x = x , . . . , x k and an R -module M we use the Koszul complexes K • ( x ; M ) PETER SCHENZEL and K • ( x ; M ) . We refer to [10, 5.2] for all the details we need. The Koszul homology and Koszul coho-mology are denoted by H i ( x ; M ) and H i ( x ; M ) resp. for i ∈ Z .We continue with the a further definition. Definition 1.6. (see [10, Section 7.3]) Let x = x , . . . , x k denote a sequence of elements of R . Let M bean R -module. The sequence x is called M -weakly proregular if for all i > n there is an integer m ≥ n such that the natural homomorphism H i ( x ( m ) ; M ) → H i ( x ( n ) ; M ) is zero. A sequence x is called weakly proregular if x is R -weakly proregular.The notion of weakly proregular sequences plays an essential r ˆole in respect to local cohomology andthe left derived functors of completion (see [1], [2] and [10]). A characterization of M -weakly proregularsequence is the following. Proposition 1.7. (see [9, Proposition 5.3])
Let x denote a sequence of elements of R. Let M be an R-module.Then the following conditions are equivalent. (i) x is M-weakly proregular. (ii)
The inverse system { H i ( x ( n ) ; M ⊗ R F ) } n ≥ is pro-zero for all i > and any flat R-module F. (iii) lim −→ H i ( x ( n ) ; Hom R ( M , I )) = for all i > and any injective R-module I. (iv) ˇ H ix ( Hom R ( M , I )) = for all i > and any injective R-module I. We shall see that an M -proregular sequence is also M -weakly proregular. The converse is not true asfollows by 1.3 since the sequence 1, x is R -regular and in particular R -proregular.2. A HOMOLOGICAL APPROACH
At first we will give an interpretation of the definition of an M -proregular sequence x = x , . . . , x k interms of Kozul complexes. For the basics about Koszul complexes we refer to [10]. We fix l ∈ {
1, . . . , k } and put x = x , . . . , x l − and y = x l . Then the natural map x ( m ) M : M y m / x ( m ) M y m − n −→ x ( n ) M : M y n / x ( n ) M for all m ≥ n ( ) coincides with the natural map H ( y m ; H ( x ( m ) ; M )) y m − n −→ H ( y n ; H ( x ( n ) ; M )) for all m ≥ n . ( ) induced by the natural map of Koszul complexes K • ( x ( m ) , y m ; M ) → K • ( x ( n ) , y n ; M ) (see [10, Section5.2] for some details). In the following we specify x i = x , . . . , x i for i =
1, . . . , k − Theorem 2.1.
Let x = x , . . . , x k denote an ordered sequence of elements of R. Let M denote an R-module. Thenthe following conditions are equivalent. (i) The sequence x is M-proregular. (ii)
The sequence x is ( M ⊗ R F ) -proregular for any flat R-module F. (iii) ˇ H x i ( Γ x i − ( Hom R ( M , I )) = for i =
1, . . . , k and any injective R-module I. (iv) Γ x i − ( Hom R ( M , I )) / Γ x i ( Hom R ( M , I )) is x i -divisible for i =
1, . . . , k and any injective R-module I.Proof.
The equivalence of (i) and (ii) holds trivially. Next we show the equivalence of (iii) and (iv). Foran R -module X and an element y ∈ R there is an exact sequence0 → Γ y ( X ) → X → X y → ˇ H y ( X ) → X y denotes the localization of X with respect to the element y ∈ R . Note that X → X y is just theˇCech complex with respect to the single element y ∈ R . Next note that ˇ H y ( X ) = X → X y is onto or equivalently X / Γ y ( X ) is y -divisible. Applying this observation to Γ x i − ( Hom R ( M , I )) provesthe equivalence of (iii) and (iv). ROREGULAR SEQUENCES 5
We use the abbreviations of the beginning of this section. Now we prove (i) = ⇒ (iii). Applying thefunctor Hom R ( · , I ) to the pro-zero sequence of Koszul homologies (4) at the beginning of this sectionyields homomorphisms that provide a direct system H ( y n ; H ( x ( n ) ; Hom R ( M , I ))) y m − n −→ H ( y m ; H ( x ( m ) ; Hom R ( M , I ))) . ( ) If the system in (4) is pro-zero it follows that0 = lim −→ H ( y n ; H ( x ( n ) ; Hom R ( M , I ))) ∼ = ˇ H y ( Γ x ( Hom R ( M , I )) .The previous isomorphism might be checked directly, or see [10, 6.1.11].For the proof of (iii) = ⇒ (i) we note that x ( n ) M : M y n / x ( n ) M ∼ = Hom R ( R / y n R , M / x ( n ) M ) . Now fix n and choose an injection f : ( x ( n ) M : M y n ) / x ( n ) M ֒ → J into an injective R -module J . This defines anelement f ∈ Hom R ( Hom R ( R / y n R , M / x ( n ) M ) , J ) ∼ = H ( y n ; H ( x ( n ) ; Hom R ( M , J )) .By the assumption lim −→ H ( y n ; H ( x ( n ) ; Hom R ( M , J )) ∼ = ˇ H y ( Γ x ( Hom R ( M , I )) =
0. Whence there is aninteger m ≥ n such that the image of f in Hom R ( Hom R ( R / y m R , M / x ( m ) M ) , J ) vanishes. In other wordsthe composite of the maps x ( m ) M : M y m / x ( m ) M y m − n −→ x ( n ) M : M y n / x ( n ) M f −→ J is zero. This finishes the proof since f is injective. (cid:3) As a first application we get extensions of some of the authors’ result in [11] to the case of an R -module M . Corollary 2.2.
Let x ∈ R denote an element and let M denote an R-module. Then the following conditions areequivalent. (i)
M is of bounded x-torsion. (ii) M ⊗ R F is of bounded x-torsion for any flat R-module. (iii) ˇ H x ( Hom R ( M , I )) = for any injective R-module. (iv) Hom R ( M , I ) / Γ x ( Hom R ( M , I )) is x-divisible.Proof. This is a particular case of 2.1 for k =
1. Namely x is M -proregular if and only if M is of bounded x -torsion. (cid:3) For an M -regular sequence x = x , . . . , x k it follows that x ( n ) = x n , . . . , x n k k is M -regular for any k -tupel ( n , . . . , n k ) ∈ ( N + ) k . A corresponding result holds for M -proregular sequences. Corollary 2.3.
Let x = x , . . . , x k be an ordered sequence of elements of R and let M be an R-module. Then thefollowing conditions are equivalent. (i)) The sequence x is M-proregular. (ii)
There is an n ∈ ( N + ) k such that the sequence x ( n ) is M-proregular. (iii) The sequence x ( n ) is M-proregular for all n ∈ ( N + ) k .Proof. The statements are easy consequences of Theorem 2.1 by condition (iii). (cid:3)
Next we provide an alternative proof of [10, A.2.3] that an M -proregular sequence is also M -weaklyprregular. This is done by the characterization of 2.1 Theorem 2.4.
Let x = x , . . . , x k be an ordered sequence in R. Let M denote an R-module. If x is M-proregular,then x is also M-weakly proregular.Proof. We fix l ∈ {
1, . . . , k } and put x = x , . . . , x l − and y = x l . Then we show by induction on l thatˇ H ix , y ( Hom R ( M , I )) = i >
0. If l = k this proves the claim by virtue of Proposition 1.7. If l = y is M -proregular if and only if M is of bounded y -torsion. Whence the claim is true by 2.3. Let l >
1. Then there is the short exact sequence0 → ˇ H y ( ˇ H i − x ( Hom R ( M , I )) → ˇ H ix , y ( Hom R ( M , I )) → ˇ H y ( ˇ H ix ( Hom R ( M , I )) → PETER SCHENZEL for all i (see [10, 6.1.11]). By the induction hypothesis ˇ H ix ( Hom R ( M , I )) = i > H ix , y ( Hom R ( M , I )) = i >
1. The vanishing for i = H y ( ˇ H x ( Hom R ( M , I )) = (cid:3)
3. ˇC
ECH ( CO -) COMPLEXES
In order to continue with the study of weakly proregular sequences let us recall a few more defini-tions.
Notation 3.1. (A) Let x = x , . . . , x k denote a sytem of elements of R . As in 1.5 we denote by ˇ C x thecorresponding ˇCech complex. In [10] and [9] there are constructions of two bounded complexes offree R -mdules ˇ L x and L x and quasi-isomorphisms ˇ L x ∼ −→ L x ∼ −→ ˇ C x . That is there are bounded freeresolutions of the ˇCech complex ˇ C x that is a complex of flat R -modules.(B) Let M denote an R -module. With the previous notation the complex RHom R ( ˇ C x , M ) in the derivedcategory has the following two representatives Hom R ( L x , M ) ∼ −→ Hom R ( ˇ L x , M ) .(C) For an integer i ∈ Z we putˇ H xi ( M ) : = H i ( Hom R ( L x , M )) ∼ = H i ( Hom R ( ˇ L x , M )) for the ˇCech homology of M with respect to x .(D) For an R -module M and an ideal a ⊂ R we write Λ a ( M ) for the a -adic completion of M , i.e. Λ a ( M ) = lim ←− M / a n M . The right derived functors of Λ a ( M ) are denoted by Λ a i ( M ) , i ∈ Z . Notethat in general Λ a ( M ) = Λ a ( M ) since Λ a is not left exact.In the following we shall prove a relative version of one of the main results of [10]. We shall recoverit for the case of M = R . Theorem 3.2.
Let x = x , . . . , x k denote a sequence of elements of R and a = xR. Let M be an R-module.Suppose that x is M-weakly proregular. (a) ˇ H ix ( Hom R ( M , I )) = for all i > and ˇ C x ⊗ R Hom R ( M , I ) is a right resolution of Γ a ( Hom R ( M , I )) for any injective R-module I. (b) ˇ H xi ( M ⊗ R F ) = for all i > and Hom R ( L x , M ⊗ R F ) is a left resolution of Λ a ( M ⊗ R F ) for any flatR-module F.Proof. The vanishing result in (a) is shown in 1.7. Moreover ˇ H x ( Hom R ( M , I )) ∼ = Γ a ( Hom R ( M , I ) . Nowthe natural morphism Γ a ( Hom R ( M , I ) → ˇ C x ⊗ R Hom R ( M , I ) proves (a).Now let us prove (b). By view of [10, 6.3.2 (b)] there are the following short exact sequences0 → lim ←− H i + ( x ( n ) ; M ⊗ R F ) → H i ( RHom R ( ˇ C x , M ⊗ R F )) → lim ←− H i ( x ( n ) ; M ⊗ R F ) → i ∈ N . By the assumption { H i ( x ( n ) ; M ⊗ R F ) } n ≥ is prozero for all i >
0. This yields thatlim ←− H i ( x ( n ) ; M ⊗ R F ) = lim ←− H i ( x ( n ) ; M ⊗ R F ) = i >
0. This proves the the vanishing part. For i = H i ( RHom R ( ˇ C x , M ⊗ R F )) ∼ = lim ←− H ( x ( n ) ; M ⊗ R F ) ∼ = Λ a ( M ⊗ R F ) as easily seen since xR = a . By [10, 8.2.1] and [9, 4.1 (B)] there is a natural morphism Hom R ( L x , M ⊗ R F ) → Λ a ( M ⊗ R F ) which completes the proof of (b). (cid:3) While the condition (a) in Theorem 3.2 is equivalent to the statement that x is M -weakly proregular(see 1.7 ) this does not hold for (b). The following example is motivated by Lipman’s example in [2]. Example 3.3.
Let R = k [ | x | ] denote the formal power series ring in the variable x over the field k . Thendefine S = ∏ n ≥ R / x n R . By the component wise operations S becomes a commutative ring. The naturalmap R → S , r → ( r + x n ) n ≥ , is a ring homomorphism and x x : = ( x + x n ) n ≥ . As a direct product of xR -complete modules S is an xR -complete R -module (see [10, 2.2.7]). Since R is a Noetherian ring x is R -weakly proregular and ˇ H xi ( S ) ∼ = H i ( Hom R ( L x , S )) = i > H x ( S ) ∼ = H ( Hom R ( L x , S )) ∼ = S . ROREGULAR SEQUENCES 7
Moreover, by the change of rings there is an isomorphism Hom R ( L x , S ) ∼ = Hom S ( L x , S ) . That is,ˇ H x i ( S ) = i > H x ( S ) ∼ = S . Now note that S is not of bounded x -torsion as easily seen. Itfollows also that S is x -adic complete and S is not a coherent ring.In the following we shall apply the previous result to the situation of a complex of flat R -modules. Itis a relative version of [10, 7.5.16]. Corollary 3.4.
We fix the notation of 3.2. Let F • denote a complex of flat R-modules. Then the natural map Hom R ( L x , M ⊗ R F • ) → Λ a ( M ⊗ R F • ) is a quasi-isomorphisms. Moreover, if F • is a complex of finitely generated free R-modules, then the natural map Hom R ( L x , M ⊗ R F • ) → Λ a ( M ) ⊗ R F • is a quasi-isomrphism.Proof. Let F • = { F q , d q } q ∈ Z be the complex of flat R -modules . Then the natural mapHom R ( L x , M ⊗ R F q ) → Λ a ( M ⊗ R F q ) is a quasi-isomorphism for all q ∈ Z (see 3.2). Since Hom R ( L x , M ⊗ R F • ) is the single complex associatedto the double complex Hom R ( L px , M ⊗ R F q ) and because L x is bounded the first claim follows (see also[10, 4.1.3]). For the second note that F q , q ∈ Z , is a finitely generated free R -module. Its tensor productcommutes with inverse limits. (cid:3) In the following we specify the previous result for a finitely generated R -module N . Corollary 3.5.
We use the assumption of 3.2. Let N denote a finitely generated R-module with L • ∼ −→ N a freeresolution by finitely generated free R-modules. Then there are isomorphisms ˇ H xi ( M ⊗ R L • ) ∼ = Tor Ri ( Λ a ( M ) , N ) for all i ∈ Z and a spectral sequenceE i , j = ˇ H xi ( Tor Rj ( M , N )) = ⇒ E ∞ i + j = Tor Ri + j ( Λ a ( M ) , N ) . Proof.
The isomorphisms are an immediate consequence of 3.4. The spectral sequence is just the spectralsequence of the double complex (see e.g. [7]). (cid:3)
In the following we shall recall a particular case of one of the main results of [10].
Corollary 3.6.
Let x = x , . . . , x k denote a sequence of elements of R and a = xR. Suppose that x is weaklyproregular. Then ˇ H xi ( N ) ∼ = Λ a i ( N ) for all i ∈ N and any R-module N.Proof. Let F • ∼ −→ N be a flat resolution of N . ThenHom R ( L x , N ) ∼ −→ Hom R ( L x , F • ) ∼ −→ Λ a ( F • ) .Since Λ a i ( N ) = H i ( Λ a ( F • )) the claim follows by taking homology. (cid:3) In fact the previous result holds for any complex X of R -modules provided x is weakly proregular(see [10] for these and more related information). Here we will continue with a further characterizationof weakly proregular sequences. Proposition 3.7.
Let x = x , . . . , x k denote a sequence of elelemts of R. Let E denote an injective cogenerator inthe category of R-modules. Then the following is equivalent. (i) The sequence x is weakly proregular. (ii) ˇ H xi ( Hom R ( I , J )) = for all i > and any two injective R-module I , J. (iii) ˇ H xi ( Hom R ( I , E )) = for all i > and any injective R-module I. PETER SCHENZEL
Proof. (i) = ⇒ (ii): By the adjointness there is the following isomorphisms of complexesHom R ( L x , Hom R ( I , J )) ∼ = Hom R ( L x ⊗ R I , J ) .Therefore (i) implies that Hom R ( H i ( L x ⊗ R I ) , J ) = i >
0. By the definition it yields (ii).(ii) = ⇒ (iii): This holds trivially.(iii) = ⇒ (i): By the previous adjointness isomorphism it implies that Hom R ( H i ( L x ⊗ R I ) , E ) = i >
0. Since E is an injective cogenerator this implies H i ( L x ⊗ R I ) ∼ = ˇ H ix ( I ) = i >
0. Now this isequivalent to the fact that x is weakly proregular (see 1.7). (cid:3) Remark 3.8. (A) If R is a coherent ring (see e.g. [10, 1.4.2]), then Hom R ( I , J ) is a flat R -module for anytwo injective R -modules (see [10, 1.4.5]).(B) Let R be a coherent ring. Then the conditions in 3.7 are equivalent to(iv) ˇ H xi ( F ) = i > R -module F .This follows because Hom R ( I , E ) is R -flat for any injective R -module I (see [10, 7.5.15] for more details).(C) There are examples of rings R and injective R -modules I , J such that Hom R ( I , J ) is not a flat R -module (see [10, A.5.7]). 4. L OCAL CONDITIONS
Let R denote a commutative ring. For an element r ∈ R we write D ( f ) = Spec R \ V ( f ) . Note that D ( f ) is an open set in the Zariski topology of Spec R . For f ∈ R there is a natural map Spec R f → Spec R that induces a homeomorphism between Spec R f and D ( f ) . Since Spec R = ∪ f ∈ R D ( f ) and since Spec R is quasi-compact there are finitely many f , . . . , f r ∈ R such that Spec R = ∪ ri = D ( f i ) . This provides thefollowing definition. Definition 4.1. (see [12]) A sequence f = f , . . . , f r of elements of R is called a covering sequence ifSpec R = ∪ ri = D ( f i ) . This is equivalent to saying that R = f R . Moreover, if f is a covering sequencethen the natural map M → ⊕ ri = M f i is injective for any R -module M as easily seen.Next we show proregularity as well as weakly proregularity are local properties. For a sequence x = x , . . . , x k of elements of R we denote by x /1 = x /1, . . . , x k /1 its image in a localization of R . Proposition 4.2.
Let f = f , . . . , f r denote a covering sequence of R. Let x = x , . . . , x k be an ordered sequenceof elements of R. For an R-module M the following conditions are equivalent. (i) x is M-weakly proregular (resp. proregular). (ii) x /1 in R f i is M f i -weakly proregular (resp. proregular) for all i =
1, . . . , r.Proof.
At first we consider the case of weakly proregular sequences. Then for all integers m ≥ n and anyinteger i > H i ( x ( m ) ; M ) → ⊕ rj = H i ( x /1 ( m ) ; M f j ) ↓ ↓ H i ( x ( n ) ; M ) → ⊕ rj = H i ( x /1 ( n ) ; M f j ) where the horizontal maps are injective since f is a covering sequence. If (i) holds the vertical maps atthe left are zero for a given n and an appropriate m ≥ n . Since localization is flat and commutes withhomology the vertical maps at the right are zero too. For the converse fix n and i and choose m ≥ n suchthat H i ( x /1 ( m ) ; M f j ) → H i ( x /1 ( n ) ; M f j ) is zero for all j =
1, . . . , r . Then the vertical map at the right iszero. Since the horizontal maps are injective it follows that the vertical map at the left is zero.The proof for the case of proregular sequence follows similar arguments by an inspection of thenatural map ( x m , . . . , x mi − ) M : M x mi / ( x m , . . . , x mi − ) M x m − ni −→ ( x n , . . . , x ni − ) M : M x ni / ( x n , . . . , x ni − ) M and the direct sum of the localizations with respect to R f i , i =
1, . . . , r . (cid:3) A corresponding local global principle is the following.
ROREGULAR SEQUENCES 9
Proposition 4.3.
Let x = x , . . . , x k be an ordered sequence of elements of R. For an R-module M the followingconditions are equivalent. (i) x is M-weakly proregular (resp. proregular). (ii) x /1 in R p is M p -weakly proregular (resp. proregular) for all p ∈ Spec R. (iii) x /1 in R m is M m -weakly proregular (resp. proregular) for all maximal ideals m ∈ Spec
R.Proof.
The proof follows easily by [6, Theorem 4.6, p. 27]. We omit the details here. (cid:3)
By view of [12] we recall the following definition and extend it with the notion of proregularity.
Definition 4.4. (A) (see [12]) Let R denote a commutative ring. An ideal I ⊂ R is called an effectiveCartier divisor if there is a covering sequence f = f , . . . , f r such that I R f i = x i R f i , i =
1, . . . , r , fornon-zerodivisors x i of R . It follows that I ⊆ ( x , . . . , x r ) R .(B) Let I denote a Cartier divisor and x ∈ R . The ideal ( I , x ) is called proregular if for any integer n there is an integer m ≥ n such that I m : x m ⊆ I n : x m − n . This is in consistence with the definition in [5](see 1.1) and is equivalent to the fact that for each n there is an integer m ≥ n such that the multiplicationmap I m : R x m / I m x m − n −→ I n : R x n / I n is the zero map.In the following we shall consider a local global principle for proregular effective Cartier divisors. Theorem 4.5.
Let
I ⊆
R an effective Cartier divisor with the covering sequence f = f , . . . , f r such that I R f i = x i R f i , i =
1, . . . , r , for non-zerodivisors x i of R. Suppose that R / I is of bounded x-torsion for somex ∈ R. (a) ( I , x ) , is proregular, i.e. for each integer n there is an m ≥ n such that the multiplication map I m : R x m / I m x m − n −→ I n : R x n / I n is the zero map. (b) Γ I ( I ) / Γ ( I , x ) ( I ) is x-divisible for any injective R-module I.Moreover, the two conditions (a) and (b) are equivalent.Proof. At first we prove (a). Since R / I is of bounded x -torsion there is an integer c such that I : R x m = I : R x c for all m ≥ c . By localization at R f i it follows that x i R f i : R fi x /1 m = x i R f i : R fi x /1 c for all m ≥ c and i =
1, . . . , r with R f i -regular elements x i /1 ∈ R f i . By view of 1.2 it follows x i /1, x /1 is an R f i -proregular sequence for all i =
1, . . . , r . For a given n and m ≥ n there is the following commutativediagram I m : R x m / I m → ⊕ rj = ( x mi R f i : R fi x /1 m ) / x mi R f i ↓ x m − n ↓ ⊕ ( x m − n /1 ) I n : R x n / I n → ⊕ rj = ( x ni R f i : R fi x /1 n ) / x ni R f i Now choose m such that multiplication at the vertical maps at the right are all zero. Then the multiplica-tion at the left is the zero map too since the horizontal maps are injective as follows by the localization(see 4.1).For the proof of (b) recall that the natural map I m : R x m / I m x m − n −→ I n : R x n / I n coincides with themap induced by the Koszul complexes, i.e., H ( x m ; R / I m ) x m − n −→ H ( x n ; R / I n ) .Now apply Hom R ( · , I ) with an arbitrary injective R -module. Then the induced map H ( x n ; Hom R ( R / I n , I )) → H ( x m ; Hom R ( R / I m , I )) is zero and 0 = lim −→ H ( x m ; Hom R ( R / I m , I )) ∼ = ˇ H x ( Γ I ( I )) which proves the statement as in the proof of2.1. For the implication (b) = ⇒ (a) we fix n and choose an injection H ( x n ; Hom R ( R / I n , I )) ∼ = I n : R x n / I n ֒ → I into an injective R -module I . By the vanishing of the direct limit there is, as in the proof of Theorem 2.1,an integer m ≥ n such that (a) holds. (cid:3) In the following we shall give a comment of the previous investigations to the recent work of Bhattand Scholze (see [3]). To this end let p ∈ N denote a prime number and let Z p : = Z p the localization atthe prime ideal ( p ) = p ∈ Spec Z . In the following let R be a Z p -algebra. Definition 4.6. (see [3, Definition 1.1]) A prism is a pair ( R , I ) consisting of a δ -ring R (see [3, Remark1.2]) and a Cartier divisor I on R satisfying the following two conditions.(a) The ring R is ( p , I ) -adic complete.(b) p ∈ I + φ R ( I ) R , where φ R is the lift of the Frobenius on R induced by its δ -structure (see [3,Remark 1.2]).With the previous definition there is the following application of our results. Corollary 4.7.
Let ( R , I ) denote a prism. Suppose that I is of bounded p-torsion. Then (a) ( I , p ) is proregular in the sense of 4.4. (b) Γ I ( I ) / Γ ( I , p ) ( I ) is p-divisible for any injective R-module I, i.e. Γ I ( I ) = p Γ I ( I ) + Γ ( I , p ) ( I ) .The conditions in (a) and (b) are equivalent.Proof. This is an immediate consequence of 4.5. (cid:3)
We note that Yekutieli (see [13, Theorem 7.3]) has slightly modified the notion of weakly proregularityand has shown that ( I , p ) is weakly proregular under the assumption of 4.7. It should be mentionedthat proregularity is more strong than weakly proregularity as shown by the example in [2].While the notion of weakly proregularity plays an essential role in the study of local (co-) homology(see [10] and the references there) the previous statements seem to be a further application of the notionof proregularity as introduced by Greenlees and May (see [5]) and by Lipman (see [1]).A CKNOWLEDGEMENT . The author thanks Anne-Marie Simon (Universit´e Libre de Bruxelles) for var-ious discussions about the subject and a careful reading of the manuscript.R
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