On Pro-zero homomorphisms and sequences in local (co-)homology

Let $\xx= x_1,\ldots,x_r$ denote a system of elements of a commutative ring $R$. For an $R$-module $M$ we investigate when $\xx$ is $M$-pro-regular resp. $M$-weakly pro-regular as generalizations of $M$-regular sequences. This is done in terms of \v{C}ech co-homology resp. homology, defined by $H^i(\check{C}_{\xx} \otimes_R \cdot)$ resp. by $H_i({\textrm{R}} \Hom_R(\check{C}_{\xx},\cdot)) \cong H_i(\Hom_R(\mathcal{L}_{\xx},\cdot))$, where $\check{C}_{\xx}$ denotes the \v{C}ech complex and $\mathcal{L}_{\xx}$ is a bounded free resolution of it as constructed in [17] resp. [16]. The property of $\xx$ being $M$-pro-regular resp. $M$-weakly pro-regular follows by the vanishing of certain \v{C}ech co-homology resp. homology modules, which is related to completions. This extends previously work by Greenlees and May (see) [5] and Lipman et al. (see [1]}). This contributes to a further understanding of \v{C}ech (co-)homology in the non-Noetherian case. As a technical tool we use one of Emmanouil's results (see [4]) about the inverse limits and its derived functor. As an application we prove a global variant of the results with an application to prisms in the sense of Bhatt and Scholze (see[3]).


INTRODUCTION
Let R denote a commutative ring with x = x 1 , . . ., x r a system of elements.For an R-module M we study generalizations of a M-regular sequence called M-pro-regular sequence and Mweakly pro-regular sequence.To this end we denote by Čx the Čech complex with respect to x (see e.g.[17, 6.1]).It is a bounded complex of flat R-modules.For an R-module M we write Čx (M) = Čx ⊗ R M. We call Ȟi x (M) = H i ( Čx (M)), i ∈ Z, the Čech cohomology of M. Dually we look at the complex R Hom R ( Čx , M) in the derived category.There is a free resolution of Čx by a bounded complex L x and Hom R (L x , M) is a representative of R Hom R ( Čx , M) (see [17] and [16]).We define Ȟx i (M) = H i (Hom R (L x , M)) ∼ = H i (R Hom R ( Čx , M)), i ∈ Z, as the Čech homology of M. For the case of R a Noetherian ring let a = xR then it follows that Ȟi x (M) ∼ = H i a (M), the i-th local cohomology of M with support in a.At first this was established by Grothendieck (see [6] and [7]).Dually, for Noetherian rings R we have Ȟx i (M) ∼ = Λ a i (M), where Λ a i (•) denotes the left derived functors of the completion Λ a (•).Contributions were done by Matlis (see [9]), Simon (see [18]), Greenlees and May (see [5]) and others.
Starting with Greenlees and May (see [5]) and Lipman et al. (see [1]) there were extensions to non-Noetherian rings with sequences x that are called pro-regular resp.weakly pro-regular (see below for the definitions).In particular, when x is weakly pro-regular the isomorphisms Ȟi x (M) ∼ = H i a (M) and Ȟx i (M) ∼ = Λ a i (M) hold for any i ∈ Z and any R-module M and more generally for any complex X ∈ D(R) (see [11], [12] , [14] and [17] for more details).
In the situation of x an R-regular sequence there is a corresponding property of x being an M-regular sequence (see e.g.[10]).This is a challenge for the study of the relative version that x is weakly M-regular for modules instead of M = R. Namely, x is called an M-weakly proregular sequence (see also [17, 7.3.1])provided the inverse system {H i (x (n) ; M)} n≥1 is pro-zero for i = 1, . . ., r, i.e. for each n there is an integer m ≥ n such that the natural map H i (x (m) ; M) → H i (x (n) ; M) is zero.Here x (n) = x n 1 , . . ., x n r and H i (x (n) ; M) denotes the Koszul homology.An R- weakly pro-regular sequence is called weakly pro-regular.For a first description of M-weakly pro-regular sequences see [15, Theorem 4.2].Let M x = Λ x (M) denote the x-adic completion of M.
Theorem 1.1.For an R-module M and a sequence x = x 1 , . . ., x r the following is equivalent: (i) x is M-weakly pro-regular.
(ii) Čx (Hom R (M, I)) is a right resolution of Hom R (Λ x (M), I) for any injective R-module I.
(iii) Hom R (L x , M ⊗ R F) is a left resolution of Λ x (M ⊗ R F) for any free R-module F.
) is a left resolution of Λ x (M[T]).
Note that the equivalence of (i), (iii) and (iv) in the particular case of M = R was shown by Positselski (see [12,Theorem 3.6]), that is in the case when x is R-weakly pro-regular (or weakly pro-regular for short).Then the complexes Hom R (L x , X) and LΛ x (X) are isomorphic in the derived category for all X ∈ D(R) (see [11] generalizing the case of bounded complexes shown in [16]).For the proof of 1.1 and the notion of left/right resolution see the comments after 3.6.
The notion of a weakly pro-regular sequence x = x 1 , . . ., x r is defined in terms of the Koszul homology of the whole sequence x.An M-regular sequence is defined by the vanishing of x i−1 M : M x i /x i−1 M for i = 1, . . ., r, where x i−1 = x 1 , . . ., x i−1 .As a generalization of that Greenlees and May (see [5]) resp.Lipman et al. (see [1]) invented the notion of an M-proregular sequence.Note that both of the definitions are equivalent (see [15,Proposition 2.2]).A sequence x is called M-pro-regular if the inverse system {x 16.1]).A characterization of pro-regular sequences in terms of Čech cohomology is known (see [15,Theorem 3.2] and 4.4).Here there is a description in the terms of Čech homology.See 4.5 for the following: Theorem 1.2.Let x = x 1 , . . ., x r denote a sequence of elements of R. For an R-module M the following conditions are equivalent: In the final section we apply the previous results to a global situation.To this end we consider a pair (I, x) consisting of an effective Cartier divisor I ⊆ R and an element x ∈ R (see 5.1 for the definitions).We call it pro-regular whenever the inverse system {H 1 (x n ; R/I n )} n≥1 is prozero.Then our investigations (see 5.5) yield the following: Corollary 1.3.With the previous notation the following conditions are equivalent: As shown in [15] this has applications to prisms in the sense of Bhatt and Scholze (see [3]).The equivalent conditions in 1.3 are improvements of the results shown in [15,Corollary 5.7].
In the paper we start with recollections about inverse limits.In particular we include a different proof of one of Emmanouil's results (see [4]) about inverse systems needed in the paper.In the third section we prove additional statements about weakly pro-regular sequences, extending those known before.In section 4 we study pro-regular sequences, continuing the results shown in [15].Moreover, we prove a necessary and sufficient condition for the isomorphism Λ x (Λ I (M)) ∼ = Λ (x,I ) (M) for an ideal I ⊂ R and an element x ∈ R generalizing a result by Greenlees and May (see [5,Lemma 1.6]).Finally in section 5 we study when a pair (I, x) consisting of an effective Cartier divisor I and an element x ∈ R is pro-regular.Finally we apply these results to prisms in the sense of [3] generalizing partial results of [15].
In the terminology we follow that of [17].In our approach we prefer to work in the category of modules instead of the derived category.For that reason we use a bounded free resolution of the Čech complex (see 3.1).

RECOLLECTIONS ABOUT INVERSE LIMITS
Notation 2.1.(A) Let R denote a commutative ring.Let {M n } n≥0 be an inverse system of R-modules with φ n,m : M m → M n for all m ≥ n.Then there is an exact sequence where Φ denotes the transition map and lim ← − 1 M n is the first left derived functor of the inverse limit (see e.g.[20, 3.5] or [17, 1.2.2]).(B) Let M denote an R-module.Let T be a variable over R. In the following we use M[|T|], the formal power series R-module over M. That is, the R-module M[|T|] consists of all formal series ∑ i≥0 x i T i with x i ∈ M for all i ≥ 0. Correspondingly, the R-module M[T] consists of all polynomials over M. Therefore, ∑ i≥0 x i T i ∈ M[T] if only finitely many x i are non-zero.Whence there is an injection 0 For more details about inverse systems we refer to Jensen's exposition in [8] and to [4].It is remarkable that the vanishing in 2.1 (C) does not imply that {M n } n≥0 is pro-zero.To this end see the example [17, 1.2.5] or the following generalization: Example 2.2.Let (R, m) denote a complete local Noetherian ring with x ∈ R a non-unit.We consider the direct system {R n } n≥0 with R n = R and ψ n,n+1 : R n → R n+1 the multiplication by x.Then lim − → R n ∼ = R x and there is a short exact sequence Now we apply Hom R (•, R) and obtain the inverse system {M n } n≥0 with M n = Hom R (R n , R) and with the multiplication M n+1 x → M n .By applying Hom R (•, R) to the previous short exact sequence it yields the exact sequence [17, 3.1.10])while the inverse system {M n } n≥0 is neither pro-zero nor satisfies the Mittag-Leffler condition.
In the following we shall discuss necessary and sufficient conditions for an inverse system to be pro-zero.This extends known results.We need a technical construction.
where T denote the shift operator defined by ∑ k n≥0 x n T n → ∑ k n≥0 x n T n+1 .The inverse system {M n } n≥0 induces a short exact sequence of inverse systems induced by the shift operator.Then we have the six-term long exact sequence associated to the inverse limit By the Example 2.2 it follows that the vanishing of lim ← − 1 M n is necessary but not sufficient for the Mittag-Leffler condition of the inverse system {M n } n≥0 .A characterization of the Mittag-Leffler condition was shown by Emmanouil (see [4]).For our purposes we recall part of Emmanouil's result (see [4,Corollary 6]).In our argument we use a certain exact sequence (see the proof of 2.4) and modify an idea of [19, tag 0CQA] as new ingredients.
Lemma 2.4.Let {M n } n≥0 denote an inverse system of R-modules.Then the following conditions are equivalent: Proof.(i) =⇒ (ii): This follows since the inverse system {M n [T]} n≥0 satisfies the Mittag-Leffler condition too.
(iii) ⇐⇒ (iv): This is a consequence of the six-term exact sequence in 2.3.
Now suppose that {M n } n≥0 does not satisfy the Mittag-Leffler condition.Then there is an integer m such that the sequence of submodules {Im φ m,k |k ≥ m} of M m does not stabilize.Whence there is an infinite sequence m = m 0 < m 1 < . . .< m i < . . .and elements As easily seen and G can not be a preimage of F ′ , a contradiction to the vanishing of lim As a consequence of 2.4 a characterization of pro-zero inverse systems follows.The vanishing lim For the proof we mod- ify Weibel's argument (see the proof [20, 3.5.7]).For an R-module M and a set S we define M (S) = ⊕ s∈S M s with M s = M. Then it is clear that conditions (iii) and (iv) hold also for the inverse system {(M n ) (S) } n≥0 when they hold for {M n } n≥0 .
Corollary 2.5.Let {M n } n≥0 denote an inverse system of R-modules.Then the following conditions are equivalent: Proof.(i) =⇒ (ii): Because {M n } n≥0 is pro-zero this holds also for the induced inverse system {M n [T]} n≥0 as easily seen.
(iii) ⇐⇒ (iv): This is a consequence of the six-term exact sequence in 2.3.(iii) =⇒ (i): By view of 2.4 the inverse system {M n } n≥1 satisfies the Mittag-Leffler condition.We define Then {N n } n≥1 becomes an inverse system with surjective maps.Because the inverse system M n = 0 and therefore N n = 0.

WEAKLY PRO-REGULAR SEQUENCES
We start with a few recalls of results and definitions of [17] and [16].As above R denotes a commutative ring.Notation 3.1.(A) For a system of elements x = x 1 , . . ., x r of R let Čx denote the Čech complex where Čx i : 0 → R → R x i → 0 (see e.g.[7] or [17, 6.1]).In the following we look at the complex R Hom R ( Čx , M) for an R-module M in the derived category.By virtue of [5] there is a finite free resolution of Čx .We follow here the one L x as given in [16].Whence Hom ) for all i ∈ Z (see [17] and [16] for more details).(B) Let U = U 1 , . . ., U r denote a sequence of r variables over R. For an R-module M we denote, as above, by M[|U|] the module of formal power series in the variables U. where is the polynomial module over M. For the sequence x = x 1 , . . ., x r we define the sequence x − U = x 1 − U 1 , . . ., x r − U r .As one of the main results of the paper [17, Section 8] the following isomorphisms are shown where K • (x − U; •) denotes the Koszul complex with respect to the sequence x − U. Moreover there are isomorphisms where M[U −1 ] denotes the module of inverse polynomials and K • (x − U; •) is the Koszul cocomplex (see [16, 4.1] for all of the details).
In the following there is technical result for the computation of Ȟx i (M) and Ȟi x (M) resp.
Lemma 3.2.We fix the notation of 3.1.Furthermore let x (n) = x n 1 , . . ., x n r and let H i (x (n) ; M) denote the Koszul homology and H i (x (n) ; M) the Koszul cohomology.
(a) There are isomorphisms Ȟi x (M) ∼ = lim − → H i (x (n) ; M) and short exact sequences Proof.For the proof of (a) we refer to [17, 6.1.4, 8.1.7]or [16, 5.6].Then (b) is a consequence of the exact sequences in (a).
Next we shall give a further characterization for an element x ∈ R such that an R-module M is of bounded x-torsion.Definition 3.3.(A) Let M denote an R-module and x ∈ R an element.Then M is called of bounded x-torsion if the family of increasing submodules {0 : M x n } n≥0 stabilizes, that is 0 : M x n = 0 : M x n+1 for all n ≫ 0.
Note that this is equivalent to the fact that the inverse system {0 : M x n } n≥0 with the multiplication map 0 : M x m x m−n −→ 0 : M x n , m ≥ n, being pro-zero.(B) It is obvious that M is of bounded x-torsion if and only if the inverse system of Koszul homology modules {H 1 (x n ; M)} n≥0 with the multiplication map H 1 (x m ; M) With this in mind Lipman (see [2]) introduced the generalization of a weakly proregular sequence for a ring R. For a generalization to an R-module M see [17, 7.3.1].That is, a sequence x = x 1 , . . ., x r is called M-weakly pro-regular, if for i > 0 the inverse system {H i (x (n) ; M)} n≥0 is pro-zero, where H i (x m ; M) → H i (x n ; M), m ≥ n, denotes the natural map induced by the Koszul complexes.A first systematic study of R-weakly pro-regular sequences has been done in [14].
For a characterization of M-weakly pro-regular sequences see [16].In fact, this is an extension of R-weakly pro-regular sequences shown in [11] which extended the results of [14] to unbounded complexes.Here we shall prove another characterization of M-weakly pro-regular sequences.It is a slight extension of Potsitselski's result see [12,Section 3]) to the case of an Rmodule M. As above, for an R-module M and a set S we define Theorem 3.4.Let x = x 1 , . . ., x r denote a sequence of elements of R. For an R-module M the following conditions are equivalent: (i) x is M-weakly pro-regular.
; M (S) ) = 0 for i > 0 and (ii) is a consequence of 3.2.(ii) =⇒ (iii) =⇒ (iv): These hold obviously.(iv) =⇒ (i): By view of 3.2 the assumptions imply that lim ← − By 2.5 this completes the proof because of In the following example we show that it is not sufficient to assume S to be finite in 3.4 for the characterization of weakly pro-regular sequences (see also [15,Example 3.3]).
More- over, by the change of rings there is an isomorphism Hom R (L x , A) ∼ = Hom A (L x , A).That is, Ȟx i (A) = 0 for i > 0 and Ȟx 0 (A) ∼ = A. Now note that A is not of bounded x-torsion as easily seen.It follows that the equivalent conditions in 3.4 do not hold for A and A[T].To be more precise, recall Therefore H 1 (x m ; A) does not stabilize under the multiplication by x m−n in H 1 (x n ; A).Note that the i-component of the image of H 1 (x m ; A) under the multiplication by x m−n in H 1 (x n ; A) is zero for i ≤ m − n < m and non-zero for i = m − n + 1. Whence {H 1 (x n ; A)} n≥1 does not satisfy the Mittag-Leffler condition.By view of 2.4 we have lim As an application we have another characterization that an R-module M is of bounded xtorsion for an element x ∈ R. Note that (iii) in 3.6 is the analogue to 3.4 (iv).Corollary 3.6.For an element x ∈ R and an R-module M the following conditions are equivalent: (i) M is of bounded x-torsion.
Proof.The equivalence of the first two conditions is a particular case of 3.4.The equivalence of the first and third condition is a particular case of 2.5.
Moreover, the proof of Theorem For an R-module X we call a complex X • : . .
With the previous results we have the following slight generalization of Potsitselski's result (see [12,Theorem 3.6]).Note that x is R-weakly pro-regular if it is R[T]-weakly pro-regular as easily seen.
Corollary 3.7.For a sequence x = x 1 , . . ., x r of a ring R the following conditions are equivalent: Remark 3.8.While the property of R-regular and M-regular sequences are quite "symmetric" this is not the case for the notion of weakly pro-regularity.Let x denote a sequence of elements of R. If it is R-weakly pro-regular it follows that Ȟx 0 (M) ∼ = Λ x 0 (M) for any R-module M (see e.g.[17,Chapter 7]).Let x be M-weakly pro-regular, then Ȟx 0 (M) ∼ = Λ x (M) as shown in 3.4.Note that the homomorphism Λ x 0 (M) → Λ x (M) is onto (see [17, 2.5.1]) but in general not an isomorphism (see e.g.Example 3.5).

PRO-REGULAR SEQUENCES
Before we shall investigate pro-regular sequences we need technical results about pro-zero inverse systems.To this end let M denote an R-module with {M n } n≥1 a decreasing sequence of submodules of M, i.e.M n+1 ⊆ M n for n ≥ 1.Then M = {M/M n } n≥1 forms an inverse system with surjective maps for the inverse limit of the induced filtration.Then there is a natural homomorphism Λ x (Λ(M)) → Λ(M/xM).In the following we will discuss when it is an isomorphism.Lemma 4.1.With the previous notation there is a short exact sequence Let m, n denote positive integers.We investigate the inverse system of Koszul complexes {K • (x (n) ; M/M m )} m≥1 .For its inverse limit there are isomorphisms The inverse system {K • (x (n) ; M/M m )} m≥1 is degree-wise surjective.Whence for its 0-th homology there is a short exact sequence [17, 1.2.8]).It forms an exact sequence of inverse systems on n.By passing to the inverse limit it provides the short exact sequence of the statement since lim ← − ; M/M m ) = 0 because of the underlying bi-countable indexed system (see the spectral sequence in [13]).Whence the statement follows.
The previous result is an extension of [5,Lemma 1.6] to the case of a sequence of elements and a more general filtration.Namely, it was shown by Greenlees and May that the vanishing of lim ← − n lim ← − 1 the vanishing is also necessary for the isomorphism.
For any set S we define also ).For an element x ∈ R we put -as before - Moreover, we study when the inverse system {M n : M x n /M n } n≥1 with the multiplication by x is pro-zero.That is, when for each n ≥ 1 there is an m ≥ n such that the multiplication map is zero.This is equivalent to the inverse system {H 1 (x n ; M/M n )} n≥1 being pro-zero, where H 1 (x n ; M/M n ) denotes the Koszul homology of M/M n with respect to the element x n .In other words, for each integer n ≥ 1 there is an m ≥ n such that M m : M x m ⊆ M n : M x m−n .Note that, if M n =: N for all n ≥ 1, then {H 1 (x n ; M/N)} n≥1 is pro-zero if and only if M/N is of bounded x-torsion.With this in mind we shall continue with an extension of 3.6.

Theorem 4.2. With the previous notation the following conditions are equivalent:
(i) The inverse system {H 1 (x n ; M/M n )} n≥1 is pro-zero.
Proof.(i) =⇒ (ii): We put X = M (S) and X n = (M n ) (S) .Then it follows that {H 1 (x n ; X/X n )} n≥1 is pro-zero too since the Koszul homology commutes with direct sums, therefore Furthermore there are isomorphisms for all n ≥ 1.We have the bi-indexed system {H 1 (x n ; X/X m )} n≥1,m≥1 and the diagonal system {H 1 (x n ; X/X n )} n≥1 cofinal in it.There are the isomorphisms and the vanishing By virtue of Roos' spectral sequence (see [13] or [20, 5.8.7]) there is a short exact sequence and a similar one with m, n reversed.This implies the vanishing lim X/X m and since the inverse limit commutes (as above) with the first Koszul homology it follows that The first vanishing implies that lim ← − H 1 (x n ; M/M n ) = 0.In order to continue note that the isomorphism of the assumption Ȟx 0 (Λ(X )) The Koszul homology commutes with direct sums.Therefore the implication follows by virtue of 2.5.
The implication (i) =⇒ (ii) in 4.2 is a generalization of [5,Proposition 1.7].Furthermore, a certain generalization of bounded torsion to the study of sequences was invented by Greenlees and May (see [5]) and Lipman et al. (see [1]), namely: Definition 4.3.(A) Let x = x 1 , . . ., x r denote a sequence of elments of R. For an R-module M it is called M-pro-regular if the inverse systems with the multiplication map by x n i {(x n 1 , . . ., x n i−1 )M : M x n i /(x n 1 , . . ., x n i−1 )M} n≥1 , i = 1, . . ., r, are pro-zero.This is equivalent to saying that the inverse systems {H 1 (x are pro-zero for i = 1, . . ., r.For a sequence of elements x = x 1 , . . ., x r we specify the subsystems The notion of pro-zero is equivalent to say that for i = 1, . . ., r and any positive integer n there is an integer m ≥ n such that For a discussion of the notions of pro-regularity of Greenlees and May (see [5]) resp.Lipman (see [1]) we refer to [15].Moreover, it follows that an M-pro-regular sequence is also M-weakly pro-regular (see e.g.[15,Theorem 2.4]), while the converse does not hold (see [2]).For a homological characterization of M-pro-regular sequences in terms of injective modules we refer to [ (i) The sequence x is M-pro-regular.
(ii) The sequence x is (M ⊗ R F)-pro-regular for any flat R-module F.
Recall that 4.4 provides a characterization of M-pro-regular sequences in terms of Čech cohomology.In the following we shall prove a characterization in terms of Čech homology.This depends upon the results of pro-zero inverse systems as shown above.
Theorem 4.5.Let x = x 1 , . . . ,x r denote a sequence of elements of R. For an R-module M the following conditions are equivalent: ) ) and Ȟx i 1 (Λ x i−1 (M (S) )) = 0 for i = 1, . . ., r and any set S. Proof.First note that x is M (S) -pro-regular for any set S. It turns out since R/x )) for all n ≥ 0 and i = 1, . . ., r, it follows that the corresponding inverse systems are isomorphic and pro-zero.Note that H 0 (x (S) .Moreover the condition and Theorem 4.2 proves the equivalence of the first three statements.(iii) ⇐⇒(iv) : By view of [16, 8.1] there are short exact sequences ) → 0 for i = 1, . . ., r and j = 0, 1.Then the equivalence is easily seen by the exact sequences.More precisely, (iii) =⇒ (iv) follows by increasing induction on i starting at i = 1.The converse follows similarly.(v) =⇒ (iii): The assumption in (v) implies the vanishing For a fixed n and j = 0, 1 we have the short exact sequences This follows since the inverse system for lim It remains to show the vanishing of lim ← − ).First note that the above short exact sequence for j = 1 provides that lim ← − Then the above sequence (#) (see proof of 4.2) with m, n reversed proves the vanishing lim ← −

A GLOBAL VARIATION
As before, let R denote a commutative ring.For an element f ∈ R we write Then we recall the following definitions (see [15]).Definition 5.1.
i=1 M f i is injective for any R-module M as easily seen.(B) An ideal I ⊂ R is called an effective Cartier divisor if there is a covering sequence f = f 1 , . . . ,f r such that I R f i = x i R f i , i = 1, . . ., r, for non-zerodivisors x i /1 of R f i with x i ∈ R. It follows that I ⊆ (x 1 , . . ., x r )R.(C) Let I denote an effective Cartier divisor and x ∈ R. The pair (I, x) is called pro-regular if for any integer n there is an integer m ≥ n such that I m : x m ⊆ I n : x m−n .This is consistent with the definition in [5] (see 4.3) and is equivalent to the fact that for each n there is an integer 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.Moreover, the pair (I, x) is pro-regular if and only if the inverse system {H 1 (x n ; R/I n )} n≥1 is pro-zero.
For the following we need a technical result about Cartier divisors and their relation to proregularity.

Lemma 5.2. Let I ⊆ R be an effective Cartier divisor with the covering sequence
(iii) x i /1, x/1 is pro-regular in R f i for i = 1, . . ., r in the sense of 4.3.
(iv) (I, x) is pro-regular in the sense of 5.1.Proof.(i) ⇐⇒ (ii): For each pair of integers m ≥ n ≥ 1 we have the following commutative diagram where the horizontal maps are injections I : R x m /I → ⊕ r j=1 (x i R f i : R f i x m /1)/x i R f i ↓ x m−n ↓ ⊕(x m−n /1) I : R x n /I → ⊕ r j=1 (x i R f i : R f i x n /1)/x i R f i which proves the equivalence.
(ii) ⇐⇒ (iii): Note that x i /1, x/1 is pro-regular if and and only if R f i /x k i R f i is of bounded x/1torsion for all k ≥ 1.The equivalence follows easily: First note that x i /1 is R f i -regular.Then use induction on the short exact sequence (iii) ⇐⇒ (iv): The equivalence comes out by the following modification of the above commutative diagram I m : R x m /I m → ⊕ r j=1 (x m i R f i : R f i x m /1)/x m i R f i ↓ x m−n ↓ ⊕(x m−n /1) I n : R x n /I n → ⊕ r j=1 (x n i R f i : R f i x n /1)/x n i R f i .Recall that the horizontal maps are injective (see also [15]).
Next we apply the previous investigations to the case when the pair (I, x) is pro-regular in the sense of 5.1.Lemma 5.3.Let I ⊆ R be an effective Cartier divisor with the covering sequence f = f 1 , . . ., f r such that I R f i = x i R f i , i = 1, . . ., r, for non-zerodivisors x i /1 of R f i .For an element x ∈ R the following conditions are equivalent: (i) R/I is of bounded x-torsion.Proof.First note that by 5.2 {H 1 (x n ; R/I )} n≥1 is pro-zero if and only if {H 1 (x k ; R/I k )} k≥1 is pro-zero.Then the equivalence of (i) and (ii) follows by 3.4.Moreover, by 4.2 the pro-zero property of the second inverse system above implies the equivalence to (iii).Finally the equivalence of (iii) and (iv) is a consequence of 4.5 and 4.1 since lim ← − n lim ← − 1 m H 1 (x n ; R/I m ) = 0.In the following we shall give a comment of the previous investigations to the recent work of Bhatt and Scholze (see [3]) completing the results of [15].To this end let p ∈ N denote a prime number and let Z p := Z p the localization at the prime ideal (p) = p ∈ Spec Z.In the following let R be a Z p -algebra.With the previous definition there is the following application of our results.

Example 3 . 5 .
Let R = k[|x|] denote the formal power series ring in the variable x over the field k.Then define A = ∏ n≥1 R/x n R. By the component wise operations A becomes a commutative ring.The natural map R By view of 3.2 and 4.1 this proves the claim.(ii) =⇒ (iii): This holds trivially.(iii) =⇒ (iv): By 3.2 the assumption implies that lim

Theorem 4 . 4 .
15, Theorem 2.1].Here we add a slight extension of [15, Theorem 2.1].Let x = x 1 , . . ., x r denote an ordered sequence of elements of R. Let M denote an R-module.Then the following conditions are equivalent.
is of bounded x i -torsion for i = 1, . . ., r and X = M, M[T].

Definition 5 . 4 .
(see [3, Definition 1.1]) A prism is a pair (R, I) consisting of a δ-ring R (see [3, Remark 1.2]) 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]).
Let{M n } n≥0 be an inverse system.Then clearly Im φ n,m ′ ⊆ Im φ n,m ⊆ M n for all m ′ ≥ m ≥ n.We say that {M n } n≥0 satisfies the Mittag-Leffler condition if for each n the sequence of submodules {Im φ n,m |m ≥ n} stabilizes.For instance, this holds if the maps φ n,m are surjective or {M n } n≥0 is an inverse system of Artinian R-modules.