Generalized Local Homology for Artinian Modules

We present some finiteness results for co-associated primes of generalized local homology modules. Let M be a finitely generated R-module and N a linearly compact R-module. If N and H I i (N ) satisfy the finiteness condition for co-associated primes for all i < k, then Coass R (H I k (M, N )) is a finite set. On the other hand, if H I i (N ) = 0 for all i < t and Tor R j (M, H I t (N )) = 0 for all j < h, then Tor R h (M, H I t (N )) ∼ = H I h+t (M, N ). Moreover, Coass(H I h+t (M, N )) is also a finite set provided N satisfies the finiteness condition for co-associated primes. Finally, N is a semi-discrete linearly compact R-module such that 0 : N I = 0. Let t = Width I (N ) and h = tor − (M, H I t (N )); it follows that Width I+Ann(M) (N ) = t + h and Coass(H I h+t (M, N )) is a finite set.

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“…We first recall the concept of generalized local homology modules ( [12]). Let I be an ideal of the ring R and M, N R-modules.…”

Section: The Finiteness Resultsmentioning

confidence: 99%

The finiteness of coassociated primes of generalized local homology modules

Nam

Yen

2015

Math Notes

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“…Then {X, ϕ i : X → X i } i is the inverse limit of the inverse system {X i , ϕ ij : X j → X i } i≤j (see [13,Chapter 2] for more details about the inverse system and inverse limit). This inverse limit is called n-th generalized local homology of M, N with respect to a and is denoted by H a n (M, N ); see [2,3,10,11] for more details and basic properties. Now, we show that the R-module X also has an R a -module structure, where…”

Section: R) These Inclusions Yield the Equation (24)mentioning

confidence: 99%

“…Since n > 1, it follows from the equivalence of (i) and (v), and also the equivalence of (vi) and (v) in the case n = 1 that Λ a (N ) ∼ = Tor R 0 (R/a t , N ) is a finitely generated R-module. Now, the exact sequence 0 → a t N → N → Λ a (N ) → 0 of Artinian R-modules induces the long exact sequences [11,Proposition 2.4]) and…”

Section: R) These Inclusions Yield the Equation (24)mentioning

confidence: 99%

“…Therefore, in general, L a 0 (•) ≇ Λ a (•) ( and so, in general, L a i (•) ≇ H a i (•)). Similarly, the ith generalized local homology H a i (M, N ) of M, N with respect to a is defined by H a i (M, N ) = lim ← − n∈N Tor R i (M/a n M, N ) ; see [11] for basic properties and more details. We use filter coregular sequences to study the finiteness of (generalized) local homology modules.…”

Section: Introductionmentioning

confidence: 99%

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On the finiteness of local homology modules

Fathi¹,

Hajikarimi²

2021

Preprint

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Let R be a commutative Noetherian ring and a an ideal of R. Let M be a finitely generated R-module and N an Artinian R-module. The concept of filter coregular sequence is introduced to determine the infimum of the integers i such that the generalized local homology H a i (M, N ) is not finitely generated as an R a -module, where R a denotes the a-adic completion of R. In particular, it is shown that H a i (M, N ) is a finitely generated R a -module for all i ∈ N 0 if and only if (0 : N a + Ann R (M )) has finite length whenever R is a complete semi-local ring.

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“…In [8], [10] we defined the i-th generalized local homology module H This definition is in some sense dual to J. Herzog's definition of generalized local cohomology modules [5] and in fact a generalization of the usual local homology modules…”

Section: Introductionmentioning

confidence: 99%