Homology of artinian and Matlis reflexive modules, I

Bethany Kubik; Micah J. Leamer; Sean Sather-Wagstaff

doi:10.1016/j.jpaa.2011.02.007

Homology of artinian and Matlis reflexive modules, I

Bethany Kubik, Micah J. Leamer, Sean Sather-Wagstaff

Mathematics & Statistics

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18 Scopus citations

Abstract

Let R be a commutative local noetherian ring, and let L and L' be R-modules. We investigate the properties of the functors ToriR(L,-) and ExtRi(L,-). For instance, we show the following: (a)if L and L' are artinian, then ToriR(L,L') is artinian, and ExtRi(L,L') is noetherian over the completion R̂;(b)if L is artinian and L' is Matlis reflexive, then ExtRi(L,L'), ExtRi(L',L), and ToriR(L,L') are Matlis reflexive. Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.

Original language	English (US)
Pages (from-to)	2486-2503
Number of pages	18
Journal	Journal of Pure and Applied Algebra
Volume	215
Issue number	10
DOIs	https://doi.org/10.1016/j.jpaa.2011.02.007
State	Published - Oct 2011

Bibliographical note

Funding Information:
✩ This material is based on work supported by North Dakota EPSCoR and National Science Foundation Grant EPS-0814442. Micah Leamer was supported

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10.1016/j.jpaa.2011.02.007

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title = "Homology of artinian and Matlis reflexive modules, I",

abstract = "Let R be a commutative local noetherian ring, and let L and L' be R-modules. We investigate the properties of the functors ToriR(L,-) and ExtRi(L,-). For instance, we show the following: (a)if L and L' are artinian, then ToriR(L,L') is artinian, and ExtRi(L,L') is noetherian over the completion {\^R};(b)if L is artinian and L' is Matlis reflexive, then ExtRi(L,L'), ExtRi(L',L), and ToriR(L,L') are Matlis reflexive. Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.",

author = "Bethany Kubik and Leamer, {Micah J.} and Sean Sather-Wagstaff",

note = "Funding Information: ✩ This material is based on work supported by North Dakota EPSCoR and National Science Foundation Grant EPS-0814442. Micah Leamer was supported",

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AU - Kubik, Bethany

AU - Leamer, Micah J.

AU - Sather-Wagstaff, Sean

N1 - Funding Information: ✩ This material is based on work supported by North Dakota EPSCoR and National Science Foundation Grant EPS-0814442. Micah Leamer was supported

PY - 2011/10

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N2 - Let R be a commutative local noetherian ring, and let L and L' be R-modules. We investigate the properties of the functors ToriR(L,-) and ExtRi(L,-). For instance, we show the following: (a)if L and L' are artinian, then ToriR(L,L') is artinian, and ExtRi(L,L') is noetherian over the completion R̂;(b)if L is artinian and L' is Matlis reflexive, then ExtRi(L,L'), ExtRi(L',L), and ToriR(L,L') are Matlis reflexive. Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.

AB - Let R be a commutative local noetherian ring, and let L and L' be R-modules. We investigate the properties of the functors ToriR(L,-) and ExtRi(L,-). For instance, we show the following: (a)if L and L' are artinian, then ToriR(L,L') is artinian, and ExtRi(L,L') is noetherian over the completion R̂;(b)if L is artinian and L' is Matlis reflexive, then ExtRi(L,L'), ExtRi(L',L), and ToriR(L,L') are Matlis reflexive. Also, we study the vanishing behavior of these functors, and we include computations demonstrating the sharpness of our results.

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Homology of artinian and Matlis reflexive modules, I

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