Homology of artinian and mini-max modules, II

Bethany Kubik; Micah Leamer; Sean Sather-Wagstaff

doi:10.1016/j.jalgebra.2014.01.016

Homology of artinian and mini-max modules, II

Bethany Kubik
, Micah Leamer
, Sean Sather-Wagstaff

Mathematics & Statistics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

Let R be a commutative ring, and let L and L' be R-modules. We investigate finiteness conditions (e.g., noetherian, artinian, mini-max, Matlis reflexive) of the modules ExtRi(L,L') and ToriR(L,L') when L and L' satisfy combinations of these finiteness conditions. For instance, if R is noetherian, then given R-modules M and M' such that M is Matlis reflexive and M' is mini-max (e.g., noetherian or artinian), we prove that ExtRi(M,M'), ExtRi(M',M), and ToriR(M,M') are Matlis reflexive over R for all i≥0 and that ExtRi(M,M')∨≅ToriR(M,M'∨) and ExtRi(M',M)∨≅ToriR(M',M∨).

Original language	English (US)
Pages (from-to)	229-272
Number of pages	44
Journal	Journal of Algebra
Volume	403
DOIs	https://doi.org/10.1016/j.jalgebra.2014.01.016
State	Published - Feb 1 2014

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 by a GAANN grant from the Department of Education. Sean Sather-Wagstaff was supported by a grant from the NSA .

Keywords

Artinian
Bass number
Betti number
Ext
Hom
Matlis duality
Mini-max
Noetherian
Primary
Secondary
Tensor product
Tor

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title = "Homology of artinian and mini-max modules, II",

abstract = "Let R be a commutative ring, and let L and L' be R-modules. We investigate finiteness conditions (e.g., noetherian, artinian, mini-max, Matlis reflexive) of the modules ExtRi(L,L') and ToriR(L,L') when L and L' satisfy combinations of these finiteness conditions. For instance, if R is noetherian, then given R-modules M and M' such that M is Matlis reflexive and M' is mini-max (e.g., noetherian or artinian), we prove that ExtRi(M,M'), ExtRi(M',M), and ToriR(M,M') are Matlis reflexive over R for all i≥0 and that ExtRi(M,M')∨≅ToriR(M,M'∨) and ExtRi(M',M)∨≅ToriR(M',M∨).",

keywords = "Artinian, Bass number, Betti number, Ext, Hom, Matlis duality, Mini-max, Noetherian, Primary, Secondary, Tensor product, Tor",

author = "Bethany Kubik and Micah Leamer 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 by a GAANN grant from the Department of Education. Sean Sather-Wagstaff was supported by a grant from the NSA . ",

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AU - Sather-Wagstaff, Sean

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PY - 2014/2/1

Y1 - 2014/2/1

N2 - Let R be a commutative ring, and let L and L' be R-modules. We investigate finiteness conditions (e.g., noetherian, artinian, mini-max, Matlis reflexive) of the modules ExtRi(L,L') and ToriR(L,L') when L and L' satisfy combinations of these finiteness conditions. For instance, if R is noetherian, then given R-modules M and M' such that M is Matlis reflexive and M' is mini-max (e.g., noetherian or artinian), we prove that ExtRi(M,M'), ExtRi(M',M), and ToriR(M,M') are Matlis reflexive over R for all i≥0 and that ExtRi(M,M')∨≅ToriR(M,M'∨) and ExtRi(M',M)∨≅ToriR(M',M∨).

AB - Let R be a commutative ring, and let L and L' be R-modules. We investigate finiteness conditions (e.g., noetherian, artinian, mini-max, Matlis reflexive) of the modules ExtRi(L,L') and ToriR(L,L') when L and L' satisfy combinations of these finiteness conditions. For instance, if R is noetherian, then given R-modules M and M' such that M is Matlis reflexive and M' is mini-max (e.g., noetherian or artinian), we prove that ExtRi(M,M'), ExtRi(M',M), and ToriR(M,M') are Matlis reflexive over R for all i≥0 and that ExtRi(M,M')∨≅ToriR(M,M'∨) and ExtRi(M',M)∨≅ToriR(M',M∨).

KW - Artinian

KW - Bass number

KW - Betti number

KW - Ext

KW - Hom

KW - Matlis duality

KW - Mini-max

KW - Noetherian

KW - Primary

KW - Secondary

KW - Tensor product

KW - Tor

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DO - 10.1016/j.jalgebra.2014.01.016

M3 - Article

AN - SCOPUS:84893369745

SN - 0021-8693

VL - 403

SP - 229

EP - 272

JO - Journal of Algebra

JF - Journal of Algebra

ER -

Homology of artinian and mini-max modules, II

Abstract

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Keywords

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