TY - JOUR
T1 - Tests for injectivity of modules over commutative rings
AU - Christensen, Lars Winther
AU - Iyengar, Srikanth B.
N1 - Funding Information:
We thank the Centre de Recerca Matemàtica, Barcelona, for hospitality during visits in Spring 2015, when part of the work reported in this article was done. L. W. C. was partly supported by NSA Grant H98230-14-0140, and S. B. I. was partly supported by NSF Grant DMS-1503044.
Publisher Copyright:
© 2016, Universitat de Barcelona.
PY - 2017/5/1
Y1 - 2017/5/1
N2 - It is proved that a module M over a commutative noetherian ring R is injective if ExtRi((R/p)p,M)=0 holds for every i⩾ 1 and every prime ideal p in R. This leads to the following characterization of injective modules: If F is faithfully flat, then a module M such that Hom R(F, M) is injective and ExtRi(F,M)=0 for all i⩾ 1 is injective. A limited version of this characterization is also proved for certain non-noetherian rings.
AB - It is proved that a module M over a commutative noetherian ring R is injective if ExtRi((R/p)p,M)=0 holds for every i⩾ 1 and every prime ideal p in R. This leads to the following characterization of injective modules: If F is faithfully flat, then a module M such that Hom R(F, M) is injective and ExtRi(F,M)=0 for all i⩾ 1 is injective. A limited version of this characterization is also proved for certain non-noetherian rings.
KW - Cosupport
KW - Injective dimension
KW - Injective module
UR - http://www.scopus.com/inward/record.url?scp=85017276551&partnerID=8YFLogxK
U2 - 10.1007/s13348-016-0176-0
DO - 10.1007/s13348-016-0176-0
M3 - Article
AN - SCOPUS:85017276551
SN - 0010-0757
VL - 68
SP - 243
EP - 250
JO - Collectanea Mathematica
JF - Collectanea Mathematica
IS - 2
ER -