Products of DG-Injective Complexes

Alina Iacob

Products of DG-Injective Complexes

Mathematical Sciences

Research output: Contribution to conference › Presentation

Abstract

<div class="line" id="line-41"> We consider a left noetherian ring R. We give a necessary and su&ffilig;cient condition in order that a complex of R-modules be DG-injective. Using this result we prove that if (Ki)i∈I is a family of DG-injective complexes of left R-modules and K is the &alefsym;1-product of (Ki)i∈I (i.e. K ⊂ Πi∈IKi is such that for each n, Kn ⊂ Πi∈IKn/i consists of all (xi)i∈I such that {i| xi =/=0} is at most countable) then K is DG-injective.</div><div class="line" id="line-44"> <br/></div><div class="line" id="line-19"> We also consider the &alefsym;0-product of a family of DG-injective complexes i.e. the direct sum. We give a necessary condition in order that every direct sum of DG-injective complexes over a left noetherian ring be DG-injective. We use this result to prove that if R is a commutative local artinian ring then every direct sum of DG-injective complexes is DG-injective if and only if gl.dim R<∞. We show that the result holds for any commutative artinian ring.</div>

Original language	American English
State	Published - Jan 5 2005
Event	Joint Meeting of the American Mathematical Society (AMS) and the Mathematical Association of America (MAA) - Atlanta, GA Duration: Jan 5 2005 → …

Conference

Conference	Joint Meeting of the American Mathematical Society (AMS) and the Mathematical Association of America (MAA)
Period	01/5/05 → …

Keywords

DG-injective complexes

DC Disciplines

Mathematics

Cite this

@conference{3bfa9a79c7044d14a87a24847ec20922,

title = "Products of DG-Injective Complexes",

abstract = " We consider a left noetherian ring R. We give a necessary and su&ffilig;cient condition in order that a complex of R-modules be DG-injective. Using this result we prove that if (Ki)i∈I is a family of DG-injective complexes of left R-modules and K is the &alefsym;1-product of (Ki)i∈I (i.e. K ⊂ Πi∈IKi is such that for each n, Kn ⊂ Πi∈IKn/i consists of all (xi)i∈I such that {i| xi =/=0} is at most countable) then K is DG-injective. We also consider the &alefsym;0-product of a family of DG-injective complexes i.e. the direct sum. We give a necessary condition in order that every direct sum of DG-injective complexes over a left noetherian ring be DG-injective. We use this result to prove that if R is a commutative local artinian ring then every direct sum of DG-injective complexes is DG-injective if and only if gl.dim R<∞. We show that the result holds for any commutative artinian ring.",

keywords = "DG-injective complexes",

author = "Alina Iacob",

year = "2005",

month = jan,

day = "5",

language = "American English",

note = "Joint Meeting of the American Mathematical Society (AMS) and the Mathematical Association of America (MAA) ; Conference date: 05-01-2005",

}

TY - CONF

T1 - Products of DG-Injective Complexes

AU - Iacob, Alina

PY - 2005/1/5

Y1 - 2005/1/5

N2 - We consider a left noetherian ring R. We give a necessary and su&ffilig;cient condition in order that a complex of R-modules be DG-injective. Using this result we prove that if (Ki)i∈I is a family of DG-injective complexes of left R-modules and K is the &alefsym;1-product of (Ki)i∈I (i.e. K ⊂ Πi∈IKi is such that for each n, Kn ⊂ Πi∈IKn/i consists of all (xi)i∈I such that {i| xi =/=0} is at most countable) then K is DG-injective. We also consider the &alefsym;0-product of a family of DG-injective complexes i.e. the direct sum. We give a necessary condition in order that every direct sum of DG-injective complexes over a left noetherian ring be DG-injective. We use this result to prove that if R is a commutative local artinian ring then every direct sum of DG-injective complexes is DG-injective if and only if gl.dim R<∞. We show that the result holds for any commutative artinian ring.

AB - We consider a left noetherian ring R. We give a necessary and su&ffilig;cient condition in order that a complex of R-modules be DG-injective. Using this result we prove that if (Ki)i∈I is a family of DG-injective complexes of left R-modules and K is the &alefsym;1-product of (Ki)i∈I (i.e. K ⊂ Πi∈IKi is such that for each n, Kn ⊂ Πi∈IKn/i consists of all (xi)i∈I such that {i| xi =/=0} is at most countable) then K is DG-injective. We also consider the &alefsym;0-product of a family of DG-injective complexes i.e. the direct sum. We give a necessary condition in order that every direct sum of DG-injective complexes over a left noetherian ring be DG-injective. We use this result to prove that if R is a commutative local artinian ring then every direct sum of DG-injective complexes is DG-injective if and only if gl.dim R<∞. We show that the result holds for any commutative artinian ring.

KW - DG-injective complexes

UR - http://jointmathematicsmeetings.org/meetings/national/jmm/2091_abstracts/1003-18-647.pdf

M3 - Presentation

T2 - Joint Meeting of the American Mathematical Society (AMS) and the Mathematical Association of America (MAA)

Y2 - 5 January 2005

ER -

Products of DG-Injective Complexes

Abstract

Conference

Keywords

DC Disciplines

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