All Modules Have Flat Covers

Bican, Ladislav; Bashir, Robert El; Enochs, Edgar E.

doi:10.1017/s0024609301008104

Cited by 330 publications

(257 citation statements)

References 0 publications

Supporting

Mentioning

251

Contrasting

Order By: Relevance

“…The three cotorsion pairs in Rmod described above are all examples of complete pairs. Proving that the "flat" pair (F, C) is complete is nontrivial, and two different proofs were recently given by the three authors of [BBE01]. Similarly we have the "flat" cotorsion pair on Sh(O), the category of sheaves of O-modules where O is a ringed space on T .…”

Section: Preliminariesmentioning

confidence: 98%

The flat model structure on complexes of sheaves

Gillespie

2006

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. Let Ch(O) be the category of chain complexes of O-modules on a topological space T (where O is a sheaf of rings on T ). We put a Quillen model structure on this category in which the cofibrant objects are built out of flat modules. More precisely, these are the dg-flat complexes. Dually, the fibrant objects will be called dg-cotorsion complexes. We show that this model structure is monoidal, solving the previous problem of not having any monoidal model structure on Ch(O). As a corollary, we have a general framework for doing homological algebra in the category Sh(O) of O-modules. I.e., we have a natural way to define the functors Ext and Tor in Sh(O).

show abstract

Section: Preliminariesmentioning

confidence: 98%

The flat model structure on complexes of sheaves

Gillespie

2006

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

show abstract

“…[11, Lemma 2.2]) Let S be a monoid and P a left collapsible (resp. right reversible) submonoid of S. Then σ = (P ×P ) , the right congruence on S generated by P ×P , is such that P ⊆ [1] σ , [1] σ is left collapsible (resp. right reversible) and S/σ is strongly flat (resp.…”

Section: Proposition 12 ([3])mentioning

confidence: 99%

“…Since S/σ is a cover for S/ρ, S/σ ∼ = S/σ and sσ → sρ under the composite epimorphism. It then follows that [1] …”

Section: Proofmentioning

confidence: 99%

“…If S is a monoid and S/ρ is a cyclic S−act then the natural map S → S/ρ is coessential if and only if [1] ρ is a subgroup of S.…”

Section: Theorem 210mentioning

confidence: 99%

“…But since u ρ 1 then s ρ us = 1 and so s ∈ [1] ρ and the result follows. Conversely if [1] ρ is a subgroup of S then for all u ∈ [1] ρ it follows that uu −1 ∈ {1} and so from Theorem 2.7, S → S/ρ given by s → sρ is coessential.…”

Section: Proofmentioning

confidence: 99%

See 2 more Smart Citations

On covers of cyclic acts over monoids

Mahmoudi

Renshaw

2008

Semigroup Forum

View full text Add to dashboard Cite

In [1] Bican, Bashir and Enochs finally solved a long standing conjecture in module theory that all modules over a unitary ring have a flat cover. The only substantial work on covers of acts over monoids seems to be that of Isbell, Fountain and Kilp ([6], [3], [8]) who only consider projective covers. To our knowledge the situation for flat covers of acts has not been addressed and this paper is an attempt to initiate such a study. We consider almost exclusively covers of cyclic acts and restrict our attention to strongly flat and condition (P ) covers. We give a necessary and sufficient condition for the existence of such covers and for a monoid to have the property that all its cyclic right acts have a strongly flat cover (resp. (P )−cover). We give numerous classes of monoids that satisfy these conditions and we also show that there are monoids that do not satisfy this condition in the strongly flat case. We give a new necessary and sufficient condition for a cyclic act to have a projective cover and provide a new proof of one of Isbell's classic results concerning projective covers. We show also that condition (P ) covers are not unique, unlike the situation for projective covers.

show abstract

Gorenstein projective and flat complexes over noetherian rings

Enochs

Estrada

Iacob

2012

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

Key words Precover, cover, Gorenstein flat complex, Gorenstein projective complex MSC (2010) 18G35, 18G25We give sufficient conditions on a class of R-modules C in order for the class of complexes of C-modules, dwC, to be covering in the category of complexes of R-modules. More precisely, we prove that if C is precovering in R − M od and if C is closed under direct limits, direct products, and extensions, then the class dwC is covering in Ch(R).Our first application concerns the class of Gorenstein flat modules. We show that when the ring R is two sided noetherian, a complex C is Gorenstein flat if and only if each module Cn is Gorenstein flat. If moreover every direct product of Gorenstein flat modules is a Gorenstein flat module, then the class of Gorenstein flat complexes is covering.We consider Gorenstein projective complexes as well. We prove that if R is a commutative noetherian ring of finite Krull dimension, then the class of Gorenstein projective complexes coincides with that of complexes of Gorenstein projective modules. We also show that if R is commutative noetherian with a dualizing complex then every right bounded complex has a Gorenstein projective precover.

show abstract

All Modules Have Flat Covers

Cited by 330 publications

References 0 publications

The flat model structure on complexes of sheaves

The flat model structure on complexes of sheaves

On covers of cyclic acts over monoids

Gorenstein projective and flat complexes over noetherian rings

Contact Info

Product

Resources

About