Flat Covers Research Articles

Flat covers and cotorsion envelopes of sheaves

In this paper we prove that any sheaf of modules over any topological space (in fact, any O \mathcal {O} -module where O \mathcal {O} is a sheaf of rings on the topological space) has a flat cover and a cotorsion envelope. This result is very useful, as we shall explain later in the introduction, in order to compute cohomology, due to the fact that the category of sheaves ( O \mathcal {O} -modules) does not have in general enough projectives.

ON THE EXISTENCE OF FLAT COVERS IN R-gr

Recently, a proof of the existence of a flat cover of any module over an arbitrary associative ring with unit has been finally given (see 4-5). In this paper we prove the existence of flat covers in the category of graded modules over a graded ring. Some graded theoretical machinery is introduced to make the proof possible and new graded homological tools are developed.

Brown representability and flat covers

We exhibit a surprising connection between the following two concepts: Brown representability which arises in stable homotopy theory, and flat covers which arise in module theory. It is shown that Brown representability holds for a compactly generated triangulated category if and only if for every additive functor from the category of compact objects into the category of abelian groups a flat cover can be constructed in a canonical way. The proof also shows that Brown representability for objects and morphisms is a consequence of Brown representability for objects and isomorphisms.

Are covering (enveloping) morphisms minimal?

We prove that for certain classes of modules F such that direct sums of F-covers (F-envelopes) are F-covers (F-envelopes), F-covering (Fenveloping) homomorphisms are always right (left) minimal. As a particular case we see that over noetherian rings, essential monomorphisms are left minimal. The same type of results are given when direct products of F-covers are F-covers. Finally we prove that over commutative noetherian rings, any direct product of flat covers of modules of finite length is a flat cover.

Flat Covers of Complexes

In this article we define and study flat complexes over any ring. Also, we prove that any complex over a commutative noetherian ring with finite Krull dimension has a flat cover and a DG-flat cover.

Injective envelopes and flat covers of modules over a commutative ring

Let R be a commutative semilocal ring and U be the minimal injective cogenerator in the category of R-modules. If R is almost noetherian, i.e., each non-minimal prime ideal of R is finitely generated, we prove that the following three statements are equivalent: (1) every U-reflexive R-module has a U-reflexive injective envelope; (2) every U-reflexive R-module has a U-reflexive flat cover; and (3) R is linearly compact and the Krull dimension of R is less than or equal to 1. Our examples show that this result is a non-trivial generalization of a recent theorem of Belshoff and Xu, and that the condition R being almost noetherian is essential.

The existence of flat covers over Noetherian rings of finite Krull dimension

Bass characterized the rings R with the property that every left R-module has a projective cover. These are the left perfect rings. A ring is left perfect if and only if the class of projective R-modules coincides with the class of flat R-modules, so the projective covers over these rings are flat covers. This prompts the conjecture that over any ring R, every left R-module has a flat cover. Known classes of rings for which the conjecture holds include Von Neumann regular rings (trivially), the left perfect rings (Bass), Prufer domains (Enochs), and then more generally, all right coherent rings of finite weak global dimension (Belshoff, Enochs, Xu). In this paper we show that the conjecture holds for all commutative Noetherian rings of finite Krull dimension and so for all local rings and all coordinate rings of affine algebraic varieties.

Injective envelopes and flat covers of Matlis reflexive modules

We show that for a commutative noetherian local ring R, every Matlis reflexive R-module has a reflexive injective envelope if and only if every Matlis reflexive R-module has a reflexive flat cover. This occurs if and only if R is complete and has Krull dimension less than or equal to 1. We also exhibit a family of Matlis reflexive R-modules whose injective envelopes are not Matlis reflexive.