Flat cover

flat covering

In [a1], R. Baer proved that every module $M$ can be embedded in an injective module $E$. In [a5], B. Eckmann and A. Schöpf defined an injective envelope of a module to be an embedding $M \subset E$ with $E$ injective and with $M$ essential in $E$, i.e. such that $S \cap M \neq 0$ for every submodule $S \subset E$, $S \neq 0$. They proved that every module has an injective envelope and that if $M \subset E _ { 1 }$ and $M \subset E _ { 2 }$ are two injective envelopes of $M$, then any morphism $E _ { 1 } \rightarrow E _ { 2 }$ which is the identity on $M$ (and such exists) is an isomorphism. So, injective envelopes are unique up to isomorphism.

In [a2], H. Bass considered projective covers of modules. The notion of a projective cover is categorically dual to that of an injective envelope. If $R$ is a ring, Bass proved that every left $R$-module has a projective cover if and only if every flat left $R$-module is projective (cf. also Projective module). Projective covers (when they exist) are also unique up to isomorphism.

In [a7] the following definition can be found: If $\mathcal{F}$ is a class of objects in a category $\mathcal{C}$, then a morphism $\phi : F \rightarrow X$ in $\mathcal{C}$ with $F \in \mathcal{F}$ is called an $\mathcal{F}$-pc if $\operatorname { Hom }( G , F ) \rightarrow \operatorname { Hom } ( G , X )$ is surjective for all $G \in \mathcal{F}$ and is called an $\mathcal{F}$-cover if, moreover, every $f : F \rightarrow F$ such that $\phi \circ f = \phi$ is an automorphism of $\mathcal{F}$.

When $\mathcal{F}$-covers exist, they are unique up to isomorphism. Both pre-covers and covers are frequently named after the class $\mathcal{F}$. So, flat covers are $\mathcal{F}$-covers with $\mathcal{F}$ the class of flat modules in the category of modules. Pre-envelopes and envelopes are defined dually. Then this terminology agrees with the earlier terminology of injective envelopes and projective covers. In this language, Bass' result says that if every module has a projective cover, then these projective covers are flat covers.

In [a7], E. Enochs proved that if a module has a flat pre-cover, then it has a flat cover. He also conjectured that every module has a flat cover. J. Xu [a12] proved that the conjecture holds for all commutative Noetherian rings of finite Krull dimension (cf. also Dimension; Noetherian ring) and L. Bican, R. El Bashir and Enochs [a4] gave two different solutions of the conjecture for any ring. One proof uses a result of El Bashir which shows that any morphism $F \rightarrow M$ of a flat module into $M$ with $F$ sufficiently large has a non-zero pure submodule of $F$ in its kernel. The other proof is an application of a theorem of P.C. Eklof and J. Trlifaj [a6], Thm. 2, guaranteeing "enough injectives and projectives" for certain cotorsion theories (as defined in [a11]). Eklof and Trlifaj attribute the inspiration for their theorem to a construction of [a9]. D. Quillen [a10], Lemma II 3.3, gives essentially the same argument in the setting of homotopical algebra.

In [a3], Bass defined what were subsequently called the Bass numbers of a module $M$ over a commutative Noetherian ring. These are computed using the minimal injective resolution of the ring. Xu [a8] showed that flat covers can be used to define dual Bass numbers of modules. These numbers have properties in some sense dual to the properties of the original Bass numbers.

References

[a1]	R. Baer, "Abelian groups which are direct summands of every containing group" Bull. Amer. Math. Soc. , 46 (1940) pp. 800–806
[a2]	H. Bass, "Finitistic dimension and a homological generalization of semiprimary rings" Trans. Amer. Math. Soc. , 95 (1960) pp. 466–488
[a3]	H. Bass, "On the ubiquity of Gorenstein rings" Math. Z. , 82 (1963) pp. 8–28
[a4]	L. Bican, R. El Bashir, E. Enochs, "All modules have flat covers" Bull. London Math. Soc. (to appear)
[a5]	B. Eckmann, A. Schöpf, "Über injektive Moduln" Archiv Math. , 4 (1953) pp. 75–78
[a6]	P. Eklof, J. Trlifaj, "How to make Ext vanish" Bull. London Math. Soc. (to appear)
[a7]	E. Enochs, "Injective and flat covers, envelopes and resolvents" Israel J. Math. , 39 (1981) pp. 33–38
[a8]	E. Enochs, J. Xu, "On invariants dual to the Bass numbers" Proc. Amer. Math. Soc. , 125 (1997) pp. 951–960
[a9]	R. Göbel, S. Shelah, "Cotorsion theories and splitters" Trans. Amer. Math. Soc. (to appear)
[a10]	D. Quillen, "Homotopical algebra" , Lecture Notes in Mathematics , 43 , Springer (1967)
[a11]	L. Salce, "Cotorsion theories for Abelian groups" , Symp. Math. , 23 , Amer. Math. Soc. (1979) pp. 11–32
[a12]	J. Xu, "The existence of flat covers over noetherian rings of finite Krull dimension" Proc. Amer. Math. Soc. , 123 (1995) pp. 27–32
[a13]	J. Xu, "Flat covers of modules" , Lecture Notes in Mathematics , 1634 , Springer (1996)