The existence of flat covers over Noetherian rings of finite Krull dimension. (English) Zbl 0840.16004

Let \(M\) be a (left) module over the ring \(R\). Then an \(R\)-homomorphism \(\phi:F\to M\) is called a flat cover of \(M\) if (i) \(F\) is flat, (ii) given any homomorphism \(\alpha:F'\to M\), where \(F'\) is flat then there is a homomorphism \(\beta: F'\to F\) such that \(\phi\beta=\alpha\) and (iii) if \(\alpha=\phi\) in (ii) then the \(\beta\) described must be an isomorphism. If we insist only on conditions (i) and (ii) then \(\phi\) is called a flat precover of \(M\) but, in this case, \(M\) will also have a flat cover by a result of E. E. Enochs [Isr. J. Math. 39, 189-209 (1981; Zbl 0464.16019)]. It is straightforward to see that over a left perfect ring every left module has a flat cover, as does (trivially) any module over a von Neumann regular ring. Moreover, R. Belshoff, E. E. Enochs and the current author have shown [in Proc. Am. Math. Soc. 122, No. 4, 985-991 (1994; see the preceding review Zbl 0840.16003)] that any left module of finite flat dimension over a right coherent ring has a flat cover. However, the existence of a flat cover for an arbitrary module over an arbitrary ring remains an open question. Here it is shown, using pullback techniques, that all modules over commutative Noetherian rings of finite Krull dimension have flat covers. En route to proving this, the author shows that if \(R\) is a right coherent ring and \(0\to A\to B\to C\to 0\) is an exact sequence of left \(R\)-modules such that \(A\) has a flat cover then \(B\) has a flat cover if and only if \(C\) has.
Reviewer: J.Clark (Dunedin)


16D40 Free, projective, and flat modules and ideals in associative algebras
16P40 Noetherian rings and modules (associative rings and algebras)
13C11 Injective and flat modules and ideals in commutative rings
16L30 Noncommutative local and semilocal rings, perfect rings
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16P70 Chain conditions on other classes of submodules, ideals, subrings, etc.; coherence (associative rings and algebras)
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