Divisibility classes are seldom closed under flat covers

Abstract

It is well-known that a class of all modules, which are torsion-free with respect to a set of ideals, is closed under injective envelopes. In this paper, we consider a kind of a dual to this statement -are the divisibility classes closed under flat covers? -and argue that this is seldom the case. More precisely, we show that the class of all divisible modules over an integral domain R is closed under flat covers if and only if R is almost perfect. Also, we show that if the class of all s-divisible modules, where s is a regular element of a commutative ring R, is closed under flat covers then the quotient ring R/sR satisfies some rather restrictive properties. The question is motivated by the recent classification [11] of tilting classes over commutative rings.Theorem 0.1. ([11, Theorem 5.3]) Let R be a commutative ring. Then the classes dual to n-tilting classes are precisely the classes of formwhere (X 0 , X 1 , . . . , X n−1 ) is a characteristic sequence in Spec(R).The terminology is explained in the next section, but for now the main point is that these classes consist of all modules M such that their i-th minimal cosyzygy