Minimal semi-flat-cotorsion replacements and cosupport

Abstract

Over a commutative noetherian ring Rof finite Krull dimension, we show that every complex of flat cotorsion R-modules decomposes as a direct sum of a minimal complex and a contractible complex. Moreover, we define the notion of a semi-flat-cotorsion complex as a special type of semi-flat complex, and provide functorial ways to construct a quasi-isomorphism from a semi-flat complex to a semi-flat-cotorsion complex. Consequently, every R-complex can be replaced by a minimal semi-flat-cotorsion complex in the derived category over R. Furthermore, we describe structure of semi-flat-cotorsion replacements, by which we recover classic theorems for finitistic dimensions. In addition, we improve some results on cosupport and give a cautionary example. We also explain that semi-flat-cotorsion replacements always exist and can be used to describe the derived category over any associative ring.