2019/07/10 by Tsutomu Nakamura, Nakamura, Tsutomu, Peder Thompson +1
Mathematics · Physics and Astronomy · #13C13 (Secondary) #13D02 (Primary) #Algebraic structures and combinatorial models #Commutative Algebra (math.AC) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Nonlinear Waves and Solitons #Rings and Algebras (math.RA)
paper · pdf · doi:10.48550/arxiv.1907.04671
openalex publication_date 2019/07/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Over a commutative noetherian ring R of 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.