The contraherent version of the theorem of Slavik and Stovicek

Abstract

This is a paper about contraherent cosheaves on non-semi-separated schemes. We prove two theorems, a negative one and a positive one. On the negative side, let $X$ be a quasi-compact, quasi-separated scheme that is not semi-separated. We present a locally cotorsion contraherent cosheaf on $X$ that does not have an admissible monomorphism into any locally injective locally contraherent cosheaf. The construction and proof follow the arguments of Slavik and Stovicek in arXiv:1902.05740. On the positive side, let $X$ be a Noetherian scheme of finite Krull dimension. We prove that every $\mathbf{W}$-locally contraherent cosheaf on $X$ has an admissible monomorphism into a locally cotorsion $\mathbf{W}$-locally contraherent cosheaf. Moreover, the cokernel is a flat contraherent cosheaf. The proof is based on the theorem of Raynaud-Gruson about the projective dimensions of flat modules and Enochs' classification of flat cotorsion modules.

LaTeX 2e with xy-pic and mathx fonts, 59 pages, 7 commutative diagrams; v.2: new Lemmas 2.1, 2.3, and 2.4 inserted, new Sections 3.4 and 6.2-6.5 inserted, new Sections 8-10 added (the second main result and its proof added)