The contraherent version of the theorem of Slavik and Stovicek

Abstract: 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. A complementary positive result, claiming that every $\mathbf W$-locally contraherent cosheaf on a Noetherian scheme $X$ of finite Krull dimension has an admissible monomorphism into a locally cotorsion $\mathbf W$-locally contraherent cosheaf, can be found in the preprint arXiv:2603.27732v5, Section 4.7.