AI 中文总结
本文针对非半分离的拟紧拟分离概形，构造了无容许单态射到局部内射余层的局部挠自由 contraherent 余层，并给出正面结果。\n
AI 中文摘要
设 $X$ 是一个拟紧、拟分离但不半分离的概形。我们在 $X$ 上构造了一个局部挠自由 contraherent 余层，它不具有到任何局部内射局部 contraherent 余层的容许单态射。该构造与证明遵循 Slavik 和 Stovicek 在 arXiv:1902.05740 中的论证。一个互补的正面结果，即关于有限 Krull 维数的 Noether 概形 $X$ 上的每个 $\mathbf W$-局部 contraherent 余层都具有到局部挠自由 $\mathbf W$-局部 contraherent 余层的容许单态射，可见于预印本 arXiv:2603.27732v5 第 4.7 节。
英文摘要
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.
CommentsLaTeX 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)
