Contramodules over pro-perfect topological rings

Abstract

For four wide classes of topological rings $\mathfrak{R}$, we show that all flat left $\mathfrak{R}$-contramodules have projective covers if and only if all flat left $\mathfrak{R}$-contramodules are projective if and only if all left $\mathfrak{R}$-contramodules have projective covers if and only if all descending chains of cyclic discrete right $\mathfrak{R}$-modules terminate if and only if all the discrete quotient rings of $\mathfrak{R}$ are left perfect. Three classes of topological rings for which this holds are the complete, separated topological associative rings with a base of neighborhoods of zero formed by open two-sided ideals such that either the ring is commutative, or it has a countable base of neighborhoods of zero, or it has only a finite number of semisimple discrete quotient rings. The fourth class consists of all the topological rings with a base of neighborhoods of zero formed by open right ideals which have a closed two-sided ideal with certain properties such that the quotient ring is a topological product of rings from the previous three classes. The key technique on which the proofs are based is the contramodule Nakayama lemma for topologically T-nilpotent ideals.

LaTeX 2e with xy-pic, 53 pages, 3 commutative diagrams; v2: this is an improved version of Sections 1-10 of v1, the rest of v1 was moved to arXiv:1907.04973 and arXiv:1907.05537; v.5: Sections 1.8, 1.9, 1.10, and 1.11 expanded; v.6: small additions and corrections, references updated, the numbering of sections (and of subsections in the introduction) shifted to agree with the journal version

References in corpus (9)

• $S$-almost perfect commutative rings
• Triangulated Matlis equivalence
• The tilting-cotilting correspondence
• Infinity-tilting theory
• Topologically semisimple and topologically perfect topological rings
• Abelian right perpendicular subcategories in module categories
• Covers and direct limits: a contramodule-based approach
• Projective covers of flat contramodules
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Cited by in corpus (11)

• Topologically semisimple and topologically perfect topological rings
• Abelian right perpendicular subcategories in module categories
• Projective covers of flat contramodules
• Closure properties of $\underrightarrow{\lim }\mathcal{C}$
• Exact categories of topological vector spaces with linear topology
• Derived decompositions of abelian categories I
• Quasi-coherent torsion sheaves, the semiderived category, and the semitensor product: Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes
• Pseudo-dualizing complexes of bicomodules and pairs of t-structures
• Enveloping Classes over Commutative Rings
• Homological full-and-faithfulness of comodule inclusion and contramodule forgetful functors
• Topologically semiperfect topological rings