Very flat, locally very flat, and contraadjusted modules. (English) Zbl 1348.13018
The authors study very flat and contraadjusted modules (introduced by L. Positselski [“Contraherent cosheaves”, Preprint, arXiv:1209.2995v5]) over a commutative ring. Here is their abstract:
“Very flat and contraadjusted modules naturally arise in algebraic geometry in the study of contraherent cosheaves over schemes. Here, we investigate the structure and approximation properties of these modules over commutative noetherian rings. Using an analogy between projective and flat Mittag-Leffler modules on one hand, and very flat and locally very flat modules on the other, we prove that each of the following statements are equivalent to the finiteness of the Zariski spectrum \mathrm{Spec}\((R)\) of a noetherian domain \(R\): (i) the class of all very flat modules is covering, (ii) the class of all locally very flat modules is precovering, and (iii) the class of all contraadjusted modules is enveloping. We also prove an analog of Pontryagin’s Criterion for locally very flat modules over Dedekind domains.”
Thus the authors prove that a module \(M\) over a Dedekind domain \(R\) is locally very flat if and only if each finite rank submodule of \(M\) is very flat.
“Very flat and contraadjusted modules naturally arise in algebraic geometry in the study of contraherent cosheaves over schemes. Here, we investigate the structure and approximation properties of these modules over commutative noetherian rings. Using an analogy between projective and flat Mittag-Leffler modules on one hand, and very flat and locally very flat modules on the other, we prove that each of the following statements are equivalent to the finiteness of the Zariski spectrum \mathrm{Spec}\((R)\) of a noetherian domain \(R\): (i) the class of all very flat modules is covering, (ii) the class of all locally very flat modules is precovering, and (iii) the class of all contraadjusted modules is enveloping. We also prove an analog of Pontryagin’s Criterion for locally very flat modules over Dedekind domains.”
Thus the authors prove that a module \(M\) over a Dedekind domain \(R\) is locally very flat if and only if each finite rank submodule of \(M\) is very flat.
Reviewer: Moshe Roitman (Haifa)
MSC:
| 13C11 | Injective and flat modules and ideals in commutative rings |
| 14F05 | Sheaves, derived categories of sheaves, etc. (MSC2010) |
| 16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |
| 13E05 | Commutative Noetherian rings and modules |
| 13G05 | Integral domains |
Keywords:
Bass module; contraadjusted module; covering class of modules; Dedekind domain; filtration; Mittag-Leffler module; Pontryagin’s criterion; precovering class of modules; projective module; very flat moduleReferences:
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