Timeline for Is there an adjoint to the inclusion of I-adically complete modules to all modules?
Current License: CC BY-SA 4.0
8 events
| when toggle format | what | by | license | comment | |
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| Nov 26, 2018 at 6:40 | vote | accept | Dasha Poliakova | ||
| Nov 26, 2018 at 0:38 | comment | added | Leonid Positselski | Moreover, from some point of view it is often not even strictly necessary to check it! If you are willing to assume a certain strong axiom in the foundations of mathematics (called Vopěnka's principle; or, which is sometimes enough, its weaker version), then Theorem 6.22 on page 257 of the same book tells that a full subcategory in the category of modules over an associative ring is reflective whenever it is closed under limits. | |
| Nov 26, 2018 at 0:35 | comment | added | Leonid Positselski | The condition on "$\lambda$-directed colimits" may look intimidating when you meet it for the first time, but it does not have to be. You can learn from the book what it means, and it is worth doing so, and it is often easy to check and use. | |
| Nov 26, 2018 at 0:27 | comment | added | Leonid Positselski | Furthermore, a very good reference source in such matters is the book by J. Adámek and J. Rosický, "Locally presentable and accessible categories", Cambridge Univ. Press 1994. For the problem at hand, "Reflection Theorem 2.48" on page 100 provides a sufficient criterion for a full subcategory in the category of modules over an associative ring to be reflective. A reflector exists provided that the full subcategory is closed under 1. all limits, and 2. all $\lambda$-directed colimits, for some big enough cardinal $\lambda$. | |
| Nov 26, 2018 at 0:17 | comment | added | Leonid Positselski | In the case of an infinitely generated ideal $I$ in a commutative ring $R$, my guess would be that the desired adjoint functor, generally speaking, does not exist. A full subcategory in a category is called reflective if its inclusion functor has a left adjoint. If this is the case, such an adjoint is called the reflector. A necessary condition for a full subcategory to be reflective is that it has to be closed under limits. I would try to show that in the situation at hand the full subcategory of complete modules is not even closed under products in the category of modules. | |
| Nov 25, 2018 at 23:59 | answer | added | Leonid Positselski | timeline score: 14 | |
| Nov 25, 2018 at 14:07 | comment | added | Laurent Moret-Bailly | The "completion" functor does not even land into complete modules! See stacks.math.columbia.edu/tag/05JA. | |
| Nov 25, 2018 at 10:21 | history | asked | Dasha Poliakova | CC BY-SA 4.0 |