Timeline for What are retracts of polynomial rings?
Current License: CC BY-SA 3.0
16 events
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| Jun 23, 2020 at 17:04 | history | edited | Vladimir Dotsenko |
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| Apr 19, 2020 at 10:39 | vote | accept | Todd Trimble | ||
| Apr 19, 2020 at 8:21 | answer | added | Vladimir Dotsenko | timeline score: 13 | |
| May 27, 2019 at 19:50 | comment | added | Todd Trimble | This question just got a downvote. What was wrong with it? | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
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| Apr 13, 2017 at 12:19 | history | edited | CommunityBot |
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| Oct 19, 2015 at 17:36 | comment | added | Todd Trimble | @JeremyRickard Thanks for your comment, which I have now incorporated into the question. | |
| Oct 19, 2015 at 17:35 | history | edited | Todd Trimble | CC BY-SA 3.0 |
incorporated a comment by Jeremy Rickard
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| Oct 18, 2015 at 10:59 | comment | added | Jeremy Rickard | Maybe your remark about what can extracted from Gupta's solution to the cancellation problem doesn't make it clear that her work shows that there are retracts of polynomial algebras over fields that are not polynomial algebras (in fact, she shows that there are examples over any field of positive characteristic). | |
| Oct 5, 2015 at 17:56 | comment | added | მამუკა ჯიბლაძე | Regarding existence of some rings with nonfree projective algebras - symmetric algebras of nonfree projective modules are such. Presumably nontrivial sphere bundles which are not equivalent to ones coming from vector bundles may give more general such examples (after switching to coordinate rings - if such bundles can be realized over algebraic varieties). But all this does not tell anything about fields or $\mathbb Z$. | |
| Oct 4, 2015 at 12:32 | comment | added | Todd Trimble | @მამუკაჯიბლაძე Thanks for clarifying further; even if you had said 'arithmetical' I probably would have misunderstood you. A simple example of non-free projective objects (not far from the case of commutative rings) occurs for the theory of Boolean rings, where any non-terminal finite Boolean algebra is projective. Meanwhile, one particular Mal'cev operation in PRA comes from thinking of the natural numbers under reverse ordering as forming a cartesian closed poset, and then putting $p(x, y, z) = x^{y^z} \wedge z^{y^x}$. | |
| Oct 4, 2015 at 11:09 | comment | added | მამუკა ჯიბლაძე | Strangely enough, although I used wrong term - "arithmetic" instead of "arithmetical" - it still turns out to be closely related (or maybe even amounting to the same thing, I don't know). Namely I meant theories with a specific Maltsev term, the one satisfying $p(x,y,y)=p(x,y,x)=p(y,y,x)=x$. | |
| Oct 4, 2015 at 10:52 | comment | added | Todd Trimble | @მამუკაჯიბლაძე Thanks for the suggestions. In particular for the suggestion to look at arithmetic theories; I understand PRA as initial among Lawvere theories in which the generating object is an NNO, but I'd never thought to consider its Cauchy completion. | |
| Oct 4, 2015 at 6:30 | comment | added | მამუკა ჯიბლაძე | Returning to commutative rings - one approach might be this: Yoneda places the category of affine schemes (i. e. the opposite of the category of commutative rings) inside the category of functors from finitely presented rings to sets, and you might also make it more tight by restricting to subtoposes of Zariski, étale, ..., canonical sheaves. Now the guys you ask about will be injectives in affine schemes, so you might hope to understand them by looking at injectives in one of these sheaf toposes, which are fairly simpler to get: they are retracts of powers of respective subobject classifiers. | |
| Oct 4, 2015 at 6:23 | comment | added | მამუკა ჯიბლაძე | To "squeeze water out" you might switch from the theory of commutative rings, which is merely Maltsev, to "much more Maltsev" theories. Now on one hand, with the theory of modules over a fixed ring you are asking about finitely generated projective modules, so the whole K-theory with its vector bundles falls on you. While zooming in in the other direction if you specialize to arithmetic theories, you are in the realm of e. g. projective formulas in the Intuitionistic Propositional Calculus or in various modal logics - there are many deep results there, and it is highly nontrivial. | |
| Oct 4, 2015 at 2:21 | history | asked | Todd Trimble | CC BY-SA 3.0 |