Timeline for Is a retract of a free object free?
Current License: CC BY-SA 3.0
21 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 5, 2015 at 8:58 | answer | added | Jeremy Rickard | timeline score: 12 | |
| S Mar 16, 2012 at 22:07 | vote | accept | Victor | ||
| S Mar 16, 2012 at 22:07 | vote | accept | Victor | ||
| S Mar 16, 2012 at 22:07 | |||||
| S Mar 14, 2012 at 16:42 | vote | accept | Victor | ||
| S Mar 16, 2012 at 22:07 | |||||
| S Mar 11, 2012 at 2:13 | vote | accept | Victor | ||
| S Mar 14, 2012 at 16:42 | |||||
| Mar 11, 2012 at 1:19 | comment | added | Bill Johnson | It is true both in $Ban_1$ (Banach spaces with morphisms the linear operators of norm at most one) and $Ban$ (Banach spaces with morphisms the bounded linear operators). In both of these analytic categories the free objects are the spaces $\ell_1(S)$ with $S$ any set. | |
| Mar 10, 2012 at 10:21 | comment | added | Vladimir Dotsenko | I think that for commutative algebras this problem is very hard (description of all retracts is related to both the cancellation conjecture and the Jacobian conjecture). For associative algebras, the only source I know uses the results in the commutative case (arxiv.org/pdf/math/9701210v1.pdf), maybe it's possible to do better. | |
| Mar 10, 2012 at 5:28 | comment | added | user6976 | The variety of groups generated by $S_3$ consists of all semidirect products of groups of abelian groups of exponent 3 and abelian groups of exponent 2. Thus the free objects in that variety have retracts that are Abelian groups of exponent 2 which are not free in that variety. If you do not know what"variety" is, read the book by Hanna Neumann "Varieties of groups". | |
| S Mar 9, 2012 at 22:21 | vote | accept | Victor | ||
| S Mar 11, 2012 at 2:13 | |||||
| Mar 9, 2012 at 21:54 | comment | added | Victor | So, what about the category of associative algebras? Is a retract of a tensor algebra free? | |
| Mar 9, 2012 at 21:53 | comment | added | Victor | @Charles. One can see from Costa, Douglas L. Retracts of polynomial rings. J. Algebra 44 (1977), no. 2, 492–502. that in 1977 it was unknown whether every retract of $K[X_1,\ldots,X_n]$ is a polynomial ring, where $K$ is a field: The author shows that an affirmative answer to this question would solve the well-known cancellation problem for polynomial rings over fields. Is it still unknown?!! | |
| Mar 9, 2012 at 21:44 | vote | accept | Victor | ||
| S Mar 9, 2012 at 22:21 | |||||
| Mar 9, 2012 at 21:44 | comment | added | Victor | @Mark. Can you explain what you mean by "groups generated by $S_3$? Indeed, by The Nielsen–Schreier theorem any subgroup of a free group is free, as the first answer below says. | |
| Mar 9, 2012 at 20:59 | comment | added | Martin Brandenburg | So the question basically asks when "free = projective". | |
| Mar 9, 2012 at 20:34 | comment | added | Buschi Sergio | In any category $\mathcal{C}$ on $Set$, If "any projactive object is free" then "any retract of a a free object is free" (where an object $X$ is free on $S$ if represent the co-presheav $X\mapsto \mathcal{C}(S, |X|)$ where $X\mapsto |X|$ is the canonical functor on $Set$. we have the inverse implication if the funtor $X\mapsto |X|$ has a left adjoint $L$ and the counit $L(|X|)\to X$ is (puntually) a epimorphism. | |
| Mar 9, 2012 at 18:42 | answer | added | Benjamin Steinberg | timeline score: 12 | |
| Mar 9, 2012 at 17:33 | comment | added | Charles Rezk | There's a literature on rectracts of polynomial rings, which is referenced in this answer: mathoverflow.net/questions/55931/… | |
| Mar 9, 2012 at 17:32 | comment | added | user6976 | It is not true in the variety of groups generated by $S_3$. | |
| Mar 9, 2012 at 17:32 | answer | added | MTS | timeline score: 11 | |
| Mar 9, 2012 at 17:25 | comment | added | Tom Leinster | Do you want to restrict to free objects on finite sets? | |
| Mar 9, 2012 at 17:17 | history | asked | Victor | CC BY-SA 3.0 |