Timeline for Infinite tensor products
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32 events
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| Sep 15, 2021 at 17:43 | history | wiki removed | Stefan Kohl♦ | ||
| Feb 28, 2020 at 10:09 | comment | added | Martin Brandenburg | A map $\beta : \prod_{i \in I} M_i \to N$ is multilinear iff for every $j \in I$ and every $u \in \prod_{i \in I \setminus \{j\}} M_i$ the induced map $\beta(-,u) : M_j \to N$ is linear. | |
| Feb 26, 2020 at 8:33 | history | edited | YCor | CC BY-SA 4.0 |
changed deprecated tag, fixed typo, romanized hom
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| Feb 5, 2020 at 16:46 | comment | added | Prince Khan | @MartinBrandenburg thank you but than what is the definition of multilinear map on the product on $∏_{i∈I}M_i?$ | |
| Feb 4, 2020 at 23:04 | comment | added | Martin Brandenburg | Read more carefully, this is not what is written there. It is said that $\bigotimes_{i \in I} M_i$ exists, and the same proof as in the finite case works. Take the free module on $\prod_{i \in I} |M_i|$ and quotient out the multilinear relations. | |
| Feb 4, 2020 at 22:51 | comment | added | Prince Khan | Can any one please make me understand the statement made in the second line that "multilinear maps defined on $∏_{i∈I}M_i$ exist by the usual construction". What does the usual construction mean by? | |
| Apr 13, 2017 at 12:58 | history | edited | CommunityBot |
replaced http://mathoverflow.net/ with https://mathoverflow.net/
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| Jul 23, 2014 at 21:12 | comment | added | ashpool | @MartinBrandenburg Actually I'm having trouble understanding many of your notations (including $K^x$). Can you give a concrete example of $\bigotimes_{i\in I}A\not\simeq A$, whose proof is as least ugly as possible? | |
| Jul 23, 2014 at 8:40 | comment | added | Martin Brandenburg | @ashpool: $U=K^x$, which is trivial when $K=\mathbb{F}_2$. | |
| Jul 23, 2014 at 0:01 | comment | added | ashpool | I'm trying to understand the vector space isomorphism $\otimes_{i\in I}K\simeq K[U^I/U^{(I)}]$ in 1.4. If $K=\mathbb{F}_2$ and $I=\mathbb{N}$, it seems that $\bigotimes_{\mathbb{N}}\mathbb{F}_2\simeq\mathbb{F}_2\varsubsetneqq\mathbb{F}_2[ \prod_{\mathbb{N}}\mathbb{F}_2/\bigoplus_{\mathbb{N}}\mathbb{F}_2]$ ? | |
| Dec 15, 2011 at 8:27 | comment | added | Buschi Sergio | I had read (student time) about infinite multiliear(and infinite tensoring) in the Claude Chevalley "Fundamental concepts of algebra". May be you just know this reference, | |
| Dec 15, 2011 at 5:54 | answer | added | Chi-Keung Ng | timeline score: 10 | |
| Jul 19, 2010 at 8:09 | comment | added | Martin Brandenburg | So basically I think it is a bad idea to impose only finitely many multilinear relations even for infinitely many factors, but every attempt to impose some sort of infinitely many multilinear relations makes every tensor product $0$. | |
| Jul 19, 2010 at 8:07 | comment | added | Martin Brandenburg | By the way, there is no possibilty of defining an infinite tensor products, which coindices with the finite one, is associative in the general sense (the tensor product over some family, which is partitioned in some subsets, is the tensor products of the individual tensor products), and which is invariant under permutations. For otherwise take some non-trivial invertible module $L$ and put $P=L \otimes L \otimes ...$. Then P is invertible with $P \otimes L \cong P$, thus $L$ is trivial. | |
| Jul 8, 2010 at 13:40 | comment | added | Martin Brandenburg | of course the direct product, otherwise every multilinear map would be trivial (with the usual definition). mathematics becomes interesting as soon as you think about something which does not follow trivially from category theory ... ;-). of course I already thought of alternative definitions (and the results above are only a selection). | |
| Jul 8, 2010 at 13:34 | comment | added | Jeff Strom | Do your multilinear maps really come out of the product and not the sum? It's categorically bad news to map out of a product. | |
| Jul 5, 2010 at 7:22 | comment | added | Martin Brandenburg | Yes if you fix some elements in the modules, you can make up a direct system of finite tensor products. If you take algebras, and each time the unit element, you get the coproduct of the algebras, which is commonly also denoted as tensor product. Please read my question more carefully (especially the last paragraph). | |
| Jul 5, 2010 at 4:01 | comment | added | ashpool | Atiyah defines a tensor product of a familiy of A-algebras as a direct limit (Introduction to Commutative algebra, p.34, Ex.23). I suspect this is also equivalent to the symmetric algebra. | |
| Jan 31, 2010 at 22:57 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Jan 31, 2010 at 14:01 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Jan 27, 2010 at 21:39 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Jan 19, 2010 at 2:25 | comment | added | Martin Brandenburg | currently I'm working on the ring structure of $K \otimes_K K \otimes_K ...$ for a field $K$. perhaps I will TeX all these results together with the facts above and replace the question by a link to this article ... | |
| Jan 16, 2010 at 20:16 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Jan 15, 2010 at 18:38 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Jan 14, 2010 at 20:57 | comment | added | Yemon Choi | I've addded a "category theory" tag since I think some of the background issues in the question might be related to things categorists have looked at. If people disagree, please feel free to retag | |
| Jan 14, 2010 at 20:56 | history | edited | Yemon Choi |
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| Jan 14, 2010 at 18:35 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Jan 14, 2010 at 18:28 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Jan 14, 2010 at 18:19 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Jan 14, 2010 at 18:12 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Jan 14, 2010 at 17:53 | history | edited | Martin Brandenburg |
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| Jan 14, 2010 at 17:47 | history | asked | Martin Brandenburg | CC BY-SA 2.5 |