Timeline for Classification of rings satisfying $a^4=a$
Current License: CC BY-SA 3.0
17 events
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| Oct 14, 2014 at 14:23 | comment | added | Martin Brandenburg | A counterexample is given in Arens-Kaplansky, "Topological representations of algebras", Section 8. | |
| Oct 13, 2014 at 19:32 | comment | added | Martin Brandenburg | I am still not convinced. I don't see any embedding without using some actions. If Neil's answer is correct (which I believe more and more), then the classification in terms of closed subsets would mean, roughly, that any $C_2$-action on a Stone spaces splits outside the closed subset of fixed points, where I call a $C_2$-space split iff it is isomorphic to $Y \coprod \sigma Y$ for some space $Y$. I don't see any reason why there should be such a splitting. And this is also where cohomology might show obstructions. | |
| Oct 13, 2014 at 18:49 | comment | added | David Handelman | It's easier to use direct limits. Every fg unital subring, $A$ of $T$, is finite and semisimple, so is a product of finitely many copies of $F_2$ and of $F_4$. Let $X_A$ be its maximal ideal space. If $A \subset A'$, there is the natural onto map $X_{A'} \to X_A$. For $E_A$, the subset of $X_S$ consisting of ideals of index two, the restriction from $X_{A'}$ to $E_{A'}$ has image in $E_A$ (not onto). Then $X = \lim_{\leftarrow} X_A$ (over finite subrings) is the maximal ideal space of $T$; set $Y = \lim_{\leftarrow} E_A$ (closed subset of $X$). Thus the embedding $T \subset S(X,Y)$, etc. | |
| Oct 13, 2014 at 18:43 | comment | added | David Handelman | To answer the last question first. For $t \in T$, $V:= t^{-1}(\{u\})$ is clopen in $X$, hence its indicator function $e = \chi_V$ belong to $R$. Then $et = u\chi_V$ is a product of elements of $T$, hence also belongs. | |
| Oct 11, 2014 at 18:02 | comment | added | Martin Brandenburg | Another question: ".. and corresponding $t_i = \chi_{V_i} t_i$ in $T$ ..." - Why can we assume that $t_i$ supported on $V_i$? | |
| Oct 9, 2014 at 9:50 | comment | added | Martin Brandenburg | And in the edit, one has to be more careful as for the transition functions of the colimit ... | |
| Oct 9, 2014 at 9:29 | comment | added | Martin Brandenburg | How do you construct the homomorphism $T \to S(X,Y)$? The residue fields embed into $\mathbb{F}_4$, but not in a canonical way. I don't see why we can choose compatible embeddings. This is the reason why I think that cohomology might enter here. | |
| Sep 29, 2014 at 23:44 | comment | added | David Handelman | Yes, I should have said fixed ring, not the image; the important thing is that all the idempotents of $R$ belong to $T$, which is obvious anyway. | |
| Sep 29, 2014 at 18:34 | comment | added | Laurent Moret-Bailly | I think you mean that $T_0$ is the fixed ring of $\alpha$, not $\alpha(R)$ (the latter is $R$ since $\alpha$ is an involution). | |
| Sep 28, 2014 at 22:54 | history | edited | David Handelman | CC BY-SA 3.0 |
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| Sep 28, 2014 at 20:34 | comment | added | Martin Brandenburg | I've just seen your edit. Thank you, I will look at it! | |
| Sep 27, 2014 at 22:26 | history | edited | David Handelman | CC BY-SA 3.0 |
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| Sep 27, 2014 at 22:07 | history | edited | David Handelman | CC BY-SA 3.0 |
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| Sep 27, 2014 at 21:57 | history | edited | David Handelman | CC BY-SA 3.0 |
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| Sep 27, 2014 at 19:13 | comment | added | Martin Brandenburg | In his book "Modules Over Commutative Regular Rings", Pierce studies as an example rings satisfying $a^n=a$ and their corresponding structure sheaves. But does the book also offer a classification? | |
| Sep 27, 2014 at 1:10 | comment | added | Martin Brandenburg | Thanks, but that seems to me what I already know. I am interested in a more explicit classification (or even better, an equivalence of categories). | |
| Sep 27, 2014 at 1:02 | history | answered | David Handelman | CC BY-SA 3.0 |