Pierre-Yves Gaillard's Weblog

The theorems of the previous section are not results about what can be proved in particular axiom systems; they are absolute statements about functions. [Section I.9, p. 39.]

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Proof of Theorem 8.7 in Atiyah-MacDonald

July 4, 2011

It seems to me that the second part of the proof of Theorem 8.7 p. 90 in Atiyah-MacDonald can be simplified. We must check the uniqueness of the decomposition of an Artin ring A as a finite product of Artin local rings A_i. To do this it suffices to observe that, for each minimal primary ideal q of A, there is a unique i such that q is the kernel of the canonical projection onto A_i.

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Diary

May 16, 2011

May 29, 2011. I texified the excerpt of the book Set Theory and the Continuum Hypothesis in which Paul Cohen defines the notion of truth. The excerpt is available here. The whole book is available there.

I think any serious discussion about Cohen’s work on the Continuum Hypothesis should focus first on the definition of truth given in this except.

May 17, 2011. Gödel and Cohen’s main results are stated under consistency assumptions, but, in my humble opinion, it is highly unlikely that any reasonable formalization of mathematics be consistent.

May 16, 2011. I think it’s interesting to compare the following two books:

* Set Theory and the Continuum Hypothesis, Paul J. Cohen — available here.

* Théorie des Ensembles, Nicolas Bourbaki — available here.

May 15, 2011. In his Set Theory book Bourbaki defines a scheme (“schéma”) as a rule having two properties. Of course, the question is

“What does one exactly mean by a rule?”

It seems at least clear that such a rule should not involve the notion of an arbitrary integer.

March 19, 2011. A couple of comments about Bourbaki’s set theory book:

I think Bourbaki could have even more strongly emphasized the following points:

Mathematical proof are never explicitly written. What is called “proof” in mathematical writing is a chain of arguments aiming at convincing the reader that a given statement, and a proof of this statement, could in principle be fully formalized.

The beginning of the book is dedicated to “proving” the metamathematical statement that the “assembly” A you get by substituting a term B for a letter X in a theorem T is again a theorem.

To be more precise than Bourbaki himself, the goal is to describe an explicit, simple and mechanical recipe turning a proof of T into a proof of A.

And, indeed, if you read these few pages carefully, you that that’s what Bourbaki does. In other words, Bourbaki “proves” much more than he claims — and what’s needed is what’s “proved”, not what’s claimed!

December 26, 2010. The main problem with Gödel and Cohen is the use of undefined terms — the main example being the term of “integer”.

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A Weird Story

May 14, 2011

In the entry “Grothendieck functor” of the “Encyclopaedia of Mathematics, Edited by Michiel Hazewinkel” [EOM] one reads:

“In the English literature, the Grothendieck functor is commonly called the Yoneda embedding or the Yoneda–Grothendieck embedding.”

On page -14 of Reprints in Theory and Applications of Categories, No. 3, 2003. Abelian Categories, by Peter J. Freyd [html page - pdf file] one reads:

“The Yoneda lemma turns out not to be in Yoneda’s paper. When, some time after both printings of the book appeared, this was brought to my (much chagrined) attention, I brought it the attention of the person who had told me that it was the Yoneda lemma. He consulted his notes and discovered that it appeared in a lecture that MacLane gave on Yoneda’s treatment of the higher Ext functors. The name “Yoneda lemma” was not doomed to be replaced.”

***

The link to the article

Grothendieck, Alexander, Technique de descente et théorèmes d’existence en géométrie algébriques. II. Le théorème d’existence en théorie formelle des modules Exp. No. 195, 22 p [Février 1960]

quoted in the EOM entry mentioned above is here.

***

As a test, here is a link to a Google Doc version of this post.

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Matrix Exponentials

April 2, 2011

A PDF version of this post (along with the LaTeX file) is available HERE.

For any element $a$ of any finite dimensional of any finite dimensional $\mathbb C$ -algebra with 1, let $f_a\in\mathbb C[X]$ be the unique polynomial of degree $<\dim\mathbb C[a]$ satisfying $f_a(a)=e^a$ . (The letter $X$ is an indeterminate.)

Let $m\in\mathbb C[X]$ be the minimal polynomial of $a$ , let $\lambda$ be a multiplicity $\mu(\lambda)$ root of $m$ , and let $x(\lambda)$ be the image of $X$ in $\mathbb C[X]/(X-\lambda)^{\mu(\lambda)}$ .

$f_a\equiv f_{x(\lambda)}\quad\bmod\quad(X-\lambda)^{\mu(\lambda)},$

where $\lambda$ runs over the roots of $m$ .

Here are some more details:

$\displaystyle f_{x(\lambda)}=e^\lambda\ \sum_{n<\mu(\lambda)}\ \frac{(X-\lambda)^n}{n!}\quad,$

$\displaystyle f_a=\sum_\lambda\ T_\lambda\left(f_{x(\lambda)}\ \frac{(X-\lambda)^{\mu(\lambda)}}{m}\right)\frac{m}{(X-\lambda)^{\mu(\lambda)}}\quad,$