So far the material is largely taken from these notes on algebraic topology.

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I added this to the chapter on completions. Also, as promised in the comments to this post, there is now some material on finite presentation in a chapter currently loosely marked various. It is far from complete, though.

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The Amitsur complex is what Tamme calls it; others just write it out. This is the sequence that you get for a faithfully flat extension of rings, whose exactness is what leads to all sorts of descent theorems, which will someday appear here. (There is a MathOverflow question on the history of it.)

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On the other hand, I’m not sure whether it would be pedantic to devote an entire chapter to them. There are probably more important things in commutative algebra proper, and the above results are really cleaner if we can use the language of schemes a bit (then we can talk about quasi-coherent sheaves on projective limits, and even derive ZMT!), though it is an open question exactly how much we should delve into algebraic geometry.

Thoughts?

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I’ve been working on the chapter on flatness in an attempt to make it look something like a portion of a book. This means, for instance, a reorganization of material like and greater material on faithful flatness. I added the canonical example of a faithfully flat algebra over a ring (the product where the generate the unit ideal); one of the nice things is that the Amitsur complex of this faithfully flat algebra (with respect to some -module ) is the Cech complex of the associated quasi-coherent sheaf with respect to the covering . In particular, the acyclicity of the Amitsur complex (which should be added soon!) lets you get Serre’s cohomological vanishing on an affine in a somewhat quicker way than Koszul complexes.

I think it would be great to include material on faithfully flat descent, though it will perhaps be hard to motivate without the language of schemes (though right now we are getting fairly close to introducing the language of schemes already!). Perhaps things like Hilbert’s Theorem 90 and Galois descent would be good items to include.

I’ve also been fleshing out the material on henselianization, following Raynaud’s *Anneaux locaux henseliens*; this is pretty fun stuff too.

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Since we have a lot of submitted material now that has yet to be edited into the project, I plan to work on that a bit during spring break. Other people should feel free to be involved in this!

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I don’t think we’ll ever use it in the project, but it is neat. Covering a bit of basic category theory for its own sake seems like a reasonable thing to do, moreover.

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In addition, various edits keep being made, slowly. There have been several articles in the Monthly that give short semi-expository pieces of mathematics; a brief account of some of them have appeared in the book. Here is one, giving a short proof of the following lemma: a field extension which is finitely generated as an algebra is algebraic.

To see this, one can use the following argument of McCabe. Let be an extension finitely generated as an algebra. There is a subring over which $k’$ is algebraic. The finite generation hypothesis implies that we can find a localization over which $k’$ is even integral. But if one has an integral extension and the top ring is a field, then so is the bottom one; this implies is a field. So must lie in every maximal ideal, meaning it is zero, contradiction.

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Lang’s *Algebra *(as well as some of his other books, too) has a lot of these kinds of isolated references to scattered results in the literature. Some of these are quite interesting; it is probably worth adding more of these. Doing so will also make the book less “canonical”!

It happens, coincidentally, that we also got a donation on euclidean domains, which has been partially merged in.

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The makefile is also fixed so that running “make” actually resolves cross-references. Apparently, you run pdflatex twice *after *invoking bibtex, and not the other way around. That makes sense.

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