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.