# Good user manuals for technical topics?

This question is motivated by this (highly recommended) comment by Emerton on Terry Tao's post "Learn and relearn your field". In particular, the following paragraphs:

In particular, the first couple of sections of Ribet’s famous Inventiones 100 article give a great example of how hundreds of pages of theory (including a lot of the theory of Néron models, and a quite a bit of SGA 7) can be summarized in ten or so pages of “working knowledge”.

and

For example, $$p$$-adic Hodge theory is another tool which plays a big role in a lot of arithmetic algebraic geometry, and which has a technically formidable underpinning. But, just like étale cohomology, it has a very nice formalism which one can learn to use comfortably without having to know all the foundations and proofs.

My question is: Do you know of other sources that provide useful manual for your favorite technical topic? (The kind of paper or section of a book which you use time and time again and would recommend for new researchers.)

As an example, where can I find the "nice formalism" of $$p$$-adic Hodge theory Emerton was referring to? (I was trying to learn more about almost ring theory from Gabber and Ramero - Almost ring theory and such a reference might be useful for bigger picture.)

Here is another example, just to fix ideas, since the question is a bit vague. At some point in graduate school I was trying to learn about "Riemann–Roch theorem for local rings". I found Fulton's intersection theory book, Chapter 18 and this paper by Kurano to be quite helpful.

This is obviously community wiki, so one topic per answer please. Answers with more description of the sources and how to use them would be very appreciated. Thanks a lot.

• Since, the question has been bumped, I will mentioned that the link to "this paper by Kurano is no longer working - perhaps somebody can find out which paper was intended. – Martin Sleziak Apr 18 at 8:58
• @MartinSleziak, I searched on MathSciNet for "Author=(Kurano) AND Anywhere=(Riemann Roch)". The only one of the results on a Springer site was Kurano and Ohta - On the limit of Frobenius in the Grothendieck group (MSN). Perhaps @‍HailongDao could confirm? – LSpice Apr 26 at 22:09
• I also changed the "almost ring theory" link to its (heh) almost certain target; it looks like maybe there was a spurious '5' in the link that prevented it from being an arXiv identifier, so I thought it was worth editing in light of @MartinSleziak's recent meta post on the arXiv front-end. – LSpice Apr 26 at 23:35

For symmetries of PDEs the book is Applications of Lie groups to differential equations by Peter Olver. In particular, in Chapters 4 and 5 one can find an excellent treatment of the celebrated theorem of Emmy Noether on the relationship of symmetries and conservation laws for Lagrangian systems. In contrast with some other books on the subject, this seminal theorem is presented there in full generality.

• Peter Olver's book is great, but I wouldn't call it a "manual" in the sense that Hailong is talking about. – Willie Wong Apr 25 '10 at 21:25
• Willie, IMHO, e.g. Chapter 5 of his book does match the OP's description ("the kind of paper or section of a book which you use time and time again and would recommend it for new researchers"). What makes you think otherwise? – mathphysicist Apr 25 '10 at 21:51

John McCleary, A User’s Guide to Spectral Sequences, Cambridge University Press