Who this is for
This is meant as a collection of suggestions for anyone wanting to write a book on advanced mathematics, and wants to minimize the confusion of their readers and make their learning experience less stressful, less confusing and hence more effective from the perspective of the student.
Make new terms easy to find
A student needs to learn a lot of definitions and their terms. And a new term will not make any sense it is highly confusing to encounter a new term unless it is precisely defined, so please define a term before it is used the first time. Most will not remember each definition right away and will need to reread it, possibly several times, before it is internalized. So, make definitions easy to find on the page (i.e. use a definition box) and every time a term is defined, add it to the index. And there absolutely should be an index.
An example of a book that uses a new term (e.g. additive functor) before it is defined is
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Rotman, J. J. (2009). An Introduction to Homological Algebra. Springer New York. https://doi.org/10.1007/b98977
Enable navigation by numbering
Speaking of definitions, when numbering definitions, theorems, lemmas, examples etc, they should have the same counter. If one is looking for a specific box, it it much easier to orient oneself with respect to where what one is looking for is, if you can turn to the approximate location and if theorem/lemma/example 1.8 is on that page, you know that definition 1.3 is on an earlier page.
An example of a book that has separate counters is
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Fulton, W. (1998). Intersection Theory. Springer New York. https://doi.org/10.1007/978-1-4612-1700-8
Do not hide important stuff
Definitions and and even simple results of a certain generality and importance should not be be marked as an example or, for that matter, a remark. A student has a to prioritize their time, and the author should make this easier rather than penalizing a reader for skipping material that (appears as if it) is not needed for a full understanding.
An example is Gathmann’s otherwise very good lecture notes on algebraic geometry https://agag-gathmann.math.rptu.de/class/alggeom-2021/alggeom-2021.pdf. His (also, otherwise very good) notes on Commutative Algebra is also an example of this phenomena.
Don’t give unhelpful hints
It has happened more than once that the hint(s) given for an exercise has made the problem harder to solve, which (I assume) is not the intention, and is supremely frustrating.
An example of a book where this sometimes occurs is
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Bosch, S. (2022). Algebraic Geometry and Commutative Algebra. Springer London. https://doi.org/10.1007/978-1-4471-7523-0
