Comments
The philosophy behind the conventions used in writing these documents is to choose those conventions that work.
Set theory
We use Zermelo-Fraenkel set theory with the axiom of choice. See [Kunen]. We do not use universes (different from SGA4). We do not stress set-theoretic issues, but we make sure everything is correct (of course) and so we do not ignore them either.
Categories
A category \(\mathcal{C}\) consists of a set of objects and, for each pair of objects, a set of morphisms between them. In other words, it is what is called a “small” category in other texts. We will use “big” categories (categories whose objects form a proper class) as well, but only those that are listed in Categories, Remark 0015.
Algebra
In these notes a ring is a commutative ring with a \(1\). Hence the category of rings has an initial object \(\mathbf{Z}\) and a final object \(\{0\}\) (this is the unique ring where \(1 = 0\)). Modules are assumed unitary. See [Eisenbud].
Notation
The natural integers are elements of \(\mathbf{N} = \{1, 2, 3, \ldots\}\). The integers are elements of \(\mathbf{Z} = \{\ldots, -2, -1, 0, 1, 2, \ldots\}\). The field of rational numbers is denoted \(\mathbf{Q}\). The field of real numbers is denoted \(\mathbf{R}\). The field of complex numbers is denoted \(\mathbf{C}\).