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\begin{document}

\title{Conventions used in the Algebraic Stacks Documents}

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\section{Comments}
\label{section-comments}

\noindent
The philosophy behind the conventions used in writing these documents is
to choose those conventions that work. In the end the precise choices we 
make here probably do not make a big difference in the resulting theory 
of algebraic stacks of finite type over the integers (or over a field, 
or over an excellent Noetherian ring). Also, perhaps the higher level
thoery is flexible enough so as to allow for different choices here, but
still we have to choose one.

\section{Set theory} 
\label{section-sets}

\noindent
We use Zermelo-Fraenkel set theory with the axiom of choice.

\subsection{Additional}
\label{subsection-sets-additional}

\noindent
A class is not a set and hence is not described by Zermelo-Fraenkel set 
theory. You can think of a class (such as the class of all sets, or the 
class of rings) as the collection of sets satisfying a rule. Or if you 
prefer you can use von Neumann-Bernays-G\"odel set theory which is a
conservative extension of Zermelo-Fraenkel set theory.

\section{Categories} 
\label{section-categories}

\noindent
A category $C$ consists of a class of objects, and for each pair of objects
a set of morphisms between them. A category is called small if the class of
objects is in fact a set. There are no set theoretic difficulties in defining
functors and natural transformations of functors. We will call a natural
transformation of functors $F \to G$ simply a morphism of functors.

\subsection{Equivalences of categories}
\label{subsection-categories-equivalences}

\noindent
Two categories $A$ and $B$ are said to be equivalent if there exist
functors $F : A \to B$ and $G : B \to A$, and isomorphisms
$\text{id}_B \to F \circ G$ and $\text{id}_A \to G \circ F$.
Recall that a functor $F : A \to B$ is fully faithfull if 
for any objects $X,Y$ of $\text{Ob}(A)$ the map
$F : \text{Mor}_A(X,Y) \to \text{Mor}(F(X), F(Y))$ is bijective.
Recall that $F$ is called essentially surjective if for any 
object $Z \in \text{Ob}(B)$ there exists an object $X \in \text{Ob}(A)$
such that $F(X)$ is isomorphic to $Z$.

\smallskip\noindent
The following lemma will be used repeatedly.

\begin{lemma}
\label{lemma-categories-small}
Suppose that $F_i : A_i \to B$, $i=1,2$ are two functors
from small categories to a category $B$. Suppose that 
$F_1$ and $F_2$ are fully faithfull and essentially surjective.
Then $A_1$ and $A_2$ are equivalent.
\end{lemma}

\begin{proof}
Obvious.
\end{proof}

\section{Algebra}
\label{section-algebra}

\noindent
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. 

\smallskip\noindent
Here are some references for this section: \cite{E}.

\subsection{Properties of modules}
\label{subsection-algebra-modules}

\noindent
This subsection recalls the definitions of some of the more common 
properties of modules over rings. Let $R$ be a ring and let $M$ be
an $R$-module.

\subsection{Properties of ring maps}
\label{subsection-algebra-ringmaps}

\noindent
This section lists various properties of ring maps. Let $R \to S$
be a homomorphism of rings.


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