Fix typos

categories.tex

@@ -9359,7 +9359,7 @@ \section{Categories of dotted arrows}
 witnessing the $2$-commutativity of the diagram.
 Given (\ref{equation-dotted-arrows}) and $\gamma$, a \emph{dotted arrow} 
 is a triple $(a, \alpha, \beta)$ consisting of a morphism 
-$a \colon T \to X$ and and $2$-isomorphisms
+$a \colon T \to X$ and $2$-isomorphisms
 $\alpha : a \circ j \to x$, $\beta : y \to f \circ a$
 such that
 $\gamma = (\text{id}_f \star \alpha) \circ (\beta \star \text{id}_j)$,

derham.tex

@@ -1493,7 +1493,7 @@ \section{de Rham cohomology of projective space}
 \end{lemma}
 
 \begin{proof}
-We have the vanishing and and freeness by
+We have the vanishing and freeness by
 Lemma \ref{lemma-twisted-hodge-cohomology-projective-space}.
 For $p = 0$ it is certainly true that
 $1 \in H^0(\mathbf{P}^n_A, \mathcal{O})$ is a generator.

injectives.tex

@@ -2555,7 +2555,7 @@ \section{The Gabriel-Popescu theorem}
 \medskip\noindent
 The first proof will use the adjoint functor theorem, see
 Categories, Theorem \ref{categories-theorem-adjoint-functor}.
-Observe that that $G : \mathcal{A} \to \text{Mod}_R$ is left exact and sends
+Observe that $G : \mathcal{A} \to \text{Mod}_R$ is left exact and sends
 products to products. Hence $G$ commutes with limits. To check the set
 theoretical condition in the theorem, suppose that $M$ is an object of
 $\text{Mod}_R$. Choose a suitably large cardinal $\kappa$ and denote $E$

modules.tex

@@ -3594,7 +3594,7 @@ \section{Internal Hom}
 $(U, \varphi)$ in $\SheafHom_{\mathcal{O}_X}(\mathcal{F}, \mathcal{G})_x$,
 where $x \in U \subset X$ is open and
 $\varphi \in \Hom_{\mathcal{O}_U}(\mathcal{F}|_U, \mathcal{G}|_U)$,
-to the the induced map on stalks at $x$, namely
+to the induced map on stalks at $x$, namely
 $\varphi_x : \mathcal{F}_x \to \mathcal{G}_x$.
 
 \medskip\noindent

more-etale.tex

@@ -1547,7 +1547,7 @@ \section{Sections with finite support}
 $b^{-1}f_!\mathcal{F}$ and show we end up with the same local
 section of $g_{1, *}f'_{1, !}a^{-1}\mathcal{F}$
 going around either way. However, in fact it suffices to check
-this for local sections which are of the the pullback by $b$ of
+this for local sections which are of the pullback by $b$ of
 a section $s = \sum (Z_i, s_i)$ of $f_{p!}\mathcal{F}(V)$
 as above (since such pullbacks generate the abelian sheaf
 $b^{-1}f_!\mathcal{F}$). Denote $V_1$, $V'_1$, and $Z_{1, i}$

pione.tex

@@ -4081,7 +4081,7 @@ \section{Restriction to a closed subscheme}
 U = \underline{\Spec}_X((\mathcal{F}, \mu))
 $$
 we obtain a finite locally free scheme $\pi : U \to X$ whose restriction
-to $Y$ is isomorphic to $U_1$. The the discriminant of $\pi$ is the zero
+to $Y$ is isomorphic to $U_1$. The discriminant of $\pi$ is the zero
 set of the section
 $$
 \det(Q_\pi) :

restricted.tex

@@ -7274,9 +7274,9 @@ \section{Application to modifications}
 By Theorem \ref{theorem-dilatations-general} (in fact we only need
 the affine case treated in Lemma \ref{lemma-dilatations-affine})
 the category (\ref{equation-modification}) for $X$ and $T$
-is equivalent to the the category of rig-\'etale morphisms
+is equivalent to the category of rig-\'etale morphisms
 $W \to X_{/T}$ of locally Noetherian formal algebraic spaces.
-Similarly, the the category (\ref{equation-modification})
+Similarly, the category (\ref{equation-modification})
 for $X_1$ and $T_1$ is equivalent to the category of rig-\'etale
 morphisms $W_1 \to X_{1, /T_1}$ of locally Noetherian formal
 algebraic spaces. Since $X_{/T} = \text{Spf}(A^\wedge)$

simplicial.tex

@@ -4557,7 +4557,7 @@ \section{Homotopies}
 from $b \circ a \circ c$ to $b \circ a' \circ c$. In this way we see that
 we obtain a new category $\text{hSimp}(\mathcal{C})$ with the same
 objects as $\text{Simp}(\mathcal{C})$ but whose morphisms are
-homotopy classes of of morphisms of $\text{Simp}(\mathcal{C})$.
+homotopy classes of morphisms of $\text{Simp}(\mathcal{C})$.
 Thus there is a canonical functor
 $$
 \text{Simp}(\mathcal{C})

stacks-properties.tex

@@ -1337,7 +1337,7 @@ \section{Properties of algebraic stacks defined by properties of schemes}
 \item for any smooth morphism $U \to \mathcal{X}$ with $U$ an algebraic space
 and $u \in |U|$ with $a(u) = x$ the algebraic space $U$ has property
 $\mathcal{P}$ at $u$, and
-\item for some smooth morphism $U \to \mathcal{X}$ with $U$ a an
+\item for some smooth morphism $U \to \mathcal{X}$ with $U$ an
 algebraic space and some $u \in |U|$ with $a(u) = x$ the algebraic space
 $U$ has property $\mathcal{P}$ at $u$.
 \end{enumerate}

weil.tex

@@ -1117,7 +1117,7 @@ \section{Chow groups of motives}
 \begin{proof}
 Any object $L$ of $M_k$ is a summand of $h(X)(m)$ for some smooth projective
 scheme $X$ over $k$ and some $m \in \mathbf{Z}$. Observe that the Chow groups
-of $M \otimes h(X)(m)$ are the same as the Chow groups of of $M \otimes h(X)$
+of $M \otimes h(X)(m)$ are the same as the Chow groups of $M \otimes h(X)$
 up to a shift in degrees. Hence our assumption implies
 that $c \otimes 1 : M \otimes L \to N \otimes L$ induces an isomorphism on
 Chow groups for every object $L$ of $M_k$. By
@@ -2116,7 +2116,7 @@ \section{Cycles over non-closed fields}
 
 \begin{proof}
 Let $k'$ be a separable algebraic closure of $k$. Suppose that we can show
-the the pullback of $[x] - [x']$ to $X_{k'}$ is divisible by $n$ in
+the pullback of $[x] - [x']$ to $X_{k'}$ is divisible by $n$ in
 $\CH_0(X_{k'})$. Then we conclude by Lemma \ref{lemma-chow-limit}.
 Thus we may and do assume $k$ is separably algebraically closed.