Fix typo ('the the' -> 'the')

curves.tex

@@ -2467,7 +2467,7 @@ \section{Riemann-Hurwitz}
 of $C$ over $\kappa(y)$ in the sense of
 Discriminants, Definition \ref{discriminant-definition-different}.
 It suffices to show: $\mathfrak{D}_{C/\kappa(y)}$
-is nonzero if and only if the the extension
+is nonzero if and only if the extension
 $\mathcal{O}_{Y, y} \subset \mathcal{O}_{X, x}$ is tamely ramified
 and in the tamely ramified case $\mathfrak{D}_{C/\kappa(y)}$
 is equal to the ideal generated by $s^{e_x - 1}$ in $C$.

dga.tex

@@ -2177,7 +2177,7 @@ \section{The canonical delta-functor}
 0 \to \bigoplus M_n \to \bigoplus M_n \to M \to 0
 $$
 by Derived Categories, Lemma \ref{derived-lemma-compute-colimit}
-(applied the the underlying complexes of abelian groups).
+(applied the underlying complexes of abelian groups).
 The direct sums are direct sums in $D(\mathcal{A})$ by
 Lemma \ref{lemma-derived-products}.
 Thus the result follows from the definition
@@ -4598,7 +4598,7 @@ \section{Derived tensor product}
 then
 \begin{enumerate}
 \item $- \otimes_A^\mathbf{L} A' : D(A, \text{d}) \to D(A', \text{d})$
-is equal the the right derived functor of
+is equal the right derived functor of
 $$
 K(A, \text{d}) \longrightarrow K(A', \text{d}),\quad
 M \longmapsto M \otimes_R R'

examples-defos.tex

@@ -2414,7 +2414,7 @@ \section{Application to isolated singularities}
 to conclude as in the case of completion.
 
 \medskip\noindent
-To get the the final case it suffices to show that
+To get the final case it suffices to show that
 $\Deformationcategory_{P_{\mathfrak m_i}} \to
 \Deformationcategory_{P_{\mathfrak m_i}^\wedge}$
 is smooth and induce isomorphisms on tangent spaces for each $i$ separately.

perfect.tex

@@ -3389,7 +3389,7 @@ \section{Compact and perfect objects}
 The following result is a strengthening of
 Proposition \ref{proposition-compact-is-perfect}.
 Let $T \subset X$ be a closed subset of a scheme $X$. As before
-$D_T(\mathcal{O}_X)$ denotes the the strictly full, saturated,
+$D_T(\mathcal{O}_X)$ denotes the strictly full, saturated,
 triangulated subcategory consisting of complexes whose
 cohomology sheaves are supported on $T$. Since taking direct
 sums commutes with taking cohomology sheaves, it follows

pic.tex

@@ -461,7 +461,7 @@ \section{Moduli of divisors on smooth curves}
 Lemma \ref{lemma-divisors-on-curves}.
 Applying
 Morphisms, Lemma \ref{morphisms-lemma-image-universally-closed-separated}
-the the surjective integral morphism $D_1 \amalg D_2 \to D$
+the surjective integral morphism $D_1 \amalg D_2 \to D$
 we find that $D \to S$ is separated. Then
 Morphisms, Lemma \ref{morphisms-lemma-image-proper-is-proper}
 implies that $D \to S$ is proper.

resolve.tex

@@ -4385,7 +4385,7 @@ \section{Contracting exceptional curves}
 elements of $\lim H^0(E_n, \mathcal{O}_n(-dE))$ which map to a basis
 of $H^0(E, \mathcal{O}_E(-dE)) = H^0(\mathbf{P}^1_k, \mathcal{O}(d))$.
 In this way we see that $A$ is separated and complete with respect
-to the linear topology defined by the the kernels
+to the linear topology defined by the kernels
 $$
 I_n = \Ker(A \longrightarrow H^0(E_n, \mathcal{O}_n))
 $$

restricted.tex

@@ -1321,7 +1321,7 @@ \section{Glueing rings along an ideal}
 \bigoplus C_0 \text{d}x_i \oplus \bigoplus C_0 \text{d}y_j
 $$
 By our direct sum decomposition of $L/L^2$ above and the fact
-that the the determinant of the partial derivatives of the $g_{0, j}$
+that the determinant of the partial derivatives of the $g_{0, j}$
 is invertible in $C_0$ we see that the natural map
 $K^\bullet \to L^\bullet$ induces a quasi-isomorphism
 $$
@@ -2306,7 +2306,7 @@ \section{Algebraization}
 
 \begin{proof}
 Consider the fibre product $E = X \times_Y Z \to X$.
-By assumption the the open immersion $X \setminus T \to X$
+By assumption the open immersion $X \setminus T \to X$
 factors through $E$ and any morphism $\varphi : X' \to X$ with
 $|\varphi|(|X'|) \subset T$ factors through $E$ as well, see
 Formal Spaces, Section \ref{formal-spaces-section-completion}.

spaces-perfect.tex

@@ -3867,7 +3867,7 @@ \section{Compact and perfect objects}
 The following result is a strengthening of
 Proposition \ref{proposition-compact-is-perfect}.
 Let $T \subset |X|$ be a closed subset where $X$ is an algebraic space.
-As before $D_T(\mathcal{O}_X)$ denotes the the strictly full, saturated,
+As before $D_T(\mathcal{O}_X)$ denotes the strictly full, saturated,
 triangulated subcategory consisting of complexes whose
 cohomology sheaves are supported on $T$. Since taking direct
 sums commutes with taking cohomology sheaves, it follows

spaces-pushouts.tex

@@ -1825,7 +1825,7 @@ \section{Coequalizers and glueing}
 \begin{remark}
 \label{remark-essentially-constant}
 The meaning of Lemma \ref{lemma-essentially-constant}
-is the the system $X_1 \to X_2 \to X_3 \to \ldots$ is essentially
+is the system $X_1 \to X_2 \to X_3 \to \ldots$ is essentially
 constant with value $X$. See Categories, Definition
 \ref{categories-definition-essentially-constant-diagram}.
 \end{remark}

stacks-limits.tex

@@ -375,7 +375,7 @@ \section{Morphisms of finite presentation}
 algebraic spaces $(U, R, s, t, c)$, see
 Algebraic Stacks, Lemma \ref{algebraic-lemma-stack-presentation}.
 Since $U$ is locally of finite presentation over $S$
-it follows that the the algebraic space $R$ is
+it follows that the algebraic space $R$ is
 locally of finite presentation over $S$.
 Recall that $[U/R]$ is the stack in groupoids over $(\Sch/S)_{fppf}$
 obtained by stackyfying the category fibred in groupoids

stacks-more-morphisms.tex

@@ -3068,7 +3068,7 @@ \section{The Keel-Mori theorem}
 $M' \times_M \mathcal{X} = \mathcal{X}'$ by
 Quotients of Groupoids, Lemma
 \ref{groupoids-quotients-lemma-base-change-quotient-stack}.
-Let $C^1$ be the the ring of $R'$-invariant functions on $U'$.
+Let $C^1$ be the ring of $R'$-invariant functions on $U'$.
 Set $M^1 = \Spec(C^1)$ and consider the diagram
 $$
 \xymatrix{

varieties.tex

@@ -7528,7 +7528,7 @@ \section{One dimensional Noetherian schemes}
 we first conclude that $H^1(Y, \mathcal{O}_Y^*) \to H^1(Y, \Im(f^\sharp))$
 is surjective and then that
 $H^1(Y, \Im(f^\sharp)) \to H^1(Y, f_*\mathcal{O}_X^*)$ is surjective.
-Combining all the the above we find that $H^1(Y, \mathcal{O}_Y^*) \to
+Combining all the above we find that $H^1(Y, \mathcal{O}_Y^*) \to
 H^1(X, \mathcal{O}_X^*)$ is surjective as desired.
 \end{proof}
 
@@ -9370,7 +9370,7 @@ \section{Degrees on curves}
 Let $s$ be a nonzero section of $\mathcal{L}$. Since $X$ is a curve, we
 see that $s$ is a regular section. Hence there is an effective
 Cartier divisor $D \subset X$ and an isomorphism
-$\mathcal{L} \to \mathcal{O}_X(D)$ mapping $s$ the the canonical
+$\mathcal{L} \to \mathcal{O}_X(D)$ mapping $s$ the canonical
 section $1$ of $\mathcal{O}_X(D)$, see
 Divisors, Lemma \ref{divisors-lemma-characterize-OD}.
 Then $\deg(\mathcal{L}) = \deg(D)$ by