Fixed the spelling of Cech

coherent.tex

@@ -29,7 +29,7 @@ \section{Introduction}
 
 
 
-\section{Cech cohomology of quasi-coherent sheaves}
+\section{{\v C}ech cohomology of quasi-coherent sheaves}
 \label{section-cech-quasi-coherent}
 
 \noindent
@@ -56,7 +56,7 @@ \section{Cech cohomology of quasi-coherent sheaves}
 In other words, $f_1, \ldots, f_n$ are elements of $A$
 which generate the unit ideal of $A$.
 Write $\mathcal{F}|_U = \widetilde{M}$ for some $A$-module $M$.
-Clearly the Cech complex
+Clearly the {\v C}ech complex
 $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$
 is identified with the complex
 $$
@@ -587,12 +587,12 @@ \section{Quasi-coherence of higher direct images}
 covering. Since $X$ is has affine diagonal the multiple intersections
 $U_{i_0 \ldots i_p}$ are all affine, see
 Lemma \ref{lemma-affine-diagonal}.
-By Lemma \ref{lemma-cech-cohomology-quasi-coherent} the Cech
+By Lemma \ref{lemma-cech-cohomology-quasi-coherent} the {\v C}ech
 cohomology groups $\check{H}^p(\mathcal{U}, \mathcal{F})$
 agree with the cohomology groups. By
 Cohomology, Lemma \ref{cohomology-lemma-alternating-usual}
-the Cech cohomology groups may be computed using the alternating
-Cech complex $\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F})$.
+the {\v C}ech cohomology groups may be computed using the alternating
+{\v C}ech complex $\check{\mathcal{C}}_{alt}^\bullet(\mathcal{U}, \mathcal{F})$.
 As the covering consists of $t$ elements we see immediately
 that $\check{\mathcal{C}}_{alt}^p(\mathcal{U}, \mathcal{F}) = 0$
 for all $p \geq t$. Hence the result follows.
@@ -1349,12 +1349,12 @@ \section{Cohomology of projective space}
 $$
 \mathcal{U} : \mathbf{P}^n_R = \bigcup\nolimits_{i = 0}^n D_{+}(T_i)
 $$
-to compute the cohomology using the Cech complex.
+to compute the cohomology using the {\v C}ech complex.
 This is permissible by Lemma \ref{lemma-cech-cohomology-quasi-coherent}
 since any intersection of finitely many affine $D_{+}(T_i)$ is also a
 standard affine open (see
 Constructions, Section \ref{constructions-section-proj}).
-In fact, we can use the alternating or ordered Cech complex according to
+In fact, we can use the alternating or ordered {\v C}ech complex according to
 Cohomology, Lemmas \ref{cohomology-lemma-ordered-alternating} and
 \ref{cohomology-lemma-alternating-usual}.
 
@@ -1398,7 +1398,7 @@ \section{Cohomology of projective space}
 NEG(\vec{e}) = \{i \in \{0, \ldots, n\} \mid e_i < 0\}.
 $$
 With this notation the weight $\vec{e}$ summand
-$\check{\mathcal{C}}^\bullet(\vec{e})$ of the Cech complex above has
+$\check{\mathcal{C}}^\bullet(\vec{e})$ of the {\v C}ech complex above has
 the following terms
 $$
 \check{\mathcal{C}}^p(\vec{e})
@@ -1446,7 +1446,7 @@ \section{Cohomology of projective space}
 \mathcal{V} : \Spec(R) = \bigcup\nolimits_{i \in \{0, \ldots, n\}} V_i
 $$
 where $V_i = \Spec(R)$ for all $i$. Consider the alternating
-Cech complex
+{\v C}ech complex
 $$
 \check{\mathcal{C}}_{ord}^\bullet(\mathcal{V}, \mathcal{O}_{\Spec(R)})
 $$
@@ -1591,7 +1591,7 @@ \section{Cohomology of projective space}
 $\mathbf{P}^n_{R'}$. Note that $\mathcal{U}'$ is the pullback of the covering
 $\mathcal{U}$ under the canonical morphism
 $\mathbf{P}^n_{R'} \to \mathbf{P}^n_R$. Hence there
-is a map of Cech complexes
+is a map of {\v C}ech complexes
 $$
 \gamma :
 \check{\mathcal{C}}_{ord}^\bullet(\mathcal{U},
@@ -1604,10 +1604,10 @@ \section{Cohomology of projective space}
 Cohomology, Lemma \ref{cohomology-lemma-functoriality-cech}.
 It is clear from the computations in the proof of
 Lemma \ref{lemma-cohomology-projective-space-over-ring}
-that this map of Cech complexes is compatible with the identifications
+that this map of {\v C}ech complexes is compatible with the identifications
 of the cohomology groups in question. (Namely the basis elements for
-the Cech complex over $R$ simply map to the corresponding basis elements
-for the Cech complex over $R'$.) Whence the first statement of the lemma.
+the {\v C}ech complex over $R$ simply map to the corresponding basis elements
+for the {\v C}ech complex over $R'$.) Whence the first statement of the lemma.
 
 \medskip\noindent
 Now fix the ring $R$ and consider two homogeneous polynomials
@@ -1631,7 +1631,7 @@ \section{Cohomology of projective space}
 \mathcal{O}_{\mathbf{P}^n_R}(d + 1)
 $$
 The statement on $H^0$ is clear. For the statement on $H^n$
-consider multiplication by $T_i$ as a map on Cech complexes
+consider multiplication by $T_i$ as a map on {\v C}ech complexes
 $$
 \check{\mathcal{C}}_{ord}^\bullet(\mathcal{U},
 \mathcal{O}_{\mathbf{P}_R}(d))

cohomology.tex

@@ -1571,7 +1571,7 @@ \section{{\v C}ech cohomology and cohomology}
 \begin{lemma}
 \label{lemma-cech-vanish}
 \begin{slogan}
-If higher Cech cohomology of an abelian sheaf vanishes for all open covers,
+If higher {\v C}ech cohomology of an abelian sheaf vanishes for all open covers,
 then higher cohomology vanishes.
 \end{slogan}
 Let $X$ be a ringed space.

descent.tex

@@ -710,7 +710,7 @@ \section{Descent for modules}
 Algebra, Lemmas \ref{algebra-lemma-standard-covering},
 \ref{algebra-lemma-cover-module} and \ref{algebra-lemma-glue-modules}.
 But the results above actually say more because of exactness
-in higher degrees. Namely, it implies that Cech cohomology of quasi-coherent
+in higher degrees. Namely, it implies that {\v C}ech cohomology of quasi-coherent
 sheaves on affines is trivial, see (insert future reference here).
 \end{remark}
 
@@ -2263,7 +2263,7 @@ \section{Quasi-coherent sheaves and topologies}
 \item $\mathcal{V} = \{\coprod_{i = 1, \ldots, n} U_i \to U\}$ is a
 $\tau$-covering of $U$,
 \item $\mathcal{U}$ is a refinement of $\mathcal{V}$, and
-\item the induced map on Cech complexes
+\item the induced map on {\v C}ech complexes
 (Cohomology on Sites,
 Equation (\ref{sites-cohomology-equation-map-cech-complexes}))
 $$

etale-cohomology.tex

@@ -1438,7 +1438,7 @@ \section{Quasi-coherent sheaves}
 
 
 
-\section{Cech cohomology}
+\section{{\v C}ech cohomology}
 \label{section-cech-cohomology}
 
 \noindent
@@ -1719,7 +1719,7 @@ \section{Cech cohomology}
 
 
 
-\section{The Cech-to-cohomology spectral sequence}
+\section{The {\v C}ech-to-cohomology spectral sequence}
 \label{section-cech-ss}
 
 \noindent
@@ -2213,7 +2213,7 @@ \section{Cohomology of quasi-coherent sheaves}
 category of all coverings of $U$. Yet it is still true that
 we can compute {\v C}ech cohomology $\check H^n(U, \mathcal{F})$ (which
 is defined as the colimit over the opposite of the category of
-coverings $\mathcal{U}$ of $U$ of the Cech cohomology groups of
+coverings $\mathcal{U}$ of $U$ of the {\v C}ech cohomology groups of
 $\mathcal{F}$ with respect to $\mathcal{U}$) in terms of the coverings
 $\{V \to U\}$. We will formulate a precise lemma (it only works for sheaves)
 and add it here if we ever need it.
@@ -3111,7 +3111,7 @@ \section{Kummer theory}
 We omit the proof of (1). Part (2) is clear from the second construction,
 since isomorphic torsors give the same cohomology classes.
 Part (3) is clear from the first construction, since the resulting
-Cech classes add up. Part (4) is clear from the second construction
+{\v C}ech classes add up. Part (4) is clear from the second construction
 since a torsor is trivial if and only if it has a global section, see
 Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-trivial-torsor}.
 

hypercovering.tex

@@ -750,7 +750,7 @@ \section{Acyclicity}
 
 
 
-\section{Cech cohomology and hypercoverings}
+\section{{\v C}ech cohomology and hypercoverings}
 \label{section-hyper-cech}
 
 \noindent
@@ -780,7 +780,7 @@ \section{Cech cohomology and hypercoverings}
 and we call it the $i$th {\v C}ech cohomology group
 of $\mathcal{F}$ with respect to $K$.
 In this section we prove analogues of some of the results for
-Cech cohomology of open coverings proved in
+{\v C}ech cohomology of open coverings proved in
 Cohomology, Sections \ref{cohomology-section-cech},
 \ref{cohomology-section-cech-functor} and
 \ref{cohomology-section-cech-cohomology-cohomology}.
@@ -1897,7 +1897,7 @@ \section{Cohomology and hypercoverings}
 
 \medskip\noindent
 This is true if $p = 0$ by Lemma \ref{lemma-h0-cech}.
-If $p = 1$, choose a Cech hypercovering $K$ of $X$ as in
+If $p = 1$, choose a {\v C}ech hypercovering $K$ of $X$ as in
 Example \ref{example-cech} starting with a covering
 $K_0 = \{U_i \to X\}$ in the site $\mathcal{C}$ such that
 $\xi|_{U_i} = 0$, see
@@ -1987,7 +1987,7 @@ \section{Cohomology and hypercoverings}
 Derived Categories, Lemma \ref{derived-lemma-higher-derived-functors}.
 In order to prove the theorem it suffices to show that
 the sequence of functors $\check{H}^i_{HC}(X, -)$ forms a
-$\delta$-functor. Namely we know that Cech cohomology
+$\delta$-functor. Namely we know that {\v C}ech cohomology
 is zero on injective sheaves (Lemma \ref{lemma-injective-trivial-cech})
 and then we can apply
 Homology, Lemma \ref{homology-lemma-efface-implies-universal}.
@@ -2047,7 +2047,7 @@ \section{Cohomology and hypercoverings}
 \medskip\noindent
 Finally, we have to verify that with this definition of $\delta$
 our short exact sequence of abelian sheaves above leads to a
-long exact sequence of Cech cohomology groups.
+long exact sequence of {\v C}ech cohomology groups.
 First we show that if $\delta(\xi) = 0$ (with $\xi$ as above) then
 $\xi$ is the image of some element
 $\xi' \in \check{H}^p_{HC}(X, \mathcal{G})$.

more-algebra.tex

@@ -19195,7 +19195,7 @@ \section{Derived Completion}
 \label{lemma-derived-completion}
 \begin{slogan}
 Derived completions along finitely generated ideals exist, and can
-be computed by a Cech procedure.
+be computed by a {\v C}ech procedure.
 \end{slogan}
 Let $I$ be a finitely generated ideal of a ring $A$.
 The inclusion functor $D_{comp}(A, I) \to D(A)$ has a

sites-cohomology.tex

@@ -43,8 +43,8 @@ \section{Topics}
 \item Cup-product.
 \item Group cohomology.
 \item Comparison of group cohomology and cohomology on $\mathcal{T}_G$.
-\item Cech cohomology on sites.
-\item Cech to cohomology spectral sequence on sites.
+\item {\v C}ech cohomology on sites.
+\item {\v C}ech to cohomology spectral sequence on sites.
 \item Leray Spectral sequence for a morphism between ringed sites.
 \item Etc, etc, etc.
 \end{enumerate}
@@ -809,7 +809,7 @@ \section{Locality of cohomology}
 
 
 
-\section{The Cech complex and Cech cohomology}
+\section{The {\v C}ech complex and {\v C}ech cohomology}
 \label{section-cech}
 
 \noindent
@@ -859,10 +859,10 @@ \section{The Cech complex and Cech cohomology}
 $U_{i_0} \times_U \ldots \times_U U_{i_p}$ exist in $\mathcal{C}$.
 Let $\mathcal{F}$ be an abelian presheaf on $\mathcal{C}$.
 The complex $\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F})$
-is the {\it Cech complex} associated to $\mathcal{F}$ and the
+is the {\it {\v C}ech complex} associated to $\mathcal{F}$ and the
 family $\mathcal{U}$. Its cohomology groups
 $H^i(\check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}))$ are
-called the {\it Cech cohomology groups} of $\mathcal{F}$ with respect
+called the {\it {\v C}ech cohomology groups} of $\mathcal{F}$ with respect
 to $\mathcal{U}$. They are denoted $\check H^i(\mathcal{U}, \mathcal{F})$.
 \end{definition}
 
@@ -900,7 +900,7 @@ \section{The Cech complex and Cech cohomology}
 Let $f : U \to V$, $\alpha : I \to J$ and $f_i : U_i \to V_{\alpha(i)}$
 be a morphism of families of morphisms with fixed target, see
 Sites, Section \ref{sites-section-refinements}.
-In this case we get a map of Cech complexes
+In this case we get a map of {\v C}ech complexes
 \begin{equation}
 \label{equation-map-cech-complexes}
 \varphi : \check{\mathcal{C}}^\bullet(\mathcal{V}, \mathcal{F})
@@ -914,7 +914,7 @@ \section{The Cech complex and Cech cohomology}
 $$
 
 
-\section{Cech cohomology as a functor on presheaves}
+\section{{\v C}ech cohomology as a functor on presheaves}
 \label{section-cech-functor}
 
 \noindent
@@ -1209,7 +1209,7 @@ \section{Cech cohomology as a functor on presheaves}
 $\mathcal{U} = \{f_i : U_i \to U\}_{i \in I}$ be a family of morphisms
 with fixed target such that all fibre products
 $U_{i_0} \times_U \ldots \times_U U_{i_p}$ exist in $\mathcal{C}$.
-The Cech cohomology functors $\check{H}^p(\mathcal{U}, -)$
+The {\v C}ech cohomology functors $\check{H}^p(\mathcal{U}, -)$
 are canonically isomorphic as a $\delta$-functor to
 the right derived functors of the functor
 $$
@@ -1308,10 +1308,10 @@ \section{Cech cohomology as a functor on presheaves}
 Both of these maps are quasi-isomorphisms by an application of
 Homology, Lemma \ref{homology-lemma-double-complex-gives-resolution}.
 Namely, the columns of the double complex are exact in positive degrees
-because the Cech complex as a functor is exact
+because the {\v C}ech complex as a functor is exact
 (Lemma \ref{lemma-cech-exact-presheaves})
 and the rows of the double complex are exact in positive degrees
-since as we just saw the higher Cech cohomology groups of the injective
+since as we just saw the higher {\v C}ech cohomology groups of the injective
 presheaves $\mathcal{I}^q$ are zero.
 Since quasi-isomorphisms become invertible
 in $D^{+}(\mathbf{Z})$ this gives the last displayed morphism
@@ -1323,14 +1323,14 @@ \section{Cech cohomology as a functor on presheaves}
 
 
 
-\section{Cech cohomology and cohomology}
+\section{{\v C}ech cohomology and cohomology}
 \label{section-cech-cohomology-cohomology}
 
 \noindent
-The relationship between cohomology and Cech cohomology comes from the fact
-that the Cech cohomology of an injective abelian sheaf is zero. To see this
+The relationship between cohomology and {\v C}ech cohomology comes from the fact
+that the {\v C}ech cohomology of an injective abelian sheaf is zero. To see this
 we note that an injective abelian sheaf is an injective abelian presheaf and
-then we apply results in Cech cohomology in the preceding section.
+then we apply results in {\v C}ech cohomology in the preceding section.
 
 \begin{lemma}
 \label{lemma-injective-abelian-sheaf-injective-presheaf}
@@ -1370,7 +1370,7 @@ \section{Cech cohomology and cohomology}
 we see that $\mathcal{I}$ is an injective object in
 $\textit{PAb}(\mathcal{C})$.
 Hence we can apply Lemma \ref{lemma-cech-cohomology-derived-presheaves}
-(or its proof) to see the vanishing of higher Cech cohomology group.
+(or its proof) to see the vanishing of higher {\v C}ech cohomology group.
 For the zeroth see Lemma \ref{lemma-cech-h0}.
 \end{proof}
 
@@ -1511,7 +1511,7 @@ \section{Cech cohomology and cohomology}
 $$
 Namely, we have $\check{H}^0(\mathcal{U}, i(\mathcal{F})) = \mathcal{F}(U)$
 by Lemma \ref{lemma-cech-h0}. We have that $i(\mathcal{I})$ is
-Cech acyclic by Lemma \ref{lemma-injective-trivial-cech}. And we
+{\v C}ech acyclic by Lemma \ref{lemma-injective-trivial-cech}. And we
 have that $\check{H}^p(\mathcal{U}, -) = R^p\check{H}^0(\mathcal{U}, -)$
 as functors on $\textit{PAb}(\mathcal{C})$
 by Lemma \ref{lemma-cech-cohomology-derived-presheaves}.
@@ -1603,11 +1603,11 @@ \section{Cech cohomology and cohomology}
 \begin{proof}
 Let $\mathcal{F}$ and $\text{Cov}$ be as in the lemma.
 We will indicate this by saying ``$\mathcal{F}$ has vanishing higher
-Cech cohomology for any $\mathcal{U} \in \text{Cov}$''.
+{\v C}ech cohomology for any $\mathcal{U} \in \text{Cov}$''.
 Choose an embedding $\mathcal{F} \to \mathcal{I}$ into an
 injective abelian sheaf.
 By Lemma \ref{lemma-injective-trivial-cech} $\mathcal{I}$
-has vanishing higher Cech cohomology for any $\mathcal{U} \in \text{Cov}$.
+has vanishing higher {\v C}ech cohomology for any $\mathcal{U} \in \text{Cov}$.
 Let $\mathcal{Q} = \mathcal{I}/\mathcal{F}$
 so that we have a short exact sequence
 $$
@@ -1619,19 +1619,19 @@ \section{Cech cohomology and cohomology}
 0 \to \mathcal{F}(U) \to \mathcal{I}(U) \to \mathcal{Q}(U) \to 0.
 $$
 for every $U \in \mathcal{B}$. Hence for any $\mathcal{U} \in \text{Cov}$
-we get a short exact sequence of Cech complexes
+we get a short exact sequence of {\v C}ech complexes
 $$
 0 \to
 \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{F}) \to
 \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{I}) \to
 \check{\mathcal{C}}^\bullet(\mathcal{U}, \mathcal{Q}) \to 0
 $$
-since each term in the Cech complex is made up out of a product of
+since each term in the {\v C}ech complex is made up out of a product of
 values over elements of $\mathcal{B}$ by assumption (1).
-In particular we have a long exact sequence of Cech cohomology
+In particular we have a long exact sequence of {\v C}ech cohomology
 groups for any covering $\mathcal{U} \in \text{Cov}$.
 This implies that $\mathcal{Q}$ is also an abelian sheaf
-with vanishing higher Cech cohomology for all
+with vanishing higher {\v C}ech cohomology for all
 $\mathcal{U} \in \text{Cov}$.
 
 \medskip\noindent
@@ -1656,7 +1656,7 @@ \section{Cech cohomology and cohomology}
 By the above we see that $H^0(U, \mathcal{I}) \to H^0(U, \mathcal{Q})$
 is surjective and hence $H^1(U, \mathcal{F}) = 0$.
 Since $\mathcal{F}$ was an arbitrary abelian sheaf with
-vanishing higher Cech cohomology for all $\mathcal{U} \in \text{Cov}$
+vanishing higher {\v C}ech cohomology for all $\mathcal{U} \in \text{Cov}$
 we conclude that also $H^1(U, \mathcal{Q}) = 0$ since $\mathcal{Q}$ is
 another of these sheaves (see above). By the long exact sequence this in
 turn implies that $H^2(U, \mathcal{F}) = 0$. And so on and so forth.
@@ -1772,7 +1772,7 @@ \section{Cohomology of modules}
 Hence $\Hom_{\textit{PMod}(\mathcal{O})}(-, \mathcal{I})$
 is an exact functor. By
 Lemma \ref{lemma-complex-tensored-still-exact} we see the vanishing of
-higher Cech cohomology groups.
+higher {\v C}ech cohomology groups.
 For the zeroth see Lemma \ref{lemma-cech-h0}.
 \end{proof}
 
@@ -2510,10 +2510,10 @@ \section{Cohomology and colimits}
 because all the $U_{j_0} \times_U \ldots \times_U U_{j_p}$
 are in $\mathcal{B}$. By
 Lemma \ref{lemma-injective-trivial-cech}
-each of the complexes in the colimit of Cech complexes is
+each of the complexes in the colimit of {\v C}ech complexes is
 acyclic in degree $\geq 1$. Hence by
 Algebra, Lemma \ref{algebra-lemma-directed-colimit-exact}
-we see that also the Cech complex
+we see that also the {\v C}ech complex
 $\check{\mathcal{C}}^\bullet(\mathcal{U}, \colim_i \mathcal{I}_i)$
 is acyclic in degrees $\geq 1$. In other words we see that
 $\check{H}^p(\mathcal{U}, \colim_i \mathcal{I}_i) = 0$