Typo hunt 2

algebra.tex

@@ -10559,7 +10559,7 @@ \section{Perfect fields}
 \begin{lemma}
 \label{lemma-perfect-reduced}
 Let $k$ be a perfect field.
-Any reduced $k$ algebra is geometrically reduced over $k$.
+Any reduced $k$-algebra is geometrically reduced over $k$.
 Let $R$, $S$ be $k$-algebras.
 Assume both $R$ and $S$ are reduced.
 Then the $k$-algebra $R \otimes_k S$ is reduced.
@@ -13374,13 +13374,13 @@ \section{Proj of a graded ring}
 $$
 
 \medskip\noindent
-Let $S = \oplus_{d \geq 0} S_d$ be a graded ring.
+Let $S = \bigoplus_{d \geq 0} S_d$ be a graded ring.
 Let $f\in S_d$ and assume that $d \geq 1$.
 We define $S_{(f)}$ to be the subring of $S_f$
 consisting of elements of the form $r/f^n$ with $r$ homogeneous and
 $\deg(r) = nd$. If $M$ is a graded $S$-module,
 then we define the $S_{(f)}$-module $M_{(f)}$ as the
-sub module of $M_f$ consisting of elements of
+submodule of $M_f$ consisting of elements of
 the form $x/f^n$ with $x$ homogeneous of degree $nd$.
 
 \begin{lemma}
@@ -13426,7 +13426,7 @@ \section{Proj of a graded ring}
 
 \begin{lemma}[Topology on Proj]
 \label{lemma-topology-proj}
-Let $S = \oplus_{d \geq 0} S_d$ be a graded ring.
+Let $S = \bigoplus_{d \geq 0} S_d$ be a graded ring.
 \begin{enumerate}
 \item The sets $D_{+}(f)$ are open in $\text{Proj}(S)$.
 \item We have $D_{+}(ff') = D_{+}(f) \cap D_{+}(f')$.
@@ -15271,7 +15271,7 @@ \section{Associated primes}
 
 \begin{lemma}
 \label{lemma-one-equation-module}
-Let $R$ is a Noetherian local ring, $M$ a finite $R$-module, and
+Let $R$ be a Noetherian local ring, $M$ a finite $R$-module, and
 $f \in \mathfrak m$ an element of the maximal ideal of $R$. Then
 $$
 \dim(\text{Supp}(M/fM)) \leq
@@ -15496,7 +15496,7 @@ \section{Associated primes}
 
 \begin{lemma}
 \label{lemma-dim-not-zero-exists-nonzerodivisor-nonunit}
-Let $k$ be a field. Let $S$ be a finite type $k$ algebra.
+Let $k$ be a field. Let $S$ be a finite type $k$-algebra.
 If $\dim(S) > 0$, then there exists an element $f \in S$
 which is a nonzerodivisor and a nonunit.
 \end{lemma}
@@ -27812,7 +27812,7 @@ \section{Dimension of finite type algebras over fields}
 \begin{lemma}
 \label{lemma-disjoint-decomposition-CM-algebra}
 Let $k$ be a field.
-Let $S$ be a finite type $k$ algebra.
+Let $S$ be a finite type $k$-algebra.
 Assume that $S$ is Cohen-Macaulay.
 Then $\Spec(S) = \coprod T_d$ is a finite disjoint union of
 open and closed subsets $T_d$ with $T_d$ equidimensional
@@ -27999,7 +27999,7 @@ \section{Noether normalization}
 \begin{lemma}
 \label{lemma-Noether-normalization-at-point}
 Let $k$ be a field.
-Let $S$ be a finite type $k$ algebra and denote $X = \Spec(S)$.
+Let $S$ be a finite type $k$-algebra and denote $X = \Spec(S)$.
 Let $\mathfrak q$ be a prime of $S$, and let $x \in X$ be the
 corresponding point. There exists a $g \in S$, $g \not \in \mathfrak q$
 such that $\dim(S_g) = \dim_x(X) =: d$ and such that
@@ -28120,7 +28120,7 @@ \section{Dimension of finite type algebras over fields, reprise}
 \begin{lemma}
 \label{lemma-dimension-prime-polynomial-ring}
 Let $k$ be a field.
-Let $S$ be a finite type $k$ algebra which is an integral domain.
+Let $S$ be a finite type $k$-algebra which is an integral domain.
 Let $K$ be the field of fractions of $S$.
 Let $r = \text{trdeg}(K/k)$ be the transcendence degree of $K$ over $k$.
 Then $\dim(S) = r$. Moreover, the local ring of $S$ at every maximal
@@ -28164,7 +28164,7 @@ \section{Dimension of finite type algebras over fields, reprise}
 \begin{lemma}
 \label{lemma-dimension-at-a-point-finite-type-field}
 Let $k$ be a field.
-Let $S$ be a finite type $k$ algebra.
+Let $S$ be a finite type $k$-algebra.
 Let $X = \Spec(S)$.
 Let $\mathfrak p \subset S$ be a prime ideal,
 and let $x \in X$ be the corresponding point.
@@ -28212,7 +28212,7 @@ \section{Dimension of finite type algebras over fields, reprise}
 \begin{lemma}
 \label{lemma-codimension}
 Let $k$ be a field.
-Let $S' \to S$ be a surjection of finite type $k$ algebras.
+Let $S' \to S$ be a surjection of finite type $k$-algebras.
 Let $\mathfrak p \subset S$ be a prime ideal,
 and let $\mathfrak p'$ be the corresponding prime ideal of $S'$.
 Let $X = \Spec(S)$, resp.\ $X' = \Spec(S')$,
@@ -33550,7 +33550,7 @@ \section{Openness of Cohen-Macaulay loci}
 
 \begin{lemma}
 \label{lemma-generic-CM}
-Let $k$ be a field. Let $S$ be a finite type $k$ algebra.
+Let $k$ be a field. Let $S$ be a finite type $k$-algebra.
 The set of Cohen-Macaulay primes forms a dense open
 $U \subset \Spec(S)$.
 \end{lemma}

constructions.tex

@@ -1630,7 +1630,7 @@ \section{Quasi-coherent sheaves on Proj}
 
 \begin{proof}
 To construct a morphism as displayed is the same as constructing
-a $\mathcal{O}_X$-bilinear map
+an $\mathcal{O}_X$-bilinear map
 $$
 \widetilde M \times \widetilde N
 \longrightarrow
@@ -2113,7 +2113,7 @@ \section{Functoriality of Proj}
 $$
 A_f \otimes_{A_{(f)}} B_{(\psi(f))}
 \longrightarrow
-B_{\psi(f)}
+B_{\psi(f)}.
 $$
 Condition (1) determines the images of all elements of $A$.
 Since $f$ is an invertible element which is mapped to $\psi(f)$
@@ -2402,7 +2402,7 @@ \section{Morphisms into Proj}
 (\ref{equation-multiply}) are isomorphisms. In particular we have
 $\mathcal{O}_{U_d}(nd) \cong \mathcal{O}_{U_d}(d)^{\otimes n}$.
 The graded ring map (\ref{equation-global-sections}) on global sections
-combined with restriction to $U_d$ give a homomorphism of graded rings
+combined with restriction to $U_d$ gives a homomorphism of graded rings
 \begin{equation}
 \label{equation-psi-d}
 \psi^d : S^{(d)} \longrightarrow \Gamma_*(U_d, \mathcal{O}_{U_d}(d)).
@@ -2475,7 +2475,7 @@ \section{Morphisms into Proj}
 see Modules, Lemma \ref{modules-lemma-s-open}. We will denote
 the inverse of this map $x \mapsto x/s$, and similarly for
 powers of $\mathcal{L}$. Using this we
-define a ring map $\psi_{(f)} : S_{(f)} \to \Gamma(Y_s, \mathcal{O})$
+define a ring map $\psi_{(f)} : S_{(f)} \to \Gamma(Y_s, \mathcal{O}_Y)$
 by mapping the fraction $a/f^n$ to $\psi(a)/s^n$.
 By Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}
 this corresponds to a morphism
@@ -2511,7 +2511,7 @@ \section{Morphisms into Proj}
 \xymatrix{
 \Gamma(Y_s, \mathcal{O}) \ar[r] &
 \Gamma(Y_{ss'}, \mathcal{O}) &
-\Gamma(Y_{s, '} \mathcal{O}) \ar[l]\\
+\Gamma(Y_{s'}, \mathcal{O}) \ar[l]\\
 S_{(f)} \ar[r] \ar[u]^{\psi_{(f)}} &
 S_{(ff')} \ar[u] &
 S_{(f')} \ar[l] \ar[u]^{\psi_{(f')}}
@@ -2602,7 +2602,7 @@ \section{Morphisms into Proj}
 \end{lemma}
 
 \begin{proof}
-This is a reformulation of Lemma \ref{lemma-converse-construction}
+This is a reformulation of Lemma \ref{lemma-converse-construction}.
 \end{proof}
 
 \begin{lemma}
@@ -3213,7 +3213,7 @@ \section{Invertible sheaves and morphisms into Proj}
 Let $\psi : A \to \Gamma_*(T, \mathcal{L})$ be a homomorphism
 of graded rings. Set
 $$
-U(\psi) = \bigcup\nolimits_{f \in A_{+}\text{ homogeneous}} T_{\psi(f)}
+U(\psi) = \bigcup\nolimits_{f \in A_{+}\text{ homogeneous}} T_{\psi(f)}.
 $$
 The morphism $\psi$ induces a canonical morphism of schemes
 $$
@@ -3443,13 +3443,15 @@ \section{Relative Proj via glueing}
 with $A = \mathcal{A}(U)$, $A' = \mathcal{A}(U')$
 is the open immersion of Lemma \ref{lemma-proj-inclusion} above.
 \end{enumerate}
+Moreover, $\underline{\text{Proj}}_S(\mathcal{A})$ is unique up to unique isomorphism over $S$.
 \end{lemma}
 
 \begin{proof}
 Follows immediately from
 Lemmas \ref{lemma-relative-glueing},
 \ref{lemma-proj-inclusion}, and
 \ref{lemma-transitive-proj}.
+Uniqueness is stated in the last sentence of Lemma \ref{lemma-relative-glueing}.
 \end{proof}
 
 \begin{lemma}
@@ -3656,7 +3658,7 @@ \section{Relative Proj as a functor}
 
 \medskip\noindent
 Let $(d, f : T \to S, \mathcal{L}, \psi)$ be a quadruple.
-We may think of $\psi$ as a $\mathcal{O}_S$-module map
+We may think of $\psi$ as an $\mathcal{O}_S$-module map
 $\mathcal{A}^{(d)} \to \bigoplus_{n \geq 0} f_*\mathcal{L}^{\otimes n}$.
 Since $\mathcal{A}^{(d)}$ is quasi-coherent this is the same
 thing as an $R$-linear homomorphism of graded rings

derived.tex

@@ -7201,7 +7201,7 @@ \section{Cartan-Eilenberg resolutions}
 
 \medskip\noindent
 Computation of the first spectral sequence. We have
-${}'E_1^{p, q} = H^q(L^{p, \bullet})$ in other words
+${}'E_1^{p, q} = H^q(L^{p, \bullet})$, in other words
 $$
 {}'E_1^{p, q} = H^q(F(I^{p, \bullet})) = R^qF(K^p)
 $$
@@ -7753,7 +7753,7 @@ \section{Right derived functors via resolution functors}
 
 \begin{lemma}
 \label{lemma-right-derived-functor}
-Let $\mathcal{A}$ be an abelian category with enough injectives
+Let $\mathcal{A}$ be an abelian category with enough injectives.
 Let $F : \mathcal{A} \to \mathcal{B}$ be an additive functor into
 an abelian category. Let $(i, j)$ be a resolution functor, see
 Definition \ref{definition-localization-functor}.
@@ -9680,7 +9680,7 @@ \section{K-injective complexes}
 \end{lemma}
 
 \begin{proof}
-Let $K^\bullet$ be an complex. Observe that the complex
+Let $K^\bullet$ be a complex. Observe that the complex
 $$
 C :
 \prod\nolimits_b \Hom(K^{-b}, I^{b - 1}) \to
@@ -9964,7 +9964,7 @@ \section{Bounded cohomological dimension}
 consisting of right acyclic objects for $F$,
 \item for $E \in D(\mathcal{A})$
 \begin{enumerate}
-\item $H^i(RF(\tau_{\leq a}E) \to H^i(RF(E))$ is an isomorphism
+\item $H^i(RF(\tau_{\leq a}E)) \to H^i(RF(E))$ is an isomorphism
 for $i \leq a$,
 \item $H^i(RF(E)) \to H^i(RF(\tau_{\geq b - n + 1}E))$ is an isomorphism
 for $i \geq b$,
@@ -11079,7 +11079,7 @@ \section{Generators of triangulated categories}
 \item if $K_i \in \mathcal{D}$, $i = 1, \ldots, r$ with
 $T(K_i)$ for $i = 1, \ldots, r$, then $T(\bigoplus K_i)$,
 \item if $K \to L \to M \to K[1]$ is a distinguished triangle and
-$T$ holds for two, then $T$ holds for the third object,
+$T$ holds for two among $K$, $L$, $M$, then $T$ holds for the third object,
 \item if $T(K \oplus L)$ then $T(K)$ and $T(L)$, and
 \item $T(E[n])$ holds for all $n$.
 \end{enumerate}
@@ -11230,11 +11230,11 @@ \section{Compact objects}
 of triangles $(E', C', C) \to (E', E'', E)$ and
 $(E', E'', E) \to (Y_{n - 1}, X_{n - 1}, X_n)$. The composition
 $C \to E \to X_n$ may not equal the given morphism $C \to X_n$, but
-the compositions into $Y_{n - 1}$ are equal. Let $C \to X_{n - 1}$
+the compositions into $Y_{n - 1}[1]$ are equal. Let $C \to X_{n - 1}$
 be a morphism that lifts the difference. By induction assumption we
 can factor this through a morphism $E''' \to X_{n - 1}$ with
-$E''$ an object of $\langle \bigoplus_{i \in I'''} E_i \rangle$
-for some finite subset $I' \subset I$. Thus we see that we get
+$E'''$ an object of $\langle \bigoplus_{i \in I'''} E_i \rangle$
+for some finite subset $I''' \subset I$. Thus we see that we get
 a solution on considering $E \oplus E''' \to X_n$ because
 $E \oplus E'''$ is an object of
 $\langle \bigoplus_{i \in I' \cup I'' \cup I'''} E_i \rangle$.

homology.tex

@@ -4584,7 +4584,7 @@ \section{Filtrations}
 \end{definition}
 
 \noindent
-This also equivalent to requiring that $f^{-1}(F^iB) = F^iA + \Ker(f)$
+This is also equivalent to requiring that $f^{-1}(F^iB) = F^iA + \Ker(f)$
 for all $i \in \mathbf{Z}$. We characterize strict morphisms
 as follows.
 
@@ -6494,15 +6494,15 @@ \section{Spectral sequences: double complexes}
 weak convergence of the first spectral sequence if for all $n$
 $$
 \text{gr}_{F_I}(H^n(\text{Tot}(K^{\bullet, \bullet}))) =
-\oplus_{p + q = n} {}'E_\infty^{p, q}
+\bigoplus_{p + q = n} {}'E_\infty^{p, q}
 $$
 via the canonical comparison of
 Lemma \ref{lemma-compute-cohomology-filtered-complex}.
 Similarly the second spectral sequence $({}''E_r, {}''d_r)_{r \geq 0}$
 weakly converges if for all $n$
 $$
 \text{gr}_{F_{II}}(H^n(\text{Tot}(K^{\bullet, \bullet}))) =
-\oplus_{p + q = n} {}''E_\infty^{p, q}
+\bigoplus_{p + q = n} {}''E_\infty^{p, q}
 $$
 via the canonical comparison of
 Lemma \ref{lemma-compute-cohomology-filtered-complex}.

injectives.tex

@@ -1755,7 +1755,7 @@ \section{K-injectives in Grothendieck categories}
 \end{lemma}
 
 \begin{proof}
-Choose a functorial injective embeddings $i_M : M \to I(M)$, see
+Choose functorial injective embeddings $i_M : M \to I(M)$, see
 Theorem \ref{theorem-injective-embedding-grothendieck}.
 For every complex $M^\bullet$ denote $J^\bullet(M^\bullet)$ the complex
 with terms $J^n(M^\bullet) = I(M^n) \oplus I(M^{n + 1})$ and differential
@@ -1866,7 +1866,7 @@ \section{K-injectives in Grothendieck categories}
 Suppose that $M \subset U$ and $\varphi : M \to \mathbf{T}^n_\alpha(M^\bullet)$
 is a morphism for some $n \in \mathbf{Z}$. By
 Proposition \ref{proposition-objects-are-small}
-we see that $\varphi$ factor through
+we see that $\varphi$ factors through
 $\mathbf{T}^n_{\alpha'}(M^\bullet)$ for some $\alpha' < \alpha$.
 In particular, by the construction of the functor
 $\mathbf{N}^\bullet(-)$ we see that $\varphi$ factors through
@@ -1879,7 +1879,7 @@ \section{K-injectives in Grothendieck categories}
 complex such that $|K^n| \leq \kappa$. Then $K^\bullet \cong K_i^\bullet$
 for some $i \in I$. Moreover, by
 Proposition \ref{proposition-objects-are-small}
-once again we see that $w$ factor through
+once again we see that $w$ factors through
 $\mathbf{T}^n_{\alpha'}(M^\bullet)$ for some $\alpha' < \alpha$.
 In particular, by the construction of the functor
 $\mathbf{M}^\bullet(-)$ we see that $w$ is homotopic to zero.
@@ -2249,7 +2249,7 @@ \section{Additional remarks on Grothendieck abelian categories}
 $F^pK^\bullet$, $\text{gr}^pK^\bullet$. Let $M \in D(\mathcal{A})$.
 Using Lemma \ref{lemma-K-injective-embedding-filtration}
 we can construct a spectral sequence $(E_r, d_r)_{r \geq 1}$
-of bigraded objects of $\mathcal{A}$ with $d_r$ of bidgree
+of bigraded objects of $\mathcal{A}$ with $d_r$ of bidegree
 $(r, -r + 1)$ and
 with
 $$
@@ -2318,7 +2318,7 @@ \section{Additional remarks on Grothendieck abelian categories}
 $M^\bullet/F^pM^\bullet$, $\text{gr}^pM^\bullet$.
 Dually to Remark \ref{remark-ext-into-filtered-complex}
 we can construct a spectral sequence $(E_r, d_r)_{r \geq 1}$
-of bigraded objects of $\mathcal{A}$ with $d_r$ of bidgree
+of bigraded objects of $\mathcal{A}$ with $d_r$ of bidegree
 $(r, -r + 1)$ and
 with
 $$

modules.tex

@@ -4289,8 +4289,8 @@ \section{Invertible modules}
 (although this is ambiguous if $\mathcal{F}$ is invertible).
 The multiplication of $\Gamma_*(\mathcal{L})$ on
 $\Gamma_*(\mathcal{F})$ is defined using the isomorphisms
-above. If $\gamma : \mathcal{F} \to \mathcal{G}$ is a $\mathcal{O}_X$-module
-map, then we get an $\Gamma_*(\mathcal{L})$-module homomorphism
+above. If $\gamma : \mathcal{F} \to \mathcal{G}$ is an $\mathcal{O}_X$-module
+map, then we get a $\Gamma_*(\mathcal{L})$-module homomorphism
 $\gamma : \Gamma_*(\mathcal{F}) \to \Gamma_*(\mathcal{G})$.
 If $\alpha : \mathcal{L} \to \mathcal{N}$ is an $\mathcal{O}_X$-module
 map between invertible $\mathcal{O}_X$-modules, then we obtain

more-algebra.tex

@@ -23505,7 +23505,7 @@ \section{Rlim of modules}
 \begin{proof}
 The proof is exactly the same as the proof of
 Lemma \ref{lemma-distinguished-triangle-Rlim}
-using Lemma \ref{lemma-compute-Rlim-modules} in stead of
+using Lemma \ref{lemma-compute-Rlim-modules} instead of
 Lemma \ref{lemma-compute-Rlim}.
 \end{proof}
 

more-etale.tex

@@ -145,7 +145,7 @@ \section{Sections with compact support}
 \label{section-compact-support}
 
 \noindent
-A reference for this section is \cite[Exposee XVII, Section 6]{SGA4}.
+A reference for this section is \cite[Exposé XVII, Section 6]{SGA4}.
 Let $f : X \to Y$ be a morphism of schemes which is separated and
 locally of finite type. In this section we define a functor
 $f_! : \textit{Ab}(X_\etale) \to \textit{Ab}(Y_\etale)$