From c1510bacba8251c99767bebe4f3310fa3570d79b Mon Sep 17 00:00:00 2001
From: Aise Johan de Jong <aise.johan.de.jong@gmail.com>
Date: Sat, 14 Nov 2020 21:48:43 -0500
Subject: [PATCH] assignes -> assigns

Thanks to R
https://stacks.math.columbia.edu/tag/0B7H#comment-5453
---
 chow.tex        | 10 +++++-----
 quot.tex        |  2 +-
 spaces-chow.tex |  8 ++++----
 3 files changed, 10 insertions(+), 10 deletions(-)

diff --git a/chow.tex b/chow.tex
index 578c5aab1..d482a84a0 100644
--- a/chow.tex
+++ b/chow.tex
@@ -5647,7 +5647,7 @@ \section{Bivariant intersection theory}
 Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
 Let $f : X \to Y$ be a flat morphism of relative dimension $r$
 between schemes locally of finite type over $S$.
-Then the rule that to $Y' \to Y$ assignes
+Then the rule that to $Y' \to Y$ assigns
 $(f')^* : \CH_k(Y') \to \CH_{k + r}(X')$ where $X' = X \times_Y Y'$
 is a bivariant class of degree $-r$.
 \end{lemma}
@@ -5666,7 +5666,7 @@ \section{Bivariant intersection theory}
 Let $X$ be locally of finite type over $S$.
 Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in
 Definition \ref{definition-gysin-homomorphism}.
-Then the rule that to $f : X' \to X$ assignes
+Then the rule that to $f : X' \to X$ assigns
 $(i')^* : \CH_k(X') \to \CH_{k - 1}(D')$ where $D' = D \times_X X'$
 is a bivariant class of degree $1$.
 \end{lemma}
@@ -5684,7 +5684,7 @@ \section{Bivariant intersection theory}
 Let $f : X \to Y$ and $g : Y \to Z$ be morphisms of
 schemes locally of finite type over $S$.
 Let $c \in A^p(X \to Z)$ and assume $f$ is proper.
-Then the rule that to $Z' \to Z$ assignes
+Then the rule that to $Z' \to Z$ assigns
 $\alpha \longmapsto f'_*(c \cap \alpha)$
 is a bivariant class denoted $f_* \circ c \in A^p(Y \to Z)$.
 \end{lemma}
@@ -5814,7 +5814,7 @@ \section{Chow cohomology and the first Chern class}
 Let $(S, \delta)$ be as in Situation \ref{situation-setup}.
 Let $X$ be locally of finite type over $S$.
 Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
-Then the rule that to $f : X' \to X$ assignes
+Then the rule that to $f : X' \to X$ assigns
 $c_1(f^*\mathcal{L}) \cap - : \CH_k(X') \to \CH_{k - 1}(X')$
 is a bivariant class of degree $1$.
 \end{lemma}
@@ -6929,7 +6929,7 @@ \section{Intersecting with Chern classes}
 Let $X$ be locally of finite type over $S$.
 Let $\mathcal{E}$ be a locally free $\mathcal{O}_X$-module
 of rank $r$. Let $0 \leq p \leq r$.
-Then the rule that to $f : X' \to X$ assignes
+Then the rule that to $f : X' \to X$ assigns
 $c_p(f^*\mathcal{E}) \cap - : \CH_k(X') \to \CH_{k - 1}(X')$
 is a bivariant class of degree $p$.
 \end{lemma}
diff --git a/quot.tex b/quot.tex
index 8bbf8f489..696af768f 100644
--- a/quot.tex
+++ b/quot.tex
@@ -3451,7 +3451,7 @@ \section{The stack of algebraic spaces}
 \medskip\noindent
 The construction from left to right in either arrow is straightforward:
 given $X \to T$ of finite type the functor
-$\mathcal{S}_T \to  \Spacesstack'_{ft}$ assignes to $U/T$ the
+$\mathcal{S}_T \to  \Spacesstack'_{ft}$ assigns to $U/T$ the
 base change $X_U \to U$. We will explain how to construct a quasi-inverse.
 
 \medskip\noindent
diff --git a/spaces-chow.tex b/spaces-chow.tex
index 95944ae4e..9264e6dd6 100644
--- a/spaces-chow.tex
+++ b/spaces-chow.tex
@@ -3927,7 +3927,7 @@ \section{Bivariant intersection theory}
 \label{lemma-cap-c1-bivariant}
 In Situation \ref{situation-setup} let $X/B$ be good.
 Let $\mathcal{L}$ be an invertible $\mathcal{O}_X$-module.
-Then the rule that to $f : X' \to X$ assignes
+Then the rule that to $f : X' \to X$ assigns
 $c_1(f^*\mathcal{L}) \cap - : \CH_k(X') \to \CH_{k - 1}(X')$
 is a bivariant class of degree $1$.
 \end{lemma}
@@ -3943,7 +3943,7 @@ \section{Bivariant intersection theory}
 \label{lemma-flat-pullback-bivariant}
 In Situation \ref{situation-setup} let $f : X \to Y$ be a morphism
 of good algebraic spaces over $B$ which is flat of relative dimension $r$.
-Then the rule that to $Y' \to Y$ assignes
+Then the rule that to $Y' \to Y$ assigns
 $(f')^* : \CH_k(Y') \to \CH_{k + r}(X')$ where $X' = X \times_Y Y'$
 is a bivariant class of degree $-r$.
 \end{lemma}
@@ -3961,7 +3961,7 @@ \section{Bivariant intersection theory}
 In Situation \ref{situation-setup} let $X/B$ be good.
 Let $(\mathcal{L}, s, i : D \to X)$ be a triple as in
 Definition \ref{definition-gysin-homomorphism}.
-Then the rule that to $f : X' \to X$ assignes
+Then the rule that to $f : X' \to X$ assigns
 $(i')^* : \CH_k(X') \to \CH_{k - 1}(D')$ where $D' = D \times_X X'$
 is a bivariant class of degree $1$.
 \end{lemma}
@@ -3978,7 +3978,7 @@ \section{Bivariant intersection theory}
 In Situation \ref{situation-setup} let $f : X \to Y$ and
 $g : Y \to Z$ be morphisms of good algebraic spaces over $B$.
 Let $c \in A^p(X \to Z)$ and assume $f$ is proper.
-Then the rule that to $X' \to X$ assignes
+Then the rule that to $X' \to X$ assigns
 $\alpha \longmapsto f_*(c \cap \alpha)$
 is a bivariant class of degree $p$.
 \end{lemma}