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Try to fix parse error on website
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aisejohan committed May 4, 2022
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Expand Up @@ -1563,7 +1563,7 @@ \section{Cycles}
Let $X$ be locally of finite type over $S$. Let $k \in \mathbf{Z}$.
Then we can write
$$
Z_k(X) = \bigoplus'\nolimits_{\delta(x) = k} K_0^M(\kappa(x))
Z_k(X) = \bigoplus\nolimits_{\delta(x) = k}' K_0^M(\kappa(x))
\quad\subset\quad
\bigoplus\nolimits_{\delta(x) = k} K_0^M(\kappa(x))
$$
Expand Down Expand Up @@ -2879,9 +2879,9 @@ \section{Rational equivalence}
Let $X$ be locally of finite type over $S$. Let $k \in \mathbf{Z}$.
Let us show that we have a presentation
$$
\bigoplus'\nolimits_{\delta(x) = k + 1} K_1^M(\kappa(x))
\bigoplus\nolimits_{\delta(x) = k + 1}' K_1^M(\kappa(x))
\xrightarrow{\partial}
\bigoplus'\nolimits_{\delta(x) = k} K_0^M(\kappa(x)) \to
\bigoplus\nolimits_{\delta(x) = k}' K_0^M(\kappa(x)) \to
\CH_k(X) \to 0
$$
Here we use the notation and conventions introduced in
Expand Down Expand Up @@ -4786,11 +4786,11 @@ \section{The key formula}
Let $X$ be locally of finite type over $S$. Let $k \in \mathbf{Z}$.
We claim that there is a complex
$$
\bigoplus'\nolimits_{\delta(x) = k + 2} K_2^M(\kappa(x))
\bigoplus\nolimits_{\delta(x) = k + 2}' K_2^M(\kappa(x))
\xrightarrow{\partial}
\bigoplus'\nolimits_{\delta(x) = k + 1} K_1^M(\kappa(x))
\bigoplus\nolimits_{\delta(x) = k + 1}' K_1^M(\kappa(x))
\xrightarrow{\partial}
\bigoplus'\nolimits_{\delta(x) = k} K_0^M(\kappa(x))
\bigoplus\nolimits_{\delta(x) = k}' K_0^M(\kappa(x))
$$
Here we use notation and conventions introduced in
Remark \ref{remark-chow-group-pointwise} and in addition
Expand Down Expand Up @@ -4853,9 +4853,9 @@ \section{The key formula}
closed subscheme). Then we find a split short exact sequence
$$
0 \to
\bigoplus'\nolimits_{y \in Y, \delta(y) = k + i} K_i^M(\kappa(y)) \to
\bigoplus'\nolimits_{x \in X, \delta(x) = k + i} K_i^M(\kappa(x)) \to
\bigoplus'\nolimits_{u \in U, \delta(u) = k + i} K_i^M(\kappa(u)) \to 0
\bigoplus\nolimits_{y \in Y, \delta(y) = k + i}' K_i^M(\kappa(y)) \to
\bigoplus\nolimits_{x \in X, \delta(x) = k + i}' K_i^M(\kappa(x)) \to
\bigoplus\nolimits_{u \in U, \delta(u) = k + i}' K_i^M(\kappa(u)) \to 0
$$
for $i = 2, 1, 0$ compatible with the boundary maps in the complexes
of Remark \ref{remark-higher-chow-pointwise}. Applying the snake lemma
Expand Down

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