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\begin{document}

\begin{center}
\textbf{COHOMOLOGY AND CENTRAL SIMPLE ALGEBRAS}

\textit{William Crawley-Boevey}

\textbf{Errata (Darij Grinberg)}

\bigskip
\end{center}

\section*{\S 1}

\begin{itemize}
\item On page 2, replace "it's \underline{homology}" by "its
\underline{homology}". (This is in the definition of "homology", at the middle
of the page.)

\item On page 2, example 1.3 (2) works only if $a\neq0$.

\item On page 4, replace "it's" by "its" in Example 1.6.

\item On page 5, in Example 1.7, "$n$-plane" means "affine $n$-plane". Maybe
you should specify this.

\item On page 5, in the definition of "simplicial complex", you require: "(1)
If $s\in K$ then so is every face of $K$." What you mean is: "... so is every
face of $s$."

\item On page 7, you write "$Z_{1}\left(  C\right)  =$ is set of [...]" (in
the middle of the page). The "is" is redundant here.

\item On page 7, you write "$B_{1}\left(  C\right)  $ is set of $\zeta\left(
\left[  24\right]  -\left[  14\right]  +\left[  12\right]  \right)  $ with
$\zeta,\eta\in\mathbb{Z}$" (again, in the middle of this page). But there is
no $\eta$ in $\zeta\left(  \left[  24\right]  -\left[  14\right]  +\left[
12\right]  \right)  $.

\item On page 9, in the commutative diagram in Definition 1.10 the lower exact
sequence should be $...\rightarrow D_{n+1}\rightarrow D_{n}\rightarrow
D_{n-1}\rightarrow...$, not $...\rightarrow C_{n+1}\rightarrow C_{n}%
\rightarrow C_{n-1}\rightarrow...$.

\item On page 11, in Corollary 1.15, the $H^{n+1}\left(  C,N\right)  $ term
should be $H^{n+1}\left(  C,L\right)  $.\newline Also, in the same Corollary,
you write "you get an exact sequence [...]". This is, in my opinion, somewhat
misleading; it would be better to say that "the sequence [...] is exact" (so
it becomes clear that the sequence always exists, even if $C_{n}$ were not
projective, but its exactness requires the projectivity of $C_{n}$). But
that's just my opinion.

\item On page 12, in Definition 1.19, "$gf$ is homotopic to
$\operatorname*{Id}\nolimits_{D}$ and $fg$ is homotopic to $\operatorname*{Id}%
\nolimits_{C}$" should be the other way round ($gf$ is homotopic to
$\operatorname*{Id}\nolimits_{C}$ and $fg$ is homotopic to $\operatorname*{Id}%
\nolimits_{D}$).

\item On page 14, the full stop after the exact sequence%
\[
0\rightarrow L\overset{f}{\rightarrow}M\overset{g}{\rightarrow}N\rightarrow0.
\]
makes no sense.

\item On page 15, you write:%
\[
=s_{n-1}\partial_{n}x+\partial_{n+1}s_{n}x+\left(  x-s_{n-1}\partial
_{n}x\right)  =x.
\]
The $\partial_{n+1}s_{n}x$ term should be removed from this.
\end{itemize}

\section*{\S 2}

\begin{itemize}
\item On page 17, in Proposition 2.3, replace "\underline{along} $\theta$" by
"\underline{along} $\psi$".

\item On page 23, in the proof of Proposition 2.15, you write:%
\[
\operatorname*{Ext}\nolimits^{n}\left(  M,N\right)  \cong\operatorname*{Ext}%
\nolimits^{n-1}\left(  M,@^{i}N\right)  \cong...\cong\operatorname*{Ext}%
\nolimits^{1}\left(  M,@^{n-1}N\right)  \ \ \ \ \ \ \ \ \ \ \left(
\text{Dimension shifting}\right)  .
\]
The $i$ here should be a $1$ instead.

\item On page 24, in Lemma 2.18, replace "Fix a projective resolution of $N$"
by "Fix a projective resolution $P_{\cdot}$ of $N$" (so that the notation
$P_{\cdot}$ is defined).

\item On page 27, in the second line from the bottom of the page, you write:
"There are maps $\theta,\phi$ as in the Comparison Theorem." Maybe you mean
the maps $\Omega^{1}\psi,\psi_{0}$ instead?

\item On page 27, in the third line from the top of the page, replace $\alpha$
by $\alpha^{\prime\prime}$.

\item On page 27, in the third line from the top of the page, replace $\theta$
by $\psi$.

\item On page 30, in Example 2.27 (1), how do you conclude that $R$ is
artinian if $R$ has global dimension $0$ ?

\item On page 31, in the part (2) of the proof, you write: "there is an exact
sequence of $R\left[  x\right]  $-modules [...]". I am totally nitpicking this
time, you should make clear that the exact sequence if $0\rightarrow R\left[
x\right]  \otimes_{R}M\overset{\alpha}{\rightarrow}R\left[  x\right]
\otimes_{R}M\overset{\beta}{\rightarrow}M\rightarrow0$, not $0\leftarrow
R\left[  x\right]  \otimes_{R}M\overset{h_{1}}{\leftarrow}R\left[  x\right]
\otimes_{R}M\overset{h_{0}}{\leftarrow}M\leftarrow0$.
\end{itemize}

\section*{\S 3}

\begin{itemize}
\item On page 33, in Definition 3.2 replace "long exact sequence sequence" by
"long exact sequence".

\item On page 35, at the very beginning of this page, you write:%
\[
=\left[  g_{0}\mid...\mid g_{n}\right]  -\sum\nolimits_{i=0}^{n}\left(
-1\right)  ^{n}\left[  g_{0}\mid...\mid\widehat{g_{i}}\mid...\mid
g_{n}\right]  .
\]
Two typos here: first, $\left(  -1\right)  ^{n}$ should be $\left(  -1\right)
^{i}$; then, $\left[  g_{0}\mid...\mid\widehat{g_{i}}\mid...\mid g_{n}\right]
$ should be $\left[  1\mid g_{0}\mid...\mid\widehat{g_{i}}\mid...\mid
g_{n}\right]  $.

\item On page 38, in Definition 3.11, you write: "A crossed homomorphism is
\underline{principal} if [...]". This should better be "A crossed homomorphism
$f$ is \underline{principal} if [...]".

\item On page 38, in the third line from the bottom of the page, you use the
term "$G$-group". (You also use it some pages later.) I assume that it just a
synonym for "multiplicative $G$-module"?
\end{itemize}

\section*{\S 4}

\begin{itemize}
\item On page 43, in Definition 4.3, you define the notion of "$A$%
-$B$-bimodule". Maybe it makes sense to add "where $A$ and $B$ are
$R$-algebras" somewhere here (although it's pretty much obvious).

\item On page 43, in Definition 4.4, replace "$S_{n}$ the the tensor product"
by "$S_{n}$ is the tensor product".

\item On page 46, when you define the notion of "equivalent" for two algebra
extensions, it is not directly clear that this is an equivalence relation.
(This could be improved by putting this notion in relation with Definition
2.1, with the only difference being that this time $E\rightarrow E^{\prime}$
is supposed to be an algebra morphism as well.)

\item On page 46, you claim: "The split extensions form one equivalence
class." Is there a quick proof for this? It is easy to see that every
extension equivalent to $0\rightarrow M\rightarrow A\oplus M\rightarrow
A\rightarrow0$ is split (where $A\oplus M$ has the obvious algebra structure),
but it took me quite a while to show that every split extension is equivalent
to $0\rightarrow M\rightarrow A\oplus M\rightarrow A\rightarrow0$. Am I
missing something obvious?

\item On page 46, in Theorem 4.11, you write: "(For comparison, $H^{1}\left(
A,M\right)  $ classifies the extensions $0\rightarrow M\rightarrow
E\rightarrow A\rightarrow0$ of $A$-$A$-bimodules.)" To be completely precise,
it doesn't classify these extensions, but it classifies their equivalence
classes. If you think the word "equivalence class" is already implicit in
"classify", then you could remove "equivalence classes" from the formulation
"$H^{2}\left(  A,M\right)  $ classifies the equivalence classes of algebra extension".

\item In the middle of page 46, the full stop at the end of "Its failure is
given by the map." makes no sense.

\item On page 46, in the third line from the bottom of the page, you write:
$\left(  a,x\right)  \left(  b,y\right)  =\left(  ab,ay+xb+f\left(
a,b\right)  \right)  $. This doesn't seem to work for me; I need%
\[
\left(  a,x\right)  \left(  b,y\right)  =\left(  ab,ay+xb-f\left(  a,b\right)
\right)  .
\]
One of us made a tiny calculation mistake.

\item On page 47, in the proof of Proposition 4.14, replace "$A\otimes_{R}A$
is projective as a left $A$-module" by "$A\otimes_{R}A$ is projective as a
right $A$-module", and replace "and all $\Omega^{n}A$ are projective left
$A$-modules" by "and all $\Omega^{n}A$ are projective right $A$-modules". Of
course, this is true both for left and for right $A$-modules, but what you
need are the right ones, not the left (since $P_{i}$ must be a right
$A$-module in order for $P_{i}\otimes_{A}M$ to make sense).

\item On page 47, on the last line of this page, replace $\operatorname*{proj}%
.\dim M$ by $\operatorname*{proj}.\dim_{A}M$.
\end{itemize}

\section*{\S 5}

\begin{itemize}
\item On page 51, in the third line from the top of the page, replace "the
separability idempotent" by "a separability idempotent $e$".

\item On page 51, in the fifth line from the top of the page, replace $\left(
a\otimes1\right)  e=\left(  a\otimes1\right)  e$ by $\left(  a\otimes1\right)
e=e\left(  a\otimes1\right)  $.

\item On page 51, in the sixth line from the top of the page, replace $z_{0}e$
by $z_{0}a$.

\item On page 51, in the proof of Proposition 5.3, read "some irreducible
polynomial $f$ in $K\left[  x\right]  $" for "some irreducible polynomial in
$K\left[  x\right]  $".

\item On page 52, in the second line from the top of the page, replace
$d=\lambda\in K$ by $d=\lambda\in\overline{K}$.

\item On page 52, in Theorem 5.7, replace%
\[
\alpha_{g}\left(  \ell z\right)  =g\left(  \ell\right)  z\text{ for }z\in
Z,\ \ell\in L
\]
by%
\[
\alpha_{g}\left(  \ell z\right)  =g\left(  \ell\right)  \alpha_{g}\left(
z\right)  \text{ for }z\in Z,\ \ell\in L.
\]


\item On page 53, the first comma in "Let $v_{1,}...,v_{n}$ be a basis of $L$
over $K$." should be one level higher (it should be on the same level with the
$v$'s, not with the indices).

\item On page 53, at the end of the proof of Theorem 5.7, the formula%
\[
\alpha_{g_{j}}=\sum\nolimits_{i}b_{ij}x_{j}%
\]
is wrong; it should be%
\[
\alpha_{g_{i}}=\sum\nolimits_{j}b_{ij}x_{j}.
\]


\item In the line directly after this formula, you write $z=\alpha_{g_{1}%
}\left(  z\right)  =\sum\nolimits_{i}b_{1j}x_{j}$. The summation index should
be $j$, not $i$.

\item On page 54, in the long computation (which proves $\rho_{\psi}\left(
gg^{\prime}\right)  =\rho_{\psi}\left(  g\right)  \left(  g\rho_{\psi}\left(
g^{\prime}\right)  \right)  $), you write:%
\[
=\rho_{\psi}\left(  g\right)  \left(  1\otimes g\right)  \rho_{Y}\left(
g^{\prime}\right)  \left(  1\otimes g^{-1}\right)  .
\]
Replace $\rho_{Y}$ by $\rho_{\psi}$ here.

\item In the bottommost line of page 54, you forgot a bracket in
$H^{1}(G,\operatorname*{Aut}\left(  X^{L}\right)  $.

\item On page 56, in the Proof of Corollary 5.11, you write: "if $x$ has the
indicated form that $N\left(  x\right)  =1$". It would be better to replace
the "that" by "then" here.

\item On page 56, in the Proof of Proposition 5.12, you write: "you can make
$K^{n}$ into a different module by making $a\in\operatorname*{M}%
\nolimits_{n}\left(  K\right)  $ act on $v\in K^{n}$ as $\theta\left(
a\right)  \left(  v\right)  $." This formulation is somewhat fishy. I propose
"[...] by making $a\in\operatorname*{M}\nolimits_{n}\left(  K\right)  $ send
$v\in K^{n}$ to $\theta\left(  a\right)  \left(  v\right)  $." or "[...] by
making $a\in\operatorname*{M}\nolimits_{n}\left(  K\right)  $ act on $K^{n}$
as $\theta\left(  a\right)  $.".

\item On page 56, in Corollary 5.13, replace $H^{1}\left(
G,\operatorname*{PGL}\nolimits_{n}\left(  K\right)  \right)  $ by
$H^{1}\left(  G,\operatorname*{PGL}\nolimits_{n}\left(  L\right)  \right)  $.
\end{itemize}

\section*{\S 6}

\begin{itemize}
\item On page 58, on the second line of Example 6.4, you write $L^{\ast}$
instead of $L^{\times}$. This mistake is repeated a few times.

\item On page 58, in Example 6.4, "As a set" should be "As a $K$-module"
(because you don't specify the additive group structure anywhere else). Maybe
it should also be said that the canonical embedding of $K$ into that
$K$-algebra $L\ast_{f}G$ is \textit{not} given by $1\mapsto\underbrace{1}_{\in
L}\cdot\underbrace{e}_{\in G}$, unless $f$ is a \textit{normalized} $2$-cocycle.

\item On page 59, two lines above Lemma 6.6, replace $\operatorname*{End}%
\nolimits_{A^{\operatorname*{e}}}\left(  K\right)  $ by $\operatorname*{End}%
\nolimits_{A^{\operatorname*{e}}}\left(  A\right)  $.

\item On page 59, in Definition 6.9, "division algebras" could better be
replaced by "division algebras (constructed in Lemma 6.2)".

\item On page 60, Theorem 6.10 would become clearer if you write "$A^{L}$ is
central simple as an $L$-algebra" instead of just "$A^{L}$ is central simple".

\item On page 60, one line above Definition 6.14, you have a typo: $M_{n_{1}%
}\left(  D_{2}\right)  $ should be $M_{n_{2}}\left(  D_{2}\right)  $.

\item On page 61, in Theorem 6.15, maybe it is better to clarify that the
division algebras are supposed to be f.d. over $K_{s}$, not over $K$.

\item On page 61, in the proof of Theorem 6.15, you write: $\delta_{x}%
:K_{s}\rightarrow D$. But this would be the zero map, since $K_{s}$ is in the
center of $D$. You want $\delta_{x}:D\rightarrow D$ instead.

\item On page 62, the proof of Theorem 6.20 talks about "central simple
$K$-algebras of dimension $n^{2}$ split by $L$". You haven't defined what
"split by $L$" means for a central simple $K$-algebra. (In fact, you call a
central simple $K$-algebra $A$ \textit{split by }$L$ if $A^{L}\cong%
\operatorname*{M}\nolimits_{n}\left(  L\right)  $ as $L$-algebras for some
$n\in\mathbb{N}$.)
\end{itemize}


\end{document}