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\mode<presentation>

\usepackage[matrix,arrow,graph,curve]{xy}
\usepackage{amssymb}
\usepackage{leftidx}
\usepackage{graphics}

\renewcommand{\O}{\mathcal{O}}
\newcommand{\place}{\mathord{-}}
\newcommand{\epi}{\twoheadrightarrow}
\newcommand{\mono}{\rightarrowtail}
\newcommand{\lMod}[1]{\leftidx{_{#1}}{\mathsf{Mod}}{}}
\newcommand{\bMod}[1]{\leftidx{_{#1}}{\mathsf{Mod}}{_{#1}}}
\newcommand{\HH}{H\!H}
\def\*{\bullet}

\DeclareMathOperator{\Ext}{Ext}
\DeclareMathOperator{\Tor}{Tor}
\DeclareMathOperator{\sOp}{sOp}

\newtheorem{proposition}{Proposition}[]

\title{Applications of the change-of-rings spectral sequence to the
computation of Hochschild cohomology}
\author{Mariano Suárez-Alvarez\\
\texttt{mariano@dm.uba.ar}}
\date{Mar del Plata, March 6--17, 2006}

\begin{document}
% Remove this to get the pauses...
\let\pause\relax

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\begin{frame}
\titlepage
\end{frame}

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\begin{frame}
\frametitle{Operations on cohomology}

\begin{theorem}
Let $A$ be an algebra, $M$, $N\in\lMod A$ and $d\geq0$.\\
Let $\O=(\O^p)_{p\geq0}$ be a sequence of natural transformations of
functors of $A$-modules
  \[
  \O^p:\Ext^p_A(N,\place) \to \Ext^{p+d}_A(M,\place).
  \]

\pause
Assume that, for each short exact sequence $P'\mono P\epi P''$, the following
diagram commutes:
  \[\xymatrix{
  \Ext^p_A(N,P'') \ar[r]^-{\partial} \ar[d]_-{\O^p} 
    & \Ext^{p+1}_A(N,P') \ar[d]_-{\O^{p+1}} 
    \\
  \Ext^{p+d}_A(M,P'') \ar[r]^-{\partial}
    & \Ext^{p+d+1}_A(M,P') 
  }\]
\pause
Then there exists exactly one $Y(\O)\in\Ext^d_A(M,N)$ such that    
  \[
  \O^p(\place) = (\place)\circ Y(\O).
  \]
\end{theorem}
\end{frame}

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\begin{frame}
\frametitle{Operations on cohomology}

\begin{corollary}
There is an isomorphism of bifunctors of $A$-modules
  \[
  Y:\sOp^\*_A(\place,\place)\cong\Ext^\*_A(\place,\place).
  \]
\end{corollary}

\pause

\begin{corollary}
Let $A$ be an algebra and $d\geq0$.
Let $\O=(\O^p)_{p\geq0}$ be a sequence of natural transformations of
functors of $A$-bimodules
  \[
  \O^p:H^p(A,\place) \to H^{p+d}(A,\place)
  \]
which commutes with boundary maps.
Then there exists exactly one $Y(\O)\in\HH^d(A)$ such that    
  \[
  \O^p(\place) = (\place)\smile Y(\O).
  \]
\end{corollary}

\end{frame}

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\begin{frame}
\frametitle{Change of rings}

\begin{corollary}
Let $\phi:A\to B$ be a map of rings, and let $M\in\lMod A$.

For each $q\geq1$ there is a unique class
  \[
  \zeta^q\in\Ext_B^2(\Tor^A_{q-1}(B,M),\Tor^A_q(B,M))
  \]
such that the differential
  \[
  d_2^{p,q}:\Ext_B^p(\Tor_q^A(B,M),\place)
            \to \Ext_B^{p+2}(\Tor_{q-1}^A(B,M),\place)
  \]
of the spectral sequence
  \[
  E_2^{p,q} = \Ext_B^p(\Tor_q^A(B,M),\place) \Rightarrow
  \Ext_A^\*(M,\place)
  \]
is given on $\alpha\in\Ext_B^p(\Tor_q^A(B,M),\place)$ by
  \[
  d_2^{p,q}(\alpha) = \alpha\circ\zeta^q.
  \]
\end{corollary}

\end{frame}

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\begin{frame}
\frametitle{Change of rings}

\begin{theorem}
Let $\phi:A\to B$ be an epimorphism of algebras. There exists a
spectral sequence, functorial on $B$-bimodules,
  \[
  E_2^{p,q}\cong\Ext_{B^e}^p(\Tor^A_q(B,B),\place)
    \Rightarrow H^\*(A,\place)
  \]
which has $E_2^{\*,0}\cong H^\*(B,\place)$.

\pause

For each $q\geq1$ there exists a unique class 
  \[
  \zeta^q\in\Ext_{B^e}^2(\Tor^A_{q-1}(B,B),\Tor^A_q(B,B))
  \]
such that $d_2^{p,q}(\place)=(\place)\circ\zeta^q$.

\pause 

If $\phi$ is surjective and $I=\ker\phi$, $\Tor^A_1(B,B)\cong I/I^2$ and
  \[
  \zeta^1\in\Ext_{B^e}^2(B,I/I^2)=H^2(B,I/I^2)
  \]
is the class of the infinitesimal extension
  \[\xymatrix{
  0 \ar[r] 
    & I/I^2 \ar[r]
    & A/I^2 \ar[r]
    & B \ar[r]
    & 0
  }\]
\end{theorem}

\end{frame}

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\begin{frame}
\frametitle{Monogenic algebras}

\begin{theorem}
Let $k$ be a field and fix a monic $f=\sum_{i=0}^Na_iX^i\in k[X]$.

Let $d=(f,f')$, pick $q\in k[X]$ such that $f=qd$, and put
  \begin{gather*}
  u = q^2\sum_{i=0}^Na_i\frac{i(i-1)}2 X^{i-2} = q^2\Delta_2(f).
  \end{gather*}
Let $A=k[X]/(f)$.

\pause

There is an isomorphism of graded commutative algebras
  \[
  \HH^\*(A) \cong \frac{k[x_0,\tau_1,\zeta_2]}
                  {(f(x), d(x)\tau ,f'(x)\zeta ,\tau^2-u(x)\zeta)}.
  \]
\end{theorem}

\end{frame}

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\begin{frame}
\frametitle{Monogenic algebras}

\begin{proposition}
Let 
  \[
  w = \sum_{i=0}^N \sum_{\substack{s,t\geq0\\s+t+1=i}}
          a_i\,\overline{(s+1)X^sq}\, X^t.
  \]
The Gerstenhaber Lie structure on $\HH^\*(A)$ is such that
  \begin{gather*}
  [\tau,x] = q(x), \\
  [\zeta,\tau] = w(x)\zeta, \\
  [x,x] = [\tau,\tau] = [\tau,\zeta] = [x,\zeta] = 0.
  \end{gather*}
\end{proposition}

\end{frame}

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\begin{frame}
\frametitle{Nice morphisms}

\begin{theorem}
Let $\phi:A\to B$ be an epimorphism of algebras. The following statements
are equivalent:
  \begin{enumerate}[a)]
  \item $\phi:A\to B$ is a homological epimorphism;
  \item $\Tor_+^A(B,M)=0$ for all $M\in\lMod A$;
  \item $\Tor_+^A(B,B)=0$;
  \item $\phi^e:A^e\to B^e$ is a homological epimorphism.
  \end{enumerate}
When they hold, there is an isomorphism of functors of $B$-bimodules
  \[
  H^\*(B,\place)\xrightarrow{\cong} H^\*(A,\place).
  \]
\end{theorem}

\end{frame}

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\begin{frame}
\frametitle{Nice ideals}

\begin{corollary}
Let $\phi:A\to B$ be a surjective homological epimorphism and let
$I=\ker\phi$. There is a long exact sequence
  \[
  \xymatrix@C-1.3em{
  \cdots \ar[r]
    & \Ext_{A^e}^p(A,I) \ar[r]
    & \HH^p(A) \ar[r]
    & \HH^p(B) \ar[r]
    & \Ext_{A^e}^{p+1}(A,I) \ar[r]
    & \cdots 
  }
  \]
\end{corollary}

\end{frame}

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\begin{frame}
\frametitle{Nice ideals}

\begin{proposition}
Let $\phi:A\to B$ be a surjective homological epimorphism such that
$I=\ker\phi$ is $A$-flat on one side. Then $H^0(B,\place)\cong
H^0(A,\place)$ on $\bMod B$ and there is a natural long exact sequence
of functors of $B$-bimodules
  \begin{multline*}
  \xymatrix@C-1ex{
  \cdots \ar[r]
    & H^p(B,\place) \ar[r]
    & H^p(A,\place) \ar[r]
    & {}
  } \\
  \xymatrix@C-1ex{
  {} \ar[r]
    & \Ext_{A^e}^{p-1}(I/I^2,\place) \ar[r]^{{}\smile\zeta}
    & H^{p+1}(B,\place) \ar[r]
    & \cdots
  }\end{multline*}
with $\zeta\in H^2(B,I/I^2)$ the class of the infinitesimal extension
  \[\xymatrix{
  0 \ar[r] 
    & I/I^2 \ar[r]
    & A/I^2 \ar[r]
    & B \ar[r]
    & 0
  }\]
\end{proposition}

\end{frame}

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\begin{frame}
\frametitle{Nice ideals}

\begin{lemma}
Let $\phi:A\to B$ be a surjective morphism of algebras and put
$I=\ker\phi$. Then
  \[
  \Tor_q^A(B,B) \cong
    \begin{cases}
    B, & \text{if $q=0$;} \\
    I/I^2, & \text{if $q=1$;} \\
    \ker\left(I\otimes_AI\xrightarrow{\mu}I\right), & \text{if $q=2$;} \\
    \Tor_{q-2}^A(I,I), & \text{if $q>2$.} 
    \end{cases}
  \]
\end{lemma}

\end{frame}

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\begin{frame}
\frametitle{Nice ideals: an example}

Let $A=kQ/J$ be an admissible quotient of the path algebra on a quiver $Q$
and let $e\in Q_0$.

Assume
\begin{itemize}
\item Every minimal relation in $J$ involving a path passing through $e$
also involves a path not passing through $e$; and
\item $e$ is on no oriented cycle of $Q$.
\end{itemize}

Then $I=AeA\lhd A$ is homological and, if $B=A/I$, there is a long exact sequence
  \begin{multline*}
  \xymatrix{
  \cdots \ar[r]
    & \Ext_{A}^p(D(eA),Ae) \ar[r]
    & \HH^p(A) \ar[r]
    & {}
  } \\
  \xymatrix{
    {} \ar[r]
    & \HH^p(B) \ar[r]
    & \Ext_{A}^{p+1}(D(eA),Ae) \ar[r]
    & \cdots 
  }
  \end{multline*}
\end{frame}

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\begin{frame}
\frametitle{Nice morphisms}

  \[
  \xygraph{
    []\bullet="a" 
    !{c+a(60)*+{\bullet}}="b1"
    [r]\bullet="c1"
    !{c-<0pt,2ex>*{e}}
    "c1"
    [r]\bullet="d1"
    !{c+a(-60)*+{\bullet}}="e"
    "d1"
    !{c+a(+60)*+{\bullet}}="y"
    "a"
    !{c+a(-60)*+{\bullet}}="b2"
    [r]\bullet="c2"
    [r]\bullet="d2"
    "b1"
    !{c+a(120)*+{\bullet}}="x"
    "c1"
    !{c+(0,-\halfrootthree)*+{\circlearrowleft}}
    %
    "a" 
      :@{->}"b1"
      :@{->}"c1"
      :@{->}"d1"
      :@{->}"e"
    "a" 
      :@{->}"b2"
      :@{->}"c2"
      :@{->}"d2"
      :@{->}"e"
    "x" 
      :@{->}"b1"
    "d1" 
      :@{->}"y"
    "x" 
      :@{-->}@/^.8em/"c1"
      :@{-->}@/^.8em/"y"
  }\]
\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Nice morphisms}

  \[\xygraph{
    []\bullet="a"
    [r]\bullet="b"
    [r]\bullet="c"
    !{c+<0pt,2ex>*{e}}
    "c"
    [r]\bullet="d"
    [r]\bullet="e"
    "a"
      :@{->}"b"
      :@{->}"c"
      :@{->}"d"
      :@{->}"e"
    "a"
      :@{-->}@/^1.3em/"c"
      :@{-->}@/^1.3em/"e"
    "b"
      :@{*{}*{}*{}}@/_1.3em/"d"
  }\]

\end{frame}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Nice morphisms}

  \[\xygraph{
    []\bullet="a"
    [r]\bullet="b"
    [r]\bullet="c"
    !{c+<0pt,2ex>*{e}}
    "c"
    [r]\bullet="d"
    [r]\bullet="e"
    "a"
      :@{->}"b"
      :@{->}"c"
      :@{->}"d"
      :@{->}"e"
    "a"
      :@{-->}@/^1.3em/"c"
      :@{-->}@/^1.3em/"e"
    "b"
      :@{->}@/_1.3em/"d"
  }\]

\end{frame}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{frame}
\frametitle{Nice morphisms}

  \[\xygraph{
    []\bullet="a"
    [r]\bullet="b"
    [r]\bullet="c"
    !{c+<0pt,2ex>*{e}}
    "c"
    [r]\bullet="d"
    [r]\bullet="e"
    "a"
      :@{->}"b"
      :@{->}"c"
      :@{->}"d"
      :@{->}"e"
    "a"
      :@{-->}@/^1.3em/"c"
      :@{-->}@/^1.3em/"e"
    "b"
      :@{->}@/_1.3em/"d"
    "a"
      :@{->}@(ul,dl)"a"
    "e"
      :@{->}@(ur,dr)"e"
  }\]

\end{frame}

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\begin{frame}
\end{frame}

\end{document}
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