A simple counterexample to the Jordan--H\"older property\\for derived categories
Alexander Kuznetsov
A counterexample to the Jordan--H\"older property for semiorthogonal decompositions of derived categories of smooth projective varieties was constructed by B\"ohning, Graf von Bothmer and Sosna. In this short note we present a simpler example by realizing Bondal's quiver in the derived category of a blowup of $\PP^3$.
1304.0903.tex
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\title[A simple counterexample to the Jordan--H\"older property for derived categories]{A simple counterexample to the Jordan--H\"older property\\for derived categories}
\author{Alexander Kuznetsov}
\address{\sloppy
\parbox{0.9\textwidth}{
% {\bf A.K.: }
Algebra Section, Steklov Mathematical Institute,
8 Gubkin str., Moscow 119991 Russia
\hfill\\[5pt]
% \phantom{{\bf A.K.: }}
The Poncelet Laboratory, Independent University of Moscow
\hfill\\[5pt]
% \phantom{{\bf A.K.: }}
Laboratory of Algebraic Geometry, SU-HSE.
\hfill
}\bigskip}
\email{akuznet@mi.ras.ru}
% \subjclass{14M15, 18E30}
% \address{\sloppy
% \parbox{0.9\textwidth}{
% Algebra Section, Steklov Mathematical Institute,
% 8 Gubkin str., Moscow 119991 Russia
% \hfill\\[5pt]
% The Poncelet Laboratory, Independent University of Moscow
% \hfill\\[5pt]
% }}
% \email{akuznet@@mi.ras.ru}
\date{}
\thanks{I was partially supported by
RFFI grants 11-01-00393, 11-01-00568, 12-01-33024, NSh-5139.2012.1,
the grant of the Simons foundation, and by AG Laboratory SU-HSE, RF government grant, ag.11.G34.31.0023.}



\begin{document}

\begin{abstract}
A counterexample to the Jordan--H\"older property for semiorthogonal decompositions
of derived categories of smooth projective varieties was constructed by B\"ohning, Graf von Bothmer and Sosna.
In this short note we present a simpler example by realizing Bondal's quiver
in the derived category of a blowup of $\PP^3$.
\end{abstract}

\maketitle

\section{Introduction}

Given a triangulated category $\CT$, a {\sf semiorthogonal decomposition for $\CT$} is a chain
$\CA_1,\dots,\CA_m \subset \CT$ of full triangulated subcategories such that 
\begin{itemize}
\item for any $j > i$ one has $\Hom(\CA_j,\CA_i) = 0$, and
\item for any object $T \in \CT$ there is a chain of morphisms
\begin{equation*}
0 = T_m \to T_{m-1} \to \dots \to T_1 \to T_0 = T
\end{equation*}
such that $\Cone(T_i \to T_{i-1}) \in \CA_i$.
\end{itemize}
We write $\CT = \langle \CA_1,\dots,\CA_m \rangle$ to denote a semiorthogonal decomposition.

It is well known \cite{BK} that the braid group acts on the set of all semiorthogonal decompositions of a given category ---
the $i$-th generator of the braid group acts as
\begin{equation*}
\langle \CA_1,\dots,\CA_r \rangle \mapsto
\langle \CA_1,\dots,\CA_{i-1},\CA_{i+1},{}^\perp \langle \CA_1,\dots,\CA_{i-1},\CA_{i+1} \rangle \cap \langle \CA_{i+2},\dots,\CA_m\rangle^\perp,\CA_{i+2},\dots,\CA_m \rangle.
\end{equation*}
So if a category $\CT$ has a semiorthogonal decomposition, it has many of them. However, 
it is also well known that the equivalence classes of the components do not change under this action ---
there is an equivalence of categories
\begin{equation*}
{}^\perp \langle \CA_1,\dots,\CA_{i-1},\CA_{i+1} \rangle \cap \langle \CA_{i+2},\dots,\CA_r\rangle^\perp \cong \CA_i
\end{equation*}
This motivates the following definition.

\begin{definition}
A triangulated category $\CT$ {\sf has the Jordan--H\"older property} if for any pair
\begin{equation*}
\CT = \langle \CA_1,\dots,\CA_m \rangle,
\qquad
\CT = \langle \CB_1,\dots,\CB_n \rangle
\end{equation*}
of semiorthogonal decompositions with indecomposable components one has $m = n$ and there is a permutation $\sigma \in S_m$
such that $\CB_i \cong \CA_{\sigma(i)}$ for each $1 \le i \le m$.
\end{definition}

Among triangulated categories of geometrical nature there are very few for which the Jordan-H\"older property has been proved.
Basically these consist of those for which all semiorthogonal decompositions can be classified
(basically these are $\BD(\PP^1)$ and its quotient stacks $\BD(\PP^1/\Gamma)$, see~\cite{Kir}
for the detailed investigation of the latter), or those
which are themselves indecomposable (connected Calabi--Yau categories~\cite{Bri}, derived categories 
of curves of positive genus~\cite{Oka}). Even for $\PP^2$ the property is questionable. 

On one hand, if the Jordan--H\"older property could be justified for derived categories
of smooth projective varieties, this would allow to define nice birational invariants,
see~\cite{KECM} and~\cite{BBS}.
On the other hand, it was known for a long time that the property is not satisfied for 
arbitrary triangulated categories. A very simple counterexample was constructed by Alexei Bondal
long ago. Namely, Bondal considered a quiver with relations
\begin{equation}\label{bq}
Q = \left(\left. \xymatrix@C=3em{ 
\bullet \ar@<1ex>[r]^{\alpha_1} \ar@<-1ex>[r]_{\alpha_2} & 
\bullet \ar@<1ex>[r]^{\beta_1} \ar@<-1ex>[r]_{\beta_2} & 
\bullet 
} \right|\
\beta_1\alpha_2 = \beta_2\alpha_1 = 0
\right)
\end{equation}
and noted that on one hand as any oriented quiver it has a full exceptional collection
\begin{equation*}
\BD(Q) = \langle P_1,P_2,P_3 \rangle
\end{equation*}
with $P_i$ being the projective module of the $i$-th vertex, and on the other hand,
it has an exceptional object
\begin{equation}\label{oe}
P = \left( \xymatrix@C=3em{ 
\kk \ar@<1ex>[r]^{1} \ar@<-1ex>[r]_{0} & 
\kk \ar@<1ex>[r]^{1} \ar@<-1ex>[r]_{0} & 
\kk
} \right)
\end{equation}
which is {\sf nonextendable}, i.e.\ does not extend to a longer exceptional collection (even numerically).

Since the category $\BD(Q)$ itself is not equivalent to the derived category of a scheme, Bondal's
counterexample does not answer the question whether the Jordan--H\"older property is true for derived 
categories of schemes, so for some time there was a little hope that by some miracle it might be true.
A recent paper of B\"ohning, Graf von Bothmer and Sosna~\cite{BBS} gave finally a negative answer
to this question. To be more precise, investigating the derived category $\BD(X)$ of the classical 
Godeaux surface 
\begin{equation*}
X = \{ x_1^5 + x_2^5 + x_3^5 + x_4^5 = 0 \}/\ZZ_5 \subset \PP^3/\ZZ_5,
\end{equation*}
where $\ZZ_5$ acts with weight $i$ on $x_i$, the authors constructed two nonextendable exceptional
collections in~$\BD(X)$, one of length 11 (the maximal possible), and the other of length 9.
The nonextendability is also checked on the numerical level. The construction of this
counterexample and the proofs are rather complicated up to such extent that at some moment
a computer computation (using Macaulay2) is used. 

The goal of this note is to give an elementary example. We note that although $\BD(Q)$ itself is not
equivalent to the derived category of an algebraic variety, it can be realized as a semiorthogonal
component of such. And this is enough to get a counterexample. The variety we consider is a two-step
blowup of $\PP^3$ in two smooth rational curves. Its derived category has a full exceptional collection
and we observe that it contains Bondal's quiver as a subcollection.


% 
% 
% 
% It was known that triangulated categories do not enjoy the Jordan--H\"older property.
% A counterexample constructed by Bondal --- the derived category of representations of
% a very simple quiver has two essentially different maximal semiorthogonal decompositions. 
% However, for some time there was a question whether one can construct a geometric example of the same sort.
% Recently, B\"onhning, Graf von Bothmer and Sosna showed that the derived category 
% of the classical Godeaux surfaces provides such. In this paper we discuss other examples,
% coming back to the original Bondal's quiver. For this we show that some algebraic varieties 
% with a full exceptional collection contain a subcollection equivalent to the Bondal' quiver.



\section{Example}

Let $A \subset V$ be a pair of vector spaces of dimensions 2 and 4 respectively, so that $\PP(A) \subset \PP(V)$
is a line $\PP^1$ in a $\PP^3$. Let $X$ be the blowup of $\PP(V)$ along $\PP(A)$ with $E$ being the exceptional divisor.
Then
\begin{equation*}
E \cong \PP(A)\times\PP(V/A) = \PP^1\times\PP^1.
\end{equation*}
We denote by $i$ the embedding of $E$ into $X$ and by $H$ the pullback to $X$ of a hyperplane on $\PP(V)$.
The Picard group of $X$ is generated by $H$ and $E$ and we have
\begin{equation}\label{oxhe}
\CO_X(H)_{E} \cong \CO_E(1,0).
\end{equation}
% 
% 
% Choose a pair of two dimensional vector spaces $A$ and $B$ and take 
% \begin{equation}\label{xe}
% X = \PP_{\PP(B)}(A\otimes \CO_{\PP(B)} \oplus \CO_{\PP(B)}(-1)),
% \qquad
% E = \PP(A) \times \PP(B) = \PP_{\PP(B)}(A\otimes \CO_{\PP(B)}) \subset X.
% \end{equation}
% The embedding of $E$ into $X$ is denoted by $i$.
% 
% It is easy to see that $X$ can also be described as a blowup of $\PP^3 = \PP(A \oplus B)$,
% the projection $X \to \PP(A\oplus B)$ is induced by the embedding 
% $A\otimes\CO_{\PP(B)} \oplus \CO_{\PP(B)} \to (A \oplus B)\otimes\CO_{\PP(B)}$.
% Clearly, $E$ is the exceptional divisor of this blowup.
% 
% We denote by $h$ and $e$ the class of a point on $B$ pulled back to $X$ and the class of the exceptional divisor~$E$.
% Together they generate the Picard group of $X$
% \begin{equation*}
% \Pic X = \ZZ h \oplus \ZZ e.
% \end{equation*}
% Note that
% \begin{equation*}
% \CO_X(h)_{|E} = \CO_E(0,1),
% \qquad
% \CO_X(e)_{|E} = \CO_E(1,-1).
% \end{equation*}
% The map to $\PP^3 = \PP(A\oplus B)$ is given by the linear system $H = h + e$. Since $\CO_X(H)_{|E} = \CO_E(1,0)$
% it contracts $E$ onto the line $\PP(A) \subset \PP(A\oplus B)$.

Let $C$ be a smooth rational curve on $X$ which intersects $E$ transversally in two points
\begin{equation*}
P_1 = (a_1,b_1),\qquad
P_2 = (a_2,b_2),
\end{equation*}
where $a_i \in \PP(A)$, $b_i \in \PP(V/A)$ and with
\begin{equation*}
a_1\ne a_2.
% ,\qquad
% b_1 \ne b_2.
\end{equation*}
For example, one can take $C$ to be the proper preimage of a conic in $\PP(V)$ intersecting the line $\PP(A)$ in two distinct points
(in this case the points $b_1$ and $b_2$ will coincide, but this does not matter).

Let $\pi:Y \to X$ be the blowup of $X$ in $C$. Let $E'$ be the exceptional divisor of this blowup,
$i':E' \to Y$ be its embedding into $Y$, $p:E' \to C$ the projection, and $j:C \to X$ the embedding of the curve.
This can be summarized in a diagram
\begin{equation*}
\xymatrix{
E' \ar[r]^{i'} \ar[d]_p & 
Y \ar[d]^\pi \\
C \ar[r]^{j} &
X
}
\end{equation*}

Recall that by Orlov's blowup formula~\cite{Or} we have a semiorthogonal decomposition
\begin{equation*}
\BD(Y) = \langle \pi^*(\BD(X)), i'_*p^*(\BD(C)) \rangle.
\end{equation*}
We take the following triple of sheaves on $Y$:
\begin{equation*}
\CE_1 = \CO_Y = \pi^*\CO_X,\qquad
\CE_2 = \pi^*i_*\CO_E(1,0),\qquad
\CE_3 = i'_*p^*\CO_C(-3).
\end{equation*}

\begin{lemma}
The triple $(\CE_1,\CE_2,\CE_3)$ is exceptional and extends to a full exceptional collection in $\BD(Y)$.
\end{lemma}
\begin{proof}
First, we extend the pair $(\CO_X,i_*\CO_E(1,0))$ to a full exceptional collection in $\BD(X)$:
\begin{equation*}
\BD(X) = \langle \CO_X(-3H),\CO_X(-2H),\CO_X(-H),\CO_X,i_*\CO_E,i_*\CO_E(1,0) \rangle.
\end{equation*}
This is just the full exceptional collection obtained by combining the standard exceptional collection 
$(\CO(-3),\CO(-2),\CO(-1),\CO)$ on $\PP(V)$ with the standard collection $(\CO,\CO(1))$ on $\PP(A)$
if we consider $X$ as the blowup of $\PP(V)$ in $\PP(A)$. Pulling it back to $Y$ and combining 
with the exceptional collection $\CO_C(-3),\CO_C(-2)$ on $C$ we obtain a full exceptional collection
in $\BD(Y)$:
\begin{equation*}
\BD(Y) = \langle \CO_Y(-3H),\CO_Y(-2H),\CO_Y(-H),
\underline{\CO_Y},\pi^*i_*\CO_E,
\underline{\pi^*i_*\CO_E(1,0)},\underline{i'_*p^*\CO_C(-3)},i'_*\CO_C(-2) \rangle
\end{equation*}
(we denote here the pullback of $H$ to $Y$ also by $H$). The underlined terms of this exceptional
collection are the objects $\CE_1$, $\CE_2$ and $\CE_3$.
\end{proof}

% Let us compute the $\Ext$-groups between the objects $\CE_i$.

\begin{lemma}
We have
\begin{equation*}
\Ext^\bullet(\CE_1,\CE_2) = A^*,
\qquad
\Ext^\bullet(\CE_2,\CE_3) = \kk^2[-1],
\qquad
\Ext^\bullet(\CE_1,\CE_3) = \kk^2[-1].
\end{equation*}
\end{lemma}
Here the brackets stand for the homological shift. In other words, it is claimed that between $\CE_1$ and $\CE_2$ there is only $\Hom$,
while from $\CE_1$ and $\CE_2$ to $\CE_3$ there are only $\Ext^1$.

\begin{proof}
Since $\pi^*$ is fully faithful we have
\begin{equation*}
\Ext^\bullet(\CE_1,\CE_2) = \Ext^\bullet(\CO_X,i_*\CO_E(1,0)) = H^\bullet(E,\CO_E(1,0)) = A^*.
\end{equation*}
Furthermore, for any $F \in \BD(X)$, $G \in \BD(C)$ we have
\begin{equation*}
\Ext^\bullet(\pi^*F,i'_*p^*G) \cong
\Ext^\bullet(F,\pi_*i'_*p^*G) \cong
\Ext^\bullet(F,j_*p_*p^*G) \cong
\Ext^\bullet(F,j_*G).
\end{equation*}
It follows that 
\begin{equation*}
\Ext^\bullet(\CE_1,\CE_3) = 
\Ext^\bullet(\CO_X,j_*\CO_C(-3)) = 
H^\bullet(C,\CO_C(-3)) = \kk^2[-1].
\end{equation*}
Finally, 
\begin{equation*}
\Ext^\bullet(\CE_2,\CE_3) = 
\Ext^\bullet(i_*\CO_E(1,0),j_*\CO_C(-3)).
\end{equation*}
To compute the latter we take the resolution
\begin{equation*}
0 \to \CO_X(H-E) \xrightarrow{\ E\ } \CO_X(H) \to i_*\CO_E(1,0) \to 0
\end{equation*}
and apply the local $\CHom(-,j_*\CO_C(-3))$ functor. We deduce that $\RCHom(i_*\CO_E(1,0),j_*\CO_C(-3))$
is quasiisomorphic to the complex
\begin{equation*}
% 0 \to \CHom(i_*\CO_E(1,0),j_*\CO_C(-3)) \to 
j_*\CO_C(-H-3p) \xrightarrow{\ E\ } j_*\CO_C(E-H-3p), 
% \to \CExt^1(i_*\CO_E(1,0),j_*\CO_C(-3)) \to 0
\end{equation*}
where $p$ stands for the class of a point on $C$, with terms in grading $0$ and $1$ respectively.
Since $C$ is a smooth curve and $E$ is a section of a line bundle on $C$
vanishing with multiplicity 1 at points $P_1$ and $P_2$ only, we see that
the above map has trivial kernel and its cokernel is just the sum of $\CO_{P_1}$ and $\CO_{P_2}$,
the structure sheaves of the points. It follows that
\begin{equation*}
\CExt^\bullet(i_*\CO_E(1,0),j_*\CO_C(-3)) \cong \CO_{P_1}[-1] \oplus \CO_{P_2}[-1].
\end{equation*}
Using the local-to-global spectral sequence we deduce that
\begin{equation}\label{e23}
\Ext^\bullet(i_*\CO_E(1,0),j_*\CO_C(-3)) = H^\bullet(Y,\CO_{P_1})[-1] \oplus H^\bullet(Y,\CO_{P_2})[-1].
\end{equation}
which gives the last claim of the Lemma.
\end{proof}

It remains to compute the multiplication map. Let $\alpha_1,\alpha_2$ be the basis of $A^*$ dual to the basis
$a_1,a_2$ of $A$ given by the first coordinates of the points $P_1$ and $P_2$. Let $\beta_1,\beta_2$
be the basis of $\Ext^1(\CE_2,\CE_3)$ given by the spaces $H^0(Y,\CO_{P_1})$ and $H^0(Y,\CO_{P_2})$ in~\eqref{e23} respectively.

\begin{proposition}
The multiplication map
\begin{equation*}
m:\Hom(\CE_1,\CE_2) \otimes \Ext^1(\CE_2,\CE_3) \to \Ext^1(E_1,E_3)
\end{equation*}
is surjective and its kernel is spanned by $\alpha_1\otimes\beta_2$ and $\alpha_2\otimes\beta_1$.
\end{proposition}

Before giving a proof let us mention the consequences.

\begin{corollary}
The algebra of homomorphisms of the exceptional collection $\CE_1,\CE_2[1],\CE_3[1]$ is isomorphic
to the path algebra of Bondal's quiver~\eqref{bq}.
\end{corollary}

\begin{corollary}
The derived category $\BD(Y)$ of $Y$ does not have the Jordan--H\"older property.
\end{corollary}


\begin{proof}[Proof of the Proposition]
First, the pullback-pushforward adjunction for the morphism $\pi$ shows that the map $m$ coincides
with the multiplication map
% 
% we reduce all the computations to $X$ by noting that the multiplication map coincides with the map
\begin{equation*}
\Hom(\CO_X,i_*\CO_E(1,0)) \otimes \Ext^1(i_*\CO_E(1,0),j_*\CO_C(-3)) \to \Ext^1(\CO_X,j_*\CO_C(-3)).
\end{equation*}
Now let us identify the bases in the spaces we are interested in.

The first space $\Hom(\CO_X,i_*\CO_E(1,0))$ has $\alpha_1,\alpha_2$ as a base. By~\eqref{oxhe} 
we can find sections $\bar\alpha_1,\bar\alpha_2$ of $\CO_X(H)$ which restrict to the sections $\alpha_1$ and $\alpha_2$ of $\CO_E(1,0)$. 
These are given just by two planes in $\PP^3 = \PP(V)$ intersecting the line $\PP^1=\PP(A)$ transversally in points 
$a_2$ and $a_1$ respectively. 

Further, denote by $\rho_i$ the canonical morphisms
\begin{equation*}
\CO_E(1,0) \xrightarrow{\ \rho_i\ } \CO_{P_i}
\end{equation*}
and by $\eta_i$ the canonical extensions
\begin{equation*}
\CO_{P_i} \xrightarrow{\ \eta_i\ } \CO_C(-3)[1].
\end{equation*}
Then
\begin{equation*}
\beta_i = j_*(\eta_i) \circ \rho_i.
\end{equation*}
To see this consider (the pushforward to $X$ of) the exact sequence
\begin{equation}\label{cext}
0 \to \CO_C(-3) \xrightarrow{\ P_1 + P_2\ } \CO_C(-1) \to \CO_{P_1} \oplus \CO_{P_2} \to 0
\end{equation}
corresponding to the sum of extension $\eta_1$ and $\eta_2$, and apply the functor $\Hom(i_*\CO_E(1,0),-)$ to it.
We will get an exact sequence
% \begin{equation*}
\begin{multline*}
\dots \to 
\Hom(i_*\CO_E(1,0),j_*\CO_C(-1)) \to \\ \to
\Hom(i_*\CO_E(1,0),\CO_{P_1}) \oplus
\Hom(i_*\CO_E(1,0),\CO_{P_2}) \to \\ \to
\Ext^1(i_*\CO_E(1,0),j_*\CO_C(-3)) \to \dots
\end{multline*}
% \end{equation*}
The first term is zero (this is proved analogously to~\eqref{e23}).
% since $E$ and $C$ intersect in two points $P_1$ and $P_2$, so any morphism
% should factor through these points, while the sheaf $\CO_C(-1)$ is locally free on $C$, so
% it has no subsheaves supported in $P_1$ and $P_2$. 
Consequently, the second map is an embedding.
Clearly, it takes the basis $(\rho_1,\rho_2)$ of the second term to $j_*(\eta_1) \circ \rho_1$
and $j_*(\eta_2) \circ \rho_2$ respectively, which thus span the second space. Clearly, these
elements coincide with $\beta_1$ and $\beta_2$.

Now we can check that the products of $\alpha_1\otimes\beta_2$ and of $\alpha_2\otimes\beta_1$ are zero.
Indeed,
\begin{equation*}
m(\alpha_1\otimes\beta_2) = 
\beta_2\circ\alpha_1 =
j_*(\eta_2) \circ \rho_2 \circ \alpha_1
\end{equation*}
and already the composition
\begin{equation*}
\rho_2\circ\alpha_1 : \CO_X \xrightarrow{\ \alpha_1\ } \CO_E(1,0) \xrightarrow{\ \rho_2\ } \CO_{P_2}
\end{equation*}
is zero since $\alpha_1$ vanishes at point $P_2$. The same argument applies to $\alpha_2\otimes\beta_1$.

So, to finish the proof of the Proposition it remains to check that the products of $\alpha_1\otimes\beta_1$ 
and $\alpha_2\otimes\beta_2$ are linearly independent in the space $\Ext^1(\CO_X,j_*\CO_C(-3))$. For this we note that 
\begin{equation*}
m(\alpha_i\otimes\beta_i) = 
\beta_i\circ\alpha_i =
j_*(\eta_i) \circ \rho_i \circ \alpha_i
\end{equation*}
and that the composition
\begin{equation*}
\rho_i\circ\alpha_i : \CO_X \xrightarrow{\ \alpha_i\ } \CO_E(1,0) \xrightarrow{\ \rho_i\ } \CO_{P_i}
\end{equation*}
is equal to the canonical evaluation map $e_i:\CO_X \to \CO_{P_i}$. Finally, applying the functor $\Hom(\CO_X,-)$ to sequence~\eqref{cext}, we obtain
an exact sequence
\begin{equation*}
\dots \to \Hom(\CO_X,j_*\CO_C(-1)) \to \Hom(\CO_X,\CO_{P_1}) \oplus \Hom(\CO_X,\CO_{P_2}) \to \Ext^1(\CO_X,j_*\CO_C(-3)) \to \dots
\end{equation*}
Again, its first term is zero since the line bundle $\CO_C(-1)$ on the rational curve $C$ is acyclic,
hence the second map is an embedding. This means that the images $j_*(\eta_i)\circ e_i$ of the 
canonical evaluation maps $e_i$ are linearly independent in $\Ext^1(\CO_X,j_*\CO_C(-3))$. This precisely
means that $m(\alpha_i\otimes\beta_i)$ are linearly independent.
\end{proof}


\begin{thebibliography}{XXXXX}

\bibitem[BBS]{BBS} Ch. B\"ohning, H.-Ch. Graf von Bothmer, P. Sosna,
{\em On the Jordan--H\"older property for geometric derived categories},
preprint arXiv:1212.4604.

\bibitem[BK]{BK} A. Bondal, M. Kapranov,
{\em Representable functors, Serre functors, and reconstructions},
(Russian)  Izv. Akad. Nauk SSSR Ser. Mat. {\bf 53} (1989), no. 6, 1183--1205, 1337;
translation in  Math. USSR-Izv. {\bf 35} (1990), no. 3, 519--541.


\bibitem[Bri]{Bri} T. Bridgeland,
{\em Equivalences of triangulated categories and Fourier--Mukai transforms},
Bulletin of the London Mathematical Society 31.1 (1999): 25--34.

\bibitem[Kir]{Kir} A. Kirillov Jr.,
{\em McKay correspondence and equivariant sheaves on $\PP^1$},
Moscow Mathematical Journal, Volume 6 (2006), Number 3, 505--529.


% \bibitem[BBKS]{BBKS} Ch. B\"ohning, H.-Ch. Graf von Bothmer, L. Katzarkov, P. Sosna,
% {\em Determinantal Barlow surfaces and phantom categories},
% preprint arXiv:1210.0343.

% \bibitem[B]{B} A. Bondal,
% {\em Representations of associative algebras and coherent sheaves},
% (Russian)  Izv. Akad. Nauk SSSR Ser. Mat. {\bf 53} (1989), no. 1, 25--44;
% translation in  Math. USSR-Izv. {\bf 34}  (1990),  no. 1, 23--42.

% \bibitem[BP]{BP} A. Bondal, A.Polishchuk,
% {\em Homological properties of associative algebras: the method of helices},
% Russian Acad. Sci. Izv. Math. 42 (1994), no.~2, 219--260.

% \bibitem[GM]{GM} L. Grunenfelder, M. Mastnak,
% {\em Cohomology of abelian matched pairs and the Kac sequence}, 
% J. Algebra 276 (2004), no. 2, 706--736.

% \bibitem[GS]{GS} S. Galkin, E. Shinder,
% {\em Exceptional collections of line bundles on the Beauville surface},
% preprint arxiv:1210.3339.



%\bibitem[SGA6]{SGA6}
%Berthelot P., Grothendieck A., Illusie L.,
%{\em SGA~6. Theorie des reductions et theoreme de Riemann-Roch}
%LNM 225, Springer, 1971.

%\bibitem[BN]{BN} B\"okstedt M., Neeman A.,
%{\em Homotopy limits in triangulated categories},
%Compositio Math.\ {\bf 86} (1993),  no.~2, 209--234.


% \bibitem[BK2]{BK-E} A. Bondal, M. Kapranov,
% {\em Framed triangulated categories},
% (Russian)  Mat. Sb.  {\bf 181}  (1990),  no. 5, 669--683;
% translation in  Math. USSR-Sb. {\bf 70} (1991),  no. 1, 93--107.
% 
% \bibitem[BO1]{BO1} A. Bondal, D. Orlov,
% {\em Semiorthogonal decomposition for algebraic varieties},
% preprint math.AG/9506012.
% 
% \bibitem[BO2]{BO2} A. Bondal, D. Orlov,
% {\em Derived categories of coherent sheaves},
% Proceedings of the International Congress of Mathematicians,
% Vol. II (Beijing, 2002), 47--56, Higher Ed. Press, Beijing, 2002.
% 
% 
%\bibitem[BO4]{BO4} A. Bondal, D. Orlov,
%{\em Reconstruction of a variety from the derived category and groups of autoequivalences},
%Compositio Math. {\bf 125} (2001), no. 3, 327--344.

% \bibitem[BV]{BV} A. Bondal, M. Van den Bergh,
% {\em Generators and representability of functors in commutative and noncommutative geometry},
% Mosc. Math. J. {\bf 3} (2003), no. 1, 1--36, 258.

%\bibitem[GM]{GM} I. Gelfand, Yu. Manin,








%\bibitem[K1]{K-V14} A. Kuznetsov,
%{\em Derived categories of cubic and $V_{14}$ threefolds},
%Proc. V.A.Steklov Inst. Math, V.~246 (2004), p.~183--207.

% \bibitem[K1]{K-HS} A. Kuznetsov,
% {\em Hyperplane sections and derived categories},
% Izvestiya RAN: Ser. Mat. 70:3 p.~23--128 (in Russian);
% translation in Izvestiya: Mathematics 70:3 p.~447--547.

%\bibitem[K3]{K3} A. Kuznetsov,
%{\em Lefschetz decompositions and categorical resolutions of singularities},
%preprint math.AG/0609240.


%\bibitem[K5]{K-CY} A. Kuznetsov,
%{\em Fractional Calabi--Yau categories from Lefschetz decompositions},
%in preparation.

% \bibitem[K2]{K-HPD} A. Kuznetsov,
% {\em Homological projective duality},
% Publications Mathematiques de L'IHES, {\bf 105}, n. 1 (2007), 157--220.

% \bibitem[K3]{K-Q} A. Kuznetsov,
% {\em Derived categories of quadric fibrations and intersections of quadrics},
% Advances in Mathematics,  V.~218 (2008), N.~5, 1340-1369.

% \bibitem[Ke]{Ke} B. Keller,
% {\em On differential graded categories},
% International Congress of Mathematicians. Vol. II, 151–190, Eur. Math. Soc., Zürich, 2006. 

% \bibitem[KS]{KS} M. Kontsevich, Ya. Soibelman,
% {\em Deformations of algebras over operads and the Deligne conjecture},
% Conf\'erence Mosh\'e Flato 1999, Vol. I (Dijon), 255–307, 
% Math. Phys. Stud., 21, Kluwer Acad. Publ., Dordrecht, 2000. 

% \bibitem[K1]{K-FBC} A. Kuznetsov,
% {\em Base change for semiorthogonal decompositions},
% Compositio Mathematica, Volume 147, Issue 03, (2011), 852--876.
% % preprint math.AG/0711.1734.

% \bibitem[K2]{K-HH} A. Kuznetsov,
% {\em Hochschild homology and semiorthogonal decompositions},
% preprint arxiv:0904.4330.


%\bibitem[K2]{K2} A. Kuznetsov,
%{\em Homological projective duality},
%to appear in Publications Mathematiques de l'IHES, preprint math.AG/0507292.
%
%\bibitem[K3]{K3} A. Kuznetsov,
%{\em Lefschetz decompositions and categorical resolutions of singularities},
%preprint math.AG/0609240.

\bibitem[Kuz]{KECM} A. Kuznetsov,
{\em Derived categories and rationality of cubic fourfolds},
slides of the talk at 5ECM, available at \verb"http://www.mi.ras.ru/~akuznet/publications/5ecm.pdf".

%
%\bibitem[K4]{FBC} Faithful base change
%
%\bibitem[K5]{V14}



\bibitem[Oka]{Oka}
Sh. Okawa, 
{\em Semi-orthogonal decomposability of the derived category of a curve}, 
Advances in Mathematics 228.5 (2011): 2869--2873.

\bibitem[Or]{Or}
D. Orlov,
{\em Projective bundles, monoidal transformations, and derived categories of coherent sheaves},
Izvestiya: Mathematics 41.1 (1993): 133--141.

\end{thebibliography}






\end{document}

1304.0903.extracted_bib
\begin{thebibliography}{XXXXX}

\bibitem[BBS]{BBS} Ch. B\"ohning, H.-Ch. Graf von Bothmer, P. Sosna,
{\em On the Jordan--H\"older property for geometric derived categories},
preprint arXiv:1212.4604.

\bibitem[BK]{BK} A. Bondal, M. Kapranov,
{\em Representable functors, Serre functors, and reconstructions},
(Russian)  Izv. Akad. Nauk SSSR Ser. Mat. {\bf 53} (1989), no. 6, 1183--1205, 1337;
translation in  Math. USSR-Izv. {\bf 35} (1990), no. 3, 519--541.


\bibitem[Bri]{Bri} T. Bridgeland,
{\em Equivalences of triangulated categories and Fourier--Mukai transforms},
Bulletin of the London Mathematical Society 31.1 (1999): 25--34.

\bibitem[Kir]{Kir} A. Kirillov Jr.,
{\em McKay correspondence and equivariant sheaves on $\PP^1$},
Moscow Mathematical Journal, Volume 6 (2006), Number 3, 505--529.


% \bibitem[BBKS]{BBKS} Ch. B\"ohning, H.-Ch. Graf von Bothmer, L. Katzarkov, P. Sosna,
% {\em Determinantal Barlow surfaces and phantom categories},
% preprint arXiv:1210.0343.

% \bibitem[B]{B} A. Bondal,
% {\em Representations of associative algebras and coherent sheaves},
% (Russian)  Izv. Akad. Nauk SSSR Ser. Mat. {\bf 53} (1989), no. 1, 25--44;
% translation in  Math. USSR-Izv. {\bf 34}  (1990),  no. 1, 23--42.

% \bibitem[BP]{BP} A. Bondal, A.Polishchuk,
% {\em Homological properties of associative algebras: the method of helices},
% Russian Acad. Sci. Izv. Math. 42 (1994), no.~2, 219--260.

% \bibitem[GM]{GM} L. Grunenfelder, M. Mastnak,
% {\em Cohomology of abelian matched pairs and the Kac sequence}, 
% J. Algebra 276 (2004), no. 2, 706--736.

% \bibitem[GS]{GS} S. Galkin, E. Shinder,
% {\em Exceptional collections of line bundles on the Beauville surface},
% preprint arxiv:1210.3339.



%\bibitem[SGA6]{SGA6}
%Berthelot P., Grothendieck A., Illusie L.,
%{\em SGA~6. Theorie des reductions et theoreme de Riemann-Roch}
%LNM 225, Springer, 1971.

%\bibitem[BN]{BN} B\"okstedt M., Neeman A.,
%{\em Homotopy limits in triangulated categories},
%Compositio Math.\ {\bf 86} (1993),  no.~2, 209--234.


% \bibitem[BK2]{BK-E} A. Bondal, M. Kapranov,
% {\em Framed triangulated categories},
% (Russian)  Mat. Sb.  {\bf 181}  (1990),  no. 5, 669--683;
% translation in  Math. USSR-Sb. {\bf 70} (1991),  no. 1, 93--107.
% 
% \bibitem[BO1]{BO1} A. Bondal, D. Orlov,
% {\em Semiorthogonal decomposition for algebraic varieties},
% preprint math.AG/9506012.
% 
% \bibitem[BO2]{BO2} A. Bondal, D. Orlov,
% {\em Derived categories of coherent sheaves},
% Proceedings of the International Congress of Mathematicians,
% Vol. II (Beijing, 2002), 47--56, Higher Ed. Press, Beijing, 2002.
% 
% 
%\bibitem[BO4]{BO4} A. Bondal, D. Orlov,
%{\em Reconstruction of a variety from the derived category and groups of autoequivalences},
%Compositio Math. {\bf 125} (2001), no. 3, 327--344.

% \bibitem[BV]{BV} A. Bondal, M. Van den Bergh,
% {\em Generators and representability of functors in commutative and noncommutative geometry},
% Mosc. Math. J. {\bf 3} (2003), no. 1, 1--36, 258.

%\bibitem[GM]{GM} I. Gelfand, Yu. Manin,








%\bibitem[K1]{K-V14} A. Kuznetsov,
%{\em Derived categories of cubic and $V_{14}$ threefolds},
%Proc. V.A.Steklov Inst. Math, V.~246 (2004), p.~183--207.

% \bibitem[K1]{K-HS} A. Kuznetsov,
% {\em Hyperplane sections and derived categories},
% Izvestiya RAN: Ser. Mat. 70:3 p.~23--128 (in Russian);
% translation in Izvestiya: Mathematics 70:3 p.~447--547.

%\bibitem[K3]{K3} A. Kuznetsov,
%{\em Lefschetz decompositions and categorical resolutions of singularities},
%preprint math.AG/0609240.


%\bibitem[K5]{K-CY} A. Kuznetsov,
%{\em Fractional Calabi--Yau categories from Lefschetz decompositions},
%in preparation.

% \bibitem[K2]{K-HPD} A. Kuznetsov,
% {\em Homological projective duality},
% Publications Mathematiques de L'IHES, {\bf 105}, n. 1 (2007), 157--220.

% \bibitem[K3]{K-Q} A. Kuznetsov,
% {\em Derived categories of quadric fibrations and intersections of quadrics},
% Advances in Mathematics,  V.~218 (2008), N.~5, 1340-1369.

% \bibitem[Ke]{Ke} B. Keller,
% {\em On differential graded categories},
% International Congress of Mathematicians. Vol. II, 151–190, Eur. Math. Soc., Zürich, 2006. 

% \bibitem[KS]{KS} M. Kontsevich, Ya. Soibelman,
% {\em Deformations of algebras over operads and the Deligne conjecture},
% Conf\'erence Mosh\'e Flato 1999, Vol. I (Dijon), 255–307, 
% Math. Phys. Stud., 21, Kluwer Acad. Publ., Dordrecht, 2000. 

% \bibitem[K1]{K-FBC} A. Kuznetsov,
% {\em Base change for semiorthogonal decompositions},
% Compositio Mathematica, Volume 147, Issue 03, (2011), 852--876.
% % preprint math.AG/0711.1734.

% \bibitem[K2]{K-HH} A. Kuznetsov,
% {\em Hochschild homology and semiorthogonal decompositions},
% preprint arxiv:0904.4330.


%\bibitem[K2]{K2} A. Kuznetsov,
%{\em Homological projective duality},
%to appear in Publications Mathematiques de l'IHES, preprint math.AG/0507292.
%
%\bibitem[K3]{K3} A. Kuznetsov,
%{\em Lefschetz decompositions and categorical resolutions of singularities},
%preprint math.AG/0609240.

\bibitem[Kuz]{KECM} A. Kuznetsov,
{\em Derived categories and rationality of cubic fourfolds},
slides of the talk at 5ECM, available at \verb"http://www.mi.ras.ru/~akuznet/publications/5ecm.pdf".

%
%\bibitem[K4]{FBC} Faithful base change
%
%\bibitem[K5]{V14}



\bibitem[Oka]{Oka}
Sh. Okawa, 
{\em Semi-orthogonal decomposability of the derived category of a curve}, 
Advances in Mathematics 228.5 (2011): 2869--2873.

\bibitem[Or]{Or}
D. Orlov,
{\em Projective bundles, monoidal transformations, and derived categories of coherent sheaves},
Izvestiya: Mathematics 41.1 (1993): 133--141.

\end{thebibliography}