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\TopicFR{Algèbre, Géométrie algébrique}
\TopicEN{Algebra, Algebraic geometry}

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\title{Tensor weight structures and t-structures on the derived categories of schemes}
\alttitle{Structures de poids tensorielles et $t$-structures sur les cat{\'e}gories d{\'e}riv{\'e}es des sch{\'e}mas}

\author{\firstname{Umesh} \middlename{V.} \lastname{Dubey}}
\address{Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj 211019, India}
\email{umeshdubey@hri.res.in}

\author{\firstname{Gopinath} \lastname{Sahoo}\IsCorresp}
\address[1]{Harish-Chandra Research Institute, A CI of Homi Bhabha National Institute, Chhatnag Road, Jhunsi, Prayagraj 211019, India}
\email{gopinathsahoo@hri.res.in}


\subjclass{14F08, 18G80}

\begin{abstract}
We give a condition which characterises those weight structures on a derived category which come from a Thomason filtration on the underlying scheme. Weight structures satisfying our condition will be called $\otimes ^c$-weight structures. More precisely, for a Noetherian separated scheme $X$, we give a bijection between the set of compactly generated $\otimes ^c$-weight structures on $\mathbf{D} (\mathrm{Qcoh}\, X)$ and the set of Thomason filtrations of $X$. We achieve this classification in two steps. First, we show that the bijection~\cite[Theorem~4.10]{SP16} restricts to give a bijection between the set of compactly generated $\otimes ^c$-weight structures and the set of compactly generated tensor t-structures. We then use our earlier classification of compactly generated tensor t-structures to obtain the desired result. We also study some immediate consequences of these classifications in the particular case of the projective line. We show that in contrast to the case of tensor t-structures, there are no non-trivial tensor weight structures on $\mathbf{D}^b (\mathrm{Coh}\, \mathbb{P}^1_k)$.
\end{abstract}

\begin{altabstract}
On d{\'e}gage une condition qui caract{\'e}rise les structures de poids sur une cat{\'e}gorie d{\'e}riv{\'e}e qui proviennent d'une filtration de Thomason sur le sch{\'e}ma sous-jacent. Les structures de poids satisfaisant notre condition s'appelleront des $\otimes^c$-structures de poids. Plus pr{\'e}cis{\'e}ment, pour tout sch{\'e}ma s{\'e}par{\'e} et noeth{\'e}rien $X$, nous construisons une bijection entre l'ensemble des $\otimes^c$-structures de poids {\`a} engendrement compact sur $\mathbf{D} (\mathrm{Qcoh}\, X)$ et l'ensemble des filtrations de Thomason sur $X$. La construction se fait en deux {\'e}tapes. On montre d'abord que la bijection de~\cite[Theorem~4.10]{SP16} donne par restriction une bijection entre l'ensemble des $\otimes^c$-structures de poids {\`a} engendrement compact et l'ensemble des t-structures tensorielles {\`a} engendrement compact. Nous utilisons ensuite notre classification pr{\'e}c{\'e}dente des $\otimes^c$-structures de poids {\`a} engendrement compact pour arriver au r{\'e}sultat. Nous {\'e}tudions aussi quelques cons{\'e}quences imm{\'e}diates dans le cas particulier de la droite projective. Nous montrons que, contrairement au cas des t-structures tensorielles, il n'y a pas de structure de poids tensorielle non-triviale sur $\mathbf{D}^b (\mathrm{Coh}\,\mathbb{P}^1_k)$.
\end{altabstract}

\begin{document}
\maketitle

\section{Introduction}

Weight structures on triangulated categories were introduced by Bondarko~\cite{Bon10} as an important natural counterpart of t-structures with applications to Voevodsky's category of motives. Pauksztello independently came up with the same notion while trying to obtain a dual version of a result due to Hoshino, Kato and Miyachi; he termed it co-t-structures, see~\cite{Pau08}. It has been observed by Bondarko that the two notions, t-structures and weight structures, are connected by interesting relations. In this vein, \v{S}\v{t}ov\'{\i}\v{c}ek and Posp\'{\i}\v{s}il have proved for a certain class of triangulated categories, the collection of compactly generated t-structures and compactly generated weight structures are in bijection~\cite[Theorem~4.10]{SP16} with each other, where the bijection goes via a duality at the compact level. In particular, this bijection holds in the derived category of a Noetherian ring $R$ and since in this case, we have the classification of compactly generated t-structures in terms of Thomason filtrations of $\spec R$~\cite[Theorem~3.11]{AJS10}, they obtain a classification of compactly generated weight structures of $\mathbf{D}(R)$.

Our aim in this short article is twofold: first to generalize the theorem of \v{S}\v{t}ov\'{\i}\v{c}ek and Posp\'{\i}\v{s}il~\cite[Theorem~4.15]{SP16} to the case of separated Noetherian schemes, and second to understand the two types of notions, in the simplest non-affine situation: the derived category of the projective line over a field $k$. Our interest in this special case arose partly from the work of Krause and Stevenson~\cite{KS19}, where the authors study the localizing subcategories of $\mathbf{D}(\Qcoh \mathbb{P} ^1 _k)$, and partly from our desire to better understand the general results.

In our earlier work~\cite{DS22}, we have shown that a t-structure on $\deriveq X$ supported on a Thomason filtration of a Noetherian scheme $X$ satisfies a tensor condition. We call them tensor t-structures. In this article, we introduce the analogous notion of tensor weight structures, also a slightly weaker notion which we call $\otimes ^c$-weight structures. We then show that the bijection~\cite[Theorem~4.10]{SP16} restricts to a bijection between tensor t-structures and $\otimes ^c$-weight structures; this can be seen as a consequence of our Lemma~\ref{preaisle and copreaisle} and Theorem~\ref{t-structure}. Next, we specialize to the case of the derived categories of separated Noetherian schemes and classify compactly generated $\otimes ^c$-weight structures in this case, see Theorem~\ref{tensor weight structure}.

In the last section, we apply all the general theory and the classification results to the derived category of the projective line over a field $k$. By our Theorem~\ref{t-structure} classifying compactly generated tensor t-structures of $\mathbf{D} (\Qcoh \mathbb{P}^1_k)$ is equivalent to classifying thick
\tensor preaisles of $\deriveb$, so we restrict our attention to \tensor preaisles of $\deriveb$. We give a complete description of the \tensor preaisles in Proposition~\ref{Main}, and in Proposition~\ref{aisle} we determine which of these are aisles or in other words give rise to t-structures on $\deriveb$. The result of Proposition~\ref{aisle} is not new, it can possibly be deduced from~\cite{Bez10}; also in~\cite{Rudakov}, the authors describe the bounded t-structures on $\deriveb$ by using the classification of t-stabilities on $\deriveb$. Finally, we consider the same question for tensor weight structures, and to our surprise, we discovered that on $\deriveb$ there are no non-trivial tensor weight structures.


\section{Preliminaries}

Let \cat T be a triangulated category and $\cat T^c$ denote the full subcategory of compact objects. We recall the definition of t-structures which was introduced in~\cite{BBD}.

\begin{defi}
A \emph{t-structure} on \cat T is a pair of full subcategories $(\cat U, \cat V)$ satisfying the following properties:
\begin{enumerate}\renewcommand*{\theenumi}{t\arabic{enumi}}\relabel
\item \label{t1} $\Sigma \cat U \subset \cat U$ and $\Sigma^{-1} \cat V \subset \cat V$.
\item \label{t2} $\cat U \perp \Sigma^{-1} \cat V$.
\item \label{t3} For any $T \in \cat T$ there is a distinguished triangle
$
U \rar T \rar V \rar \Sigma U
$
where $U \in \cat U$ and $V \in \Sigma^{-1} \cat V$. We call such a triangle
\emph{truncation decomposition} of $T$.
\end{enumerate}
\end{defi}

Next, we quote the definition of weight structures from~\cite{Bondarko18}.

\begin{defi}
A \emph{weight structure} on \cat T is a pair of full subcategories $(\cat X, \cat Y)$ satisfying the following properties:
\begin{enumerate}\renewcommand*{\theenumi}{w\arabic{enumi}}\relabel
\setcounter{enumi}{-1}
\item \label{w0} \cat X and \cat Y are closed under direct summands.
\item \label{w1} $\Sigma^{-1} \cat X \subset \cat X$ and $ \Sigma \cat Y \subset \cat Y$.
\item \label{w2} $\cat X \perp \Sigma \cat Y$.
\item \label{w3} For any object $T\in \cat T$ there is a distinguished triangle
$
X \rar T \rar Y \rar \Sigma X
$
where $X \in \cat X$ and $Y \in \Sigma \cat Y$. The above triangle is called a \emph{weight decomposition} of $T$.
\end{enumerate}
\end{defi}

Note that if $(\cat U, \cat V)$ is a t-structure on \cat T then $(\cat V, \cat U)$ is a t-structure on $\cat T ^{op}$. Similarly, If $(\cat X, \cat Y)$ is a weight structure on \cat T then $(\cat Y, \cat X)$ is a weight structure on$\cat T ^{op}$.

For any subcategory \cat U of \cat T, we denote $\cat U ^{\perp}$ to be the full subcategory consisting of objects $B \in \cat T$ such that $\Hom (A, B) = 0$ for all $A \in \cat U$. Analogously we define $^{\perp} \cat U $ to be the full subcategory of objects $B \in \cat T$ such that $\Hom (B, A) = 0$ for all $A \in \cat U$.


\begin{defi}
We say a t-structure $(\cat U, \cat V)$ is \emph{compactly generated} if there is a set of compact objects $\cat S$ such that $\cat U = { }^{\perp}(\cat S ^{\perp})$. A weight structure $(\cat X, \cat Y)$ is \emph{compactly generated} if there is a set of compact objects $\cat S$ such that $\cat X = {}^{\perp}(\cat S ^{\perp})$.
\end{defi}

\begin{defi}
A subcategory $\cat U$ of $\cat T$ is a \emph{preaisle} if it is closed under positive shifts and extensions. Dually, we say $\cat U$ is a \emph{copreaisle} of $\cat T$, if $\cat U$ is a preaisle of $\cat T^{op}$.

A preaisle is called \emph{thick} if it is closed under direct summands. We say a preaisle is \emph{cocomplete} if it is closed under coproducts in $\cat T$, and \emph{complete} if it is closed under products. Similarly, we define
\emph{thick}, \emph{cocomplete} and \emph{complete} copreaisles.
\end{defi}

For a t-structure $(\cat U, \cat V)$ the subcategory $\cat U$ is a cocomplete preaisle of $\cat T$, and for a weight structure $(\cat X, \cat Y)$ the subcategory $\cat X$ is a cocomplete copreaisle of $\cat T$.

We need the notion of stable derivators to formulate the next theorem but our requirement of the theory of derivators is the bare minimum. We will not go into the precise lengthy definition here, instead, we refer the reader to~\cite[Section~2.1]{SP16} and references therein.

\begin{theo}[{\cite[Theorem~4.5]{SP16}}]\label{SP theorem}
Let $\cat T = \mathbb{D}(e)$, where $\mathbb{D}$ is a stable derivator such that for each small category $I$, $\mathbb{D}(I)$ has all small coproducts. Then,
\begin{enumerate}\romanenumi
\item \label{5i} There is a bijection between the set of compactly generated t-structures of $\cat T$ and the set of thick preaisles of $\cat T^c$ given by
\[
(\cat U, \cat V) \mapsto \cat U \cap \cat T^c
\qquad 
\cat P \mapsto (^{\perp} (\cat P ^{\perp}), \Sigma \cat P ^{\perp}).
\]

\item \label{5ii} There is a bijection between the set of compactly generated weight structures of $\cat T$ and the set of thick copreaisles of $\cat T^c$ given by
\[
(\cat X, \cat Y) \mapsto \cat X \cap \cat T^c
\qquad
\cat P \mapsto (^{\perp} (\cat P ^{\perp}), \Sigma^{-1} \cat P ^{\perp}).
\]
\end{enumerate}
\end{theo}


\section{Tensor weight and t-structures}

We recall the definition of tensor triangulated category from~\cite[Definition~A.2.1]{HPS97}.

\begin{defi}
A \emph{tensor triangulated category} $(\cat T, \otimes, \mathbf{1})$ is a triangulated category with a compatible closed symmetric monoidal structure. This means there is a functor $- \otimes - \colon \cat T \times \cat T \to \cat T$ which is triangulated in both the variables and satisfies certain compatibility conditions. Moreover, for each $B \in \cat T$ the functor $- \otimes B$ has a right adjoint which we denote by $\scrHom (B, -) $. The functor $\scrHom (-,-)$ is triangulated in both the variables, and for any $A$, $B$, and $C$ in $\cat T$ we have natural isomorphisms $ \Hom (A \otimes B, C) \rar
\Hom (A, \scrHom (B, C))$.
\end{defi}

\begin{defi}\label{defi}
Let $\cat T$ be a tensor triangulated category given with a preaisle $\aisle T$ satisfying
\[
\aisle T \otimes \aisle T \subset \aisle T\text{ and }\mathbf{1} \in \aisle T.
\]

A preaisle $\cat U$ of $\cat T$ is a \emph {\tensor preaisle (with respect to $\aisle T$)} if $\aisle T \otimes \cat U \subset \cat U$. We say a copreaisle $\cat X$ of $\cat T$ is a \emph{\tensor copreaisle (with respect to $\aisle T$) } if $\scrHom(\aisle T, \cat X) \subset \cat X$. A t-structure $(\cat U, \cat V)$ is a called a \emph{tensor t-structure} if $\cat U$ is a \tensor preaisle, and a weight structure $(\cat X,
\cat Y)$ is a \emph{tensor weight structure} if $\cat X$ is a \tensor copreaisle.
\end{defi}

Let \cat S $\subset$ \cat T be a class of objects. We denote the smallest cocomplete preaisle containing \cat S by \preaisle S and call it the \emph{cocomplete preaisle generated by \cat S.} If $\cat T$ does not have coproducts we denote $\preaisle S$ to be the smallest preaisle containing $\cat S$.

\begin{lemm}\label{tensor generator 1}
Let \aisle {\cat T} be generated by a set of objects \cat K, that is, \aisle {\cat T} =
\preaisle{\cat K}. Then
\begin{enumerate}\romanenumi
\item \label{8i} a cocomplete preaisle \cat U of \cat T is a $\otimes$-preaisle if and only if \cat K $\otimes$ \cat U $\subset \cat U$.

\item \label{8ii} a complete copreaisle $\cat X$ of $\cat T$ is a \tensor copreaisle if and only if $\scrHom(\cat K, \cat X) \subset \cat X$.
\end{enumerate}
\end{lemm}

\begin{proof}
\begin{proof}[Part~\meqref{8i}]
Suppose \cat K $\otimes$ \cat U $\subset \cat U$. We define $\cat B$ = $\{ X \in \aisle T
\mid X \otimes \cat U \subset \cat U\}$. Since $\cat U$ is a cocomplete preaisle we can observe that $\cat B$ is also a cocomplete preaisle. Now, by our assumption $\cat K \subset
\cat B$ so we get $\aisle T \subset \cat B$ which proves $\cat U$ is \tensor preaisle. The converse is immediate.
\let\qed\relax
\end{proof}

\begin{proof}[Part~\meqref{8ii}]
Let $\cat B = \{ X \in \aisle T \mid \scrHom (X, \cat X)
\subset \cat X\}$. Since $\cat X$ is a copreaisle we can see that $\cat B$ is a preaisle. Completeness of $\cat X$ implies $\cat B$ is cocomplete. Now, by following a similar argument as in \eqref{8i} we get $\cat X$ is \tensor copreaisle.
\end{proof}
\let\qed\relax
\end{proof}

An immediate consequence of the above lemma is if $\aisle T = \langle \mathbf{1} \rangle ^{\leq 0}$ then every cocomplete preaisle of $\cat T$ is a \tensor preaisle and every complete copreaisle of $\cat T$ is a \tensor copreaisle. In particular, for a commutative ring $R$, all cocomplete preaisle and complete copreaisle of $\mathbf{D}(R)$ satisfy the tensor condition.

\begin{defi}
We say an object $X \in \cat T$ is \emph{rigid} or \emph{strongly dualizable} if for each $Y \in \cat T$ the natural map $\mu: \scrHom(X, \mathbf{1}) \otimes Y \rar \scrHom(X, Y)$ is an isomorphism. A tensor triangulated category $(\cat T, \otimes, \mathbf{1})$ is \emph{rigidly compactly generated} if the following conditions hold:
\begin{enumerate}\romanenumi
\item \label{9i} \cat T is compactly generated;
\item \label{9ii} $\mathbf{1}$ is compact;
\item \label{9iii} every compact object is rigid.
\end{enumerate}
\end{defi}

Let $\cat T$ be a rigidly compactly generated tensor triangulated category. For such a triangulated category the tensor product on $\cat T$ restricts to $\cat T^c$ therefore $(\cat T^c, \otimes, \mathbf{1})$ is also a tensor triangulated category. Suppose $\cat T$ is given with a preaisle $\aisle T$ satisfying the condition of Definition~\ref{defi} then so does the preaisle $\cat T^c \cap \aisle T$ of $\cat T^c$. So we can define \tensor preaisles and \tensor copreaisles of $\cat T^c$ with respect to $\cat T^c \cap \aisle T$.

\begin{lemm}\label{preaisle and copreaisle}
Let $\cat T$ be a rigidly compactly generated tensor triangulated category given with a preaisle $\aisle T$ satisfying the condition of Definition~\ref{defi}.

Then, there is a one-to-one correspondence between the set of \tensor preaisles and the set of \tensor copreaisles of $\cat T^c$.
\end{lemm}

\begin{proof}

Let $\cat U$ be a full subcategory of $\cat T ^c$. We denote $\cat U^*$ the full subcategory given by
\[
\cat U^* = \bigl\{ X \in \cat T^c \,\big|\, X \cong \scrHom (Y,
\mathbf{1}) \text{ for some } Y \in \cat U \bigr\}.
\]

The assignment $\cat U \mapsto \cat U^*$ induces an equivalence between the preaisles and copreaisles of $\cat T ^c$; see~\cite[Lemma~4.9]{SP16}. We only need to show that the above assignment preserves the tensor condition.

Let $\cat U$ be a preaisle of $\cat T^c$, $X \in \cat U^*$ and $T \in \cat T^c \cap \cat T ^{\leq 0}$. We have
\[
\scrHom(T, X) \cong \scrHom (T, \scrHom (Y, \mathbf{1}))
\cong \scrHom (T \otimes Y, \mathbf{1}).
\]

If we assume $\cat U$ is a \tensor preaisle then $T \otimes Y \in \cat U$. Hence $ \scrHom(T, X) \in \cat U^*$, this proves $\cat U^* $ is a \tensor copreaisle.

Now, suppose $\cat U$ is a copreaisle. Let $X \in \cat U^*$ and $T \in \cat T^c \cap \cat T ^{\leq 0}$. Then,
\begin{align*}
T \otimes X & \cong T \otimes \scrHom(Y,\mathbf{1})\\
& \cong \scrHom(Y,T)\\
& \cong \scrHom(Y \otimes \scrHom(T, \mathbf{1}), \mathbf{1})\\
& \cong \scrHom(\scrHom(T,Y), \mathbf{1}).
\end{align*}

If we assume $\cat U$ is a \tensor copreaisle then $\scrHom(T,Y) \in \cat U$. We get, $T \otimes X \in \cat U^*$, which proves $\cat U^*$ is a \tensor preaisle.
\end{proof}

\begin{defi}
A preaisle\/ \cat U is \emph{compactly generated} if\/ $\cat U = \preaisle S$ for a set of compact objects \cat S.
\end{defi}

\begin{defi}\label{*}
We say a triangulated category $\cat T$ has the property $(*)$ if:
\begin{enumerate}\romanenumi
\item \label{12i} $\cat T$ is rigidly compactly generated;
\item \label{12ii} $\cat T$ has a preaisle $\aisle T$ satisfying $\aisle T \otimes \aisle T \subset \aisle T\text{ and }\mathbf{1} \in \aisle T$;
\item \label{12iii} $\aisle T$ is compactly generated, that is, $\aisle T = \langle \cat T^c \cap
\aisle T \rangle ^{\leq 0}$.
\end{enumerate}
\end{defi}

For a triangulated category $\cat T$ having the property $(*)$, we define a weaker notion than \tensor (co)preaisle.

\begin{defi}
Let $\cat T$ have the property $(*)$.

A preaisle $\cat U$ of $\cat T$ is a \emph{$\otimes ^c$-preaisle} if for any $T \in \cat T^c
\cap \aisle T$ and $U \in \cat U$ we have $T \otimes U \in \cat U$. Similarly a copreaisle $\cat X$ of $\cat T$ is a \emph{$\otimes ^c$-copreaisle} if for any $T \in \cat T^c \cap \aisle T $ and $X \in \cat X$ we have $\scrHom(T, X) \in \cat X$.

A t-structure $(\cat U, \cat V)$ on $\cat T$ is a \emph{$\otimes ^c$-t-structure} if $\cat U$ is a $\otimes ^c$-preaisle and a weight structure $(\cat X, \cat Y)$ on $\cat T$ is a
\emph{$\otimes ^c$-weight structure} if $\cat X$ is a $\otimes ^c$-copreaisle.
\end{defi}

\begin{rema}\label{weak}
This weaker notion gives something new only for preaisles(resp. copreaisles) which are not cocomplete(resp. complete) since by Lemma~\ref{tensor generator 1} it can easily be observed that for $\cat T$ having the property $(*)$: (i) every cocomplete $\otimes ^c$-preaisles of $\cat T$ is a $\otimes$-preaisle of $\cat T$, and (ii) every complete $\otimes ^c$-copreaisle of $\cat T$ is a \tensor copreaisle of $\cat T$.
\end{rema}

With this weaker notion, we now prove the tensor analogue of Theorem~\ref{SP theorem}.

\begin{theo}\label{t-structure}
Let $\cat T$ have the property $(*)$ (see Definition~\ref{*}) and $\cat T =
\mathbb{D}(e)$, where $\mathbb{D}$ is a stable derivator such that for each small category $I$, $\mathbb{D}(I)$ has all small coproducts. Then,
\begin{enumerate}\romanenumi
\item \label{15i} There is a bijective correspondence between the set of compactly generated tensor t-structures of $\cat T$ and the set of thick \tensor preaisles of $\cat T^c$ given by
\[
(\cat U, \cat V) \mapsto \cat U \cap \cat T^c
\qquad
\cat P \mapsto (^{\perp} (\cat P ^{\perp}), \Sigma \cat P ^{\perp}).
\]
\item \label{15ii} There is a bijective correspondence between the set of compactly generated $\otimes ^c$-weight structures of $\cat T$ and the set of thick \tensor copreaisles of $
\cat T^c$ given by
\[
(\cat X, \cat Y) \mapsto \cat X \cap \cat T^c
\qquad
\cat P \mapsto (^{\perp} (\cat P ^{\perp}), \Sigma^{-1} \cat P ^{\perp}).
\]
\end{enumerate}
\end{theo}

Before proving the theorem, we will make some comments about the lack of symmetry in the above statement:

\begin{rema}\label{remark 1}
It is easy to observe that given a subcategory $\cat P$ the subcategory ${}^{\perp} (\cat P ^{\perp})$ is always cocomplete, that is, closed under coproducts. Therefore, by Remark~\ref{weak} saying ${}^{\perp} (\cat P ^{\perp})$ is a $\otimes ^c$- preaisle of $\cat T$ is equivalent to saying it is a \tensor preaisle. However, compactly generated copreaisles are not closed under products in general, so Remark~\ref{weak} is not applicable here.
\end{rema}

\begin{exam}\!\!\!\!\footnote{We thank the referee for suggesting this example.}\,
Consider the derived category $\mathbf{D}(\Z)$, it is equivalent to the homotopy category of K-projectives or dg-projectives $\mathbf{K}(dg$-$\proj \Z)$. The copreaisle $\mathbf{K}^{\geq 0}(dg$-$\proj \Z)$ is compactly generated but it is not closed under products. Indeed, if $\mathbf{K}^{\geq 0}(dg$-$\proj \Z)$ is closed under product then the countable product of $\Z$, that is, $\prod_i \Z \in \mathbf{K}^{\geq 0}(dg$-$\proj \Z)$. Let $P^{\cdot} $ be a projective (hence K-projective) resolution of $\prod_i \Z$. Now, take the brutal truncations of $P^{\cdot}$ at degree~$0$, $\sigma ^{\geq 0}(P ^{\cdot})$ and $\sigma ^{< 0}(P^{\cdot})$. As $\sigma ^{< 0} (P^{\cdot}) \in \mathbf{K}^{< 0}(dg$-$\proj \Z)$, we have $\Hom(\prod_i \Z, \sigma ^{< 0} (P^{\cdot})) = 0$ which implies $\sigma ^{\geq 0}(P ^{\cdot})$ is isomorphic to $\prod_i \Z$. Since $\sigma ^{\geq 0}(P ^{\cdot})$ is projective we get $\prod_i \Z$ is projective, which is a contradiction; see for instance~\cite[Corollary to Theorem~3.1, p.~466]{Chase60}.
\end{exam}

\begin{proof}[Proof of Theorem~\ref{t-structure}]
\begin{proof}[Part \meqref{15i}]
It has already been shown in~\cite[Theorem~4.5$\MK$(i)]{SP16} that the above assignments are bijections between the set of compactly generated t-structures of $\cat T$ and the set of thick preaisles of $\cat T^c$. We only need to show that the assignments preserve the tensor conditions.

From the definition of \tensor preaisle, it is easy to observe that if $\cat U$ is a
\tensor preaisle of $\cat T$ then $\cat U \cap \cat T^c$ is a \tensor preaisle of $\cat T^c$. Suppose $\cat P$ is a \tensor preaisle of $\cat T^c$. Since $^{\perp} (\cat P ^{\perp})$ is a cocomplete preaisle of $\cat T$ by Remark~\ref{weak} it is enough to show $^{\perp} (\cat P ^{\perp})$ is a $\otimes ^c$-preaisle of $\cat T$. Let $\cat B = \{ X \in$ $ ^{\perp} (\cat P ^{\perp}) \mid (\cat T^c \cap \aisle T) \otimes X
\subset$ $ ^{\perp} (\cat P ^{\perp}) \}$. We note that $\cat B$ is a cocomplete preaisle containing $\cat P$. Since $^{\perp} (\cat P ^{\perp})$ is the smallest cocomplete preaisle containing $\cat P$ by~\cite[Lemma~1.9]{DS22} we get $\cat B = $ $^{\perp} (\cat P^{\perp})$.
\let\qed\relax
\end{proof}

\begin{proof}[Part \meqref{15ii}]
In view of~\cite[Theorem~4.5$\MK$(ii)]{SP16}, again we only need to show that the assignments preserve the appropriate tensor conditions. If $\cat X$ is a $\otimes ^c$-copreaisle of $\cat T$ then it is easy to observe that $\cat X \cap \cat T^c$ is a \tensor copreaisle of $\cat T^c$. Suppose $\cat P$ is a \tensor copreaisle of $\cat T^c$ we need to show that $^{\perp} (\cat P ^{\perp})$ is a $\otimes ^c$-copreaisle of $\cat T$. By~\cite[Theorem~3.7]{SP16} an object $A$ of $\cat T$ belongs to $ ^{\perp} (\cat P ^{\perp})$ if and only if $A$ is a summand of a homotopy colimit of a sequence.
\[
\begin{tikzcd}
&0 = Y_0 \arrow[r, "f_0"] & Y_1 \rar ["f_1"] & Y_2 \rar ["f_2"] &\cdots
\end{tikzcd}
\]
where each $f_i$ occurs in a triangle $ Y_i \rar Y_{i+1} \rar S_i \rar \Sigma Y_i$ with $S_i \in \Add\cat P$. First, we observe that for any compact object $T$ the functor $\scrHom(T, -)$ preserves small coproducts therefore $\scrHom(T, -)$ takes homotopy sequences to homotopy sequences. Since $\cat P$ is \tensor copreaisle of $\cat T^c$ for any $T \in \cat T^c \cap \aisle T$ we have $\scrHom(T, S_i) \in \Add \cat P$. Thus applying~\cite[Theorem~3.7]{SP16} again we get $\scrHom(T, A) \in $ $^{\perp} (\cat P ^{\perp}).$
\end{proof}
\let\qed\relax
\end{proof}


\section{The classification theorem for weight structures}

Let $X$ be a Noetherian separated scheme. $\deriveq X$ denotes the derived category of complexes of quasi coherent $\CO _X$-modules. The derived category $(\deriveq X, \otimes ^{L} _{\CO _X},\CO_X)$ is a tensor triangulated category with the derived tensor product $\otimes ^{L} _{\CO _X}$ and the structure sheaf $\CO_X$ as the unit. The full subcategory of complexes whose cohomologies vanish in positive degree $\mathbf{D}^{\leq 0}(\Qcoh X)$ is a preaisle of $\deriveq X$ satisfying the conditions of Definition~\ref{defi}. We define the \tensor preaisles and \tensor copreaisles of $\deriveq X$ with respect to $\mathbf{D}^{\leq 0}(\Qcoh X)$. Similarly, the \tensor preaisles and \tensor copreaisles of $\perfect X$ are defined with respect to $ \Perf^{\leq 0} (X)$. Note that $\deriveq X$ has the property $(*)$ (see Definition~\ref{*}), so we can define $\otimes ^c$-(co)preaisles of $\deriveq X$.

\begin{defi}
\label{D Thomason subset}
A subset $Z$ is a \emph{specialization closed} subset of $X$ if for each $x \in Z$ the closure of the singleton set $\{x\}$ is contained in $Z$, that is, $\bar{\{x\}} \subset Z$. Note that a specialization closed subset is a union of closed subsets of $X$.

A subset $Y$ is a \emph{Thomason} subset of $X$ if $Y = \bigcup _{\alpha} Y_{\alpha}$ is a union of closed subsets $Y_{\alpha}$ such that $X \setminus Y_{\alpha}$ is quasi compact. Note that if $X$ is Noetherian then the two notions coincide.
\end{defi}

\begin{defi}
A \emph{Thomason filtration} of $X$ is a map $\phi : \Z \rar 2^X$ such that $\phi(i)$ is a Thomason subset of $X$ and $\phi(i) \supset \phi (i+1)$ for all $i \in \Z$.
\end{defi}

In our earlier work we have mentioned without proof (see~\cite[Remark~4.13]{DS22}) about the following result, here we explicitly state it for future reference. This is a generalization of Thomason's classification~\cite[Theorem~3.15]{Thomason} of \tensor ideals to \tensor preaisles of $\perfect X$, for separated Noetherian scheme $X$.

\begin{prop}\label{tensor preaisle}
Let $X$ be a separated Noetherian scheme. The assignment sending a Thomason filtration $ \phi$ to $ \cat S_{\phi} = \{ E \in \perfect X \:|\, \Supp (H^i(E)) \subset \phi (i) \} $ provides a one-to-one correspondence between the following sets:
\begin{enumerate}\romanenumi
\item \label{20i} the set of Thomason filtrations of $X$;
\item \label{20ii} the set of thick \tensor preaisles of $\perfect X$.
\end{enumerate}
\end{prop}

\begin{proof}
In~\cite[Theorem~4.11]{DS22} we have shown that sending $\phi$ to
\[
\cat U_{\phi} = \{ E \in \deriveq X \mid \Supp (H^i(E)) \subset \phi (i) \}
\]
provides a bijection between the set of Thomason filtration of $X$ and the set of compactly generated tensor t-structure of $\deriveq X$. From part~\eqref{15i} of Theorem~\ref{t-structure}, we conclude that the above assignment provides a bijection between Thomason filtrations of $X$ and thick \tensor preaisles of $\perfect X$.
\end{proof}

\begin{theo}\label{tensor weight structure}
Let $X$ be a separated Noetherian scheme. There is a one-to-one correspondence between the following sets:
\begin{enumerate}\romanenumi
\item \label{21i} the set of Thomason filtrations of $X$;
\item \label{21ii} the set of compactly generated $\otimes ^c$-weight structures of\/ $\deriveq X$.
\end{enumerate}
The assignment is given by
\[
\phi \mapsto (\cat A_{\phi}, \cat B_{\phi})
\]
where
\begin{align*}
\cat B _{\phi} &= \bigl\{ B \in \deriveq X \,\big|\, \Hom (\CO_X, S \otimes ^{L} _{\CO _X} B) = 0 \text{ for all } S \in \cat S_\phi \bigr\},\\
\cat S_\phi &= \bigl\{S \in \perfect X \,\big|\, \Supp (H^iS) \subset \phi (i) \bigr\},\text{ and}\\
\cat A_\phi &= \bigl\{ A \in \deriveq X \,\big|\, \Hom (A, B) = 0 \text{ for all } B \in \cat B_{\phi} \bigr\}.
\end{align*}
\end{theo}

\begin{proof}
Let $\phi$ be a Thomason filtration of $X$. By Proposition~\ref{tensor preaisle} we know $\phi \mapsto \cat S_{\phi}$ is a bijection. Now sending $\cat S_{\phi}$ to $\cat S_{\phi} ^*$ is again a bijection by Lemma~\ref{preaisle and copreaisle}. Since $\cat S_{\phi} ^*$ is a \tensor copreaisle of $\perfect X$, the assignment $
\cat S_{\phi} ^* \mapsto (^{\perp}((\cat S_{\phi} ^*) ^{\perp}),(\cat S_{\phi} ^*)^{\perp})$ is a bijection by Theorem~\ref{t-structure}. We only need to show that $\cat B_{\phi} = (\cat S_{\phi} ^*)^{\perp}$ which is the consequence of the tensor-hom adjunction.
\end{proof}


\section{In the case of projective line}

In this section, we specialize to the case of projective line $\mathbb{P}^1_k$ over a field $k$. By the results of earlier sections, classifying compactly generated tensor t-structures of $\mathbf{D}(\Qcoh\mathbb{P}^1_k)$ is equivalent to classifying thick
\tensor preaisles of $\perfect {\mathbb{P} ^1 _k}$. For any smooth Noetherian scheme $X$ the inclusion functor from $\perfect X$ to the derived category of bounded complexes of coherent sheaves $\mathbf{D}^b(\Coh X)$ is an equivalence. Therefore, we restrict our attention to $\deriveb$. Note that we define \tensor preaisles of $\deriveb$ with respect to the standard preaisle
\[
\mathbf{D}^{b, \leq 0} (\Coh \mathbb{P}^1_k) \colonequals \bigl\{ E \in
\deriveb \,\big|\, H^i(E) = 0 \forall i > 0\bigr\}.
\]

\begin{lemm}\label{ch lemma}
A thick preaisle $\cat A$ of\/ $\deriveb$ is a \tensor preaisle if and only if
\[
\CO (-1) \otimes \cat A \subset \cat A.
\]
\end{lemm}

\begin{proof}
Suppose $\cat A$ is a \tensor preaisle then $\CO (-1) \otimes \cat A \subset \cat A$ is true by definition. Conversely, suppose $\cat A$ is a preaisle of $\deriveb$. Take $\cat B \colonequals \{ B \in \mathbf{D}^{b, \leq 0} (\Coh \mathbb{P}^1_k) \mid B \otimes \cat A \subset \cat A\}$. From our assumption, we have $\CO (-1) \in \cat B$. It is now easy to see that for every $n \geq 0$ we have $
\CO (-n) \in \cat B$.

As $\Coh\mathbb{P}^1 _k$ has homological dimension one, every complex of $\deriveb$ is quasi isomorphic to the direct sum of its cohomology sheaves, see~\cite[Proposition~6.1]{Rudakov}. Also, every coherent sheaf over $\mathbb{P}^1_k$ is the direct sum of line bundles and torsion sheaves. Since $\cat B$ is a preaisle, to show $\cat B = \mathbf{D}^{b, \leq 0} (\Coh \mathbb{P}^1_k)$ it is enough to show that $\cat B$ contains all the line bundles and torsion sheaves.

For any $m \geq 0$ consider the following triangle coming from the corresponding short exact sequence in $\Coh \mathbb{P}^1 _k$; see for instance~\cite[Equation 6.3]{Rudakov},
\[
\CO (-2) ^{\oplus (m +1)} \longrightarrow \CO (-1)^{\oplus (m+2)} \longrightarrow \CO (m)
\longrightarrow \CO (-2)[1].
\]
Since $\cat B$ is closed under extensions and positive shifts we have $\CO (m) \in \cat B$.

Next, for any indecomposable torsion sheaf of degree $d$ say $ T_x$, which is supported on a closed point $x \in \mathbb{P}^1_k$, consider the following triangle coming from the corresponding short exact sequence in $\Coh \mathbb{P}^1 _k$; see~\cite[Equation 6.5]{Rudakov},
\[
\CO (-2)^{\oplus d} \longrightarrow \CO (-1) ^{\oplus d} \longrightarrow T_x
\longrightarrow \CO (-2)[1].
\]

Again using the fact that $\cat B$ is closed under extensions and positive shifts we have $T_x \in \cat B$.
\end{proof}

Recall that for a set of objects $\cat S$ of $\cat T$ we denote the smallest cocomplete preaisle containing $\cat S $ by $\preaisle S$. If $\cat T$ does not have coproducts, for instance $\deriveb$, we denote $\preaisle S$ to be the smallest preaisle containing $\cat S$. Similarly we denote $\langle \cat S \rangle ^{\geq 0}$ to be the smallest copreaisle containing $\cat S$. Also recall that for any subcategory $\cat U$ we denote $\cat U^*$ the full subcategory given~by
\[
\cat U^* = \bigl\{ X \in \cat T \,\big|\, X \cong \scrHom (Y,
\mathbf{1}) \text{ for some } Y \in \cat U \bigr\}.
\]

\begin{exam}
For a fixed $n \in \Z$ we denote
\begin{align*}
& \cat B_n \colonequals \langle \CO (n) \rangle ^{\leq 0}; \text{ and} \\
& \cat C_n \colonequals \langle \CO (n), \CO (n+1) \rangle ^{\leq 0}.
\end{align*}
Using Lemma~\ref{ch lemma}, we can check that $\cat B_n$ and $\cat C_n$ are not
\tensor preaisles of $\deriveb$. Similarly, $\cat B_n ^*$ and $\cat C_n ^*$ provide examples of copreaisles of $\deriveb$ which are not \tensor copreaisles. This can be observed using Lemma~\ref{preaisle and copreaisle}.
\end{exam}

Recall that a Thomason filtration of $X$ is a map $\phi : \Z \rar 2^X$ such that $\phi(i)$ is a Thomason subset of $X$ and $\phi(i) \supset \phi (i+1)$ for all $i \in \Z$. We say $\phi $ is of \emph{type-1} if $\bigcup _i \phi (i) \neq X$; and we say $\phi$ is of \emph{type-2} if $\bigcup _i \phi (i) = X$ but not all $\phi(i) =X$.

Let $x \in \mathbb{P} ^1 _k$ be a closed point. We denote the simple torsion sheaf supported on $x$ by $k(x)$. Now, we give an explicit description of the \tensor preaisles of $\deriveb$ in terms of simple torsion sheaves and line bundles.

\goodbreak
\begin{prop}\label{Main}
Any proper thick \tensor preaisle of $\deriveb$ is one of the following forms:
\begin{enumerate}\romanenumi
\item \label{24i} $\langle k(x)[-i] \mid x \in \phi (i) \rangle ^{\leq 0};$\\
where $\phi$ is a type-1 Thomason filtration of $\mathbb{P}^1_k$.
\item \label{24ii} $\langle \CO (n) [-i_0], k(x)[-i] \mid \forall n \in \Z \text{ and } x \in \phi (i) \rangle ^{\leq 0}$\\
where $\phi$ is a type-2 Thomason filtration of $\mathbb{P}^1_k$ and $i_0$ a fixed integer.
\end{enumerate}
\end{prop}

\begin{proof}
Suppose $\cat A$ is a thick \tensor preaisle of $\deriveb $. By Proposition~\ref{tensor preaisle} there is a unique Thomason filtration $\phi$ such that
\[
\cat A = \bigl\{ E \in \deriveb \,\big|\, \Supp H^i(E) \subset \phi (i) \bigr\}.
\]

Since $\Coh \mathbb{P}^1 _k$ has homological dimension one, every complex of $\deriveb$ is quasi isomorphic to the direct sum of its cohomology sheaves. Therefore, we can write $\cat A$ in terms of coherent sheaves alone,
\[
\cat A = \bigl\langle F[-i] \,\big|\, F \in \Coh \mathbb{P} ^1 _k \text{ and }
\Supp F \subset \phi (i) \bigr\rangle ^{\leq 0}.
\]

\begin{proof}[Case 1. $\phi$ is type-1]
Note that $\phi (i) \subsetneq \mathbb{P}^1 _k $ for all $i$. Every coherent sheaf over $\mathbb{P}^1_k$ is the direct sum of line bundles and torsion sheaves. Since the support of any line bundle is whole $\mathbb{P}^1 _k$. In this case, $\cat A$ contains only torsion sheaves. As torsion sheaves can be generated by simple torsion sheaves we have,
\[
\cat A = \langle k(x)[-i] \mid x \in \phi (i) \rangle ^{\leq 0}.
\]
\let\qed\relax
\end{proof}

\begin{proof}[Case 2. $\phi$ is type-2]
Since $\bigcup _i \phi (i) = X$ there is an integer $i_0$ such that $\phi(i_0)$ contains the generic point of $\mathbb{P}^1 _k$. We can take $i_0$ to be the largest such integer. Observe that $\phi (i) = \mathbb{P}^1 _k $ for all $i \leq i_0$ and $\phi(i_0+1) \subsetneq \mathbb{P}^1 _k$. Here we can check that
\[
\cat A = \langle \CO (n) [-i_0], k(x)[-i] \mid \forall n \in \Z \text{ and } x \in \phi (i) \rangle ^{\leq 0}.\qedhere
\]
\end{proof}
\let\qed\relax
\end{proof}

Next, we will show which of these \tensor preaisles of $\deriveb$ are t-structures on $\deriveb$. First, we prove a few lemmas.

\begin{lemm}\label{delta lemma}
Let $E$ be a complex in $\deriveb$ such that $H^{-1}(E)$ is a torsion sheaf. Let $\mathscr{L}$ be a line bundle over $\mathbb{P}^1_k$ and $\delta : E \rar\mathscr{L}[1]$ be a map in $\deriveb$. Then the following statements hold:
\begin{enumerate}\romanenumi
\item \label{25i} If $T \in \Coh \mathbb{P}^1 _k$ is a torsion sheaf then $\Hom (T, \cone(\delta)) \neq 0$.
\item \label{25ii} If $\cat A$ is a subcategory of $\deriveb$ containing all line bundles, then $\cone(\delta) \notin \cat A^{\perp}$.
\end{enumerate}
\end{lemm}

\begin{proof}
Let $\delta : E \rightarrow \scr{L}[1]$ be a map. The abelian category $\Coh \mathbb{P}^1 _k$ is hereditary, hence $E$ is quasi-isomorphic to $\oplus _i H^i(E)[-i]$, see~\cite[Proposition~6.1]{Rudakov}. Therefore, $\Hom(E, \scr{L}[1]) = \oplus_ i \Hom (H^i (E)[-i], \scr{L}[1]) = \oplus_i \Ext^{i+1}(H^i(E), \scr{L})$. Again since $\Ext^n(-,-)$ groups vanish for all $n > 1$ as well as for $n < 0$, it is enough to consider the cone of the map $H^{-1}(E)[1] \oplus H^0(E) \rar \scr{L}[1]$; other factors being mapped to zero. By assumption the factor $H^{-1}(E)$ is a torsion sheaf, hence the map $H^{-1}(E)[1] \rar
\scr{L}[1]$, from a torsion sheaf to torsion-free is zero.

Let us denote the map $H^0(E) \rar \scr{L}[1]$ by $\delta$ and $H^0(E)$ by $A$. As we know $\Hom (A, \mathscr{L}[1]) \cong \Ext^1 (A,
\mathscr{L})$, a map $\delta : A \rar \mathscr{L}[1]$ corresponds to an element of the group $\Ext ^1(A,\mathscr{L})$. By abuse of notation, we denote the corresponding element in $\Ext ^1(A,\mathscr{L})$ by $\delta$.

Now we take the short exact sequence corresponding to $\delta \in \Ext^1(A, \mathscr{L})$, say
\[
\begin{tikzcd}
0 \rar & \mathscr{L} \rar & B \rar & A \rar & 0.
\end{tikzcd}
\]

Since $\scr{L}$ injects into $B$, and every coherent sheaf over $\mathbb{P}^1_k$ is direct sum of its torsion and torsion-free part, $B$ has a torsion-free summand, in particular has a line bundle as a summand, say $\scr{M}$. And the short exact sequence gives rise to a distinguished triangle
\[
\begin{tikzcd}
\mathscr{L} \rar & B \rar & A \rar [" \delta "] & \mathscr{L} [1].
\end{tikzcd}
\]

Therefore, $\cone(\delta)$ being isomorphic to $B[1]$ has a summand isomorphic to $\scr{M}[1]$. Now claim~\eqref{25i} follows, since $\Ext ^1(T, \scr{M}) \neq 0$ is factor of $\Hom(T, \cone(\delta))$. And claim~\eqref{25ii} follows, since by assumption $\cat A$ contains all the line bundles, in particular, contains twists of $\scr{M}$ and $\Ext ^1(\scr{M}(2), \scr{M}) \neq 0$ implies $\Hom(\scr{M}(2), \cone(\delta)) \neq 0$.
\end{proof}

Recall that a preaisle $\cat A$ is an aisle if $(\cat A, \cat A^{\perp} [1])$ is a t-structure.

\begin{lemm}\label{one step}
Let $\cat A$ be a \tensor preaisle of $\deriveb$ and $\phi$ be its corresponding Thomason filtration. If $\cat A$ is an aisle and $\phi (i) \neq \emptyset$ for some $i$, then $\phi (i-1) = \mathbb{P}^1 _k$.
\end{lemm}

\begin{proof}
Without loss of generality, we may assume $i = 0$. If $\phi (0) = \mathbb{P} ^1 _k$ then $\phi (-1) = \mathbb{P} ^1 _k$ and there is nothing to prove. Let $\mathscr{L}$ be a line bundle on $\Coh \mathbb{P}^1 _k$. If $\phi (-1) \neq \mathbb{P}^1 _k$ then $\mathscr{L}[1] \notin \cat A$. Since $\phi (0) \neq
\emptyset$ there is a closed point $x \in \phi (0) $ and $k(x) \in \cat A$. As $\Hom (k(x), \scr{L}[1]) = \Ext^1 (k(x), \mathscr{L}) \neq 0$ we also have $\mathscr{L}[1] \notin \cat A ^{\perp}$. We must have a t-decomposition of $\mathscr{L}[1]$ as $\cat A$ is given to be an aisle.

Let $E \in \cat A$ and $\delta : E \rightarrow \scr{L}[1]$ be a map. As $\phi (-1) \neq \mathbb{P}^1 _k$, the cohomology sheaf $H^{-1}(E)$ is a torsion sheaf, so by Lemma~\ref{delta lemma}(i), $\Hom (k(x), \cone(\delta)) \neq 0$ which implies $\cone(\delta) \notin \cat A^{\perp} $. This shows that there is no distinguished triangle
\[
\begin{tikzcd}
E \rar & \scr{L}[1] \rar & F \rar & E [1]
\end{tikzcd}
\]
such that $E \in \cat A$ and $F \in \cat A ^{\perp}$. This contradicts the fact that $\cat A$ is an aisle.
\end{proof}

\begin{defi}
We say $\phi$ is a \emph{one-step} Thomason filtration of $\mathbb{P}^1 _k$ if there is an integer $i_0$ and a Thomson subset $Z_{i_0}$ such that
\begin{equation*}
\phi(j)=
\begin{cases}
\mathbb{P}^1 _k &\text{if\/ } j < i_0;\\
Z_{i_0} &\text{if\/ } j = i_0;\\
\emptyset &\text{if\/ } j > i_0.
\end{cases}
\end{equation*}
\end{defi}

\begin{prop}\label{aisle}
Let $\cat A$ be a \tensor preaisle of $\deriveb$. Then, $\cat A$ is an aisle if and only if the corresponding Thomason filtration is a one-step filtration.
\end{prop}

\begin{proof}
If $\cat A$ is a \tensor preaisle which is also an aisle then by Lemma~\ref{one step} the corresponding filtration is a one-step filtration.Conversely, suppose the filtration is one step, we will show that every complex of $\deriveb$ can be decomposed into a triangle where the first term is in $\cat A$ and the third term is in $\cat A ^{\perp}$. Without loss of generality we may assume the one step occurs at $i_0 = 0$, and $\phi (0) = Z_0$ is a Thomason subset.

If $Z_0 = \mathbb{P}^1 _k$, then $\cat A = \mathbf{D}^{b, \leq 0} (\Coh \mathbb{P}^1_k)$ and we get the standard t-structure. Now, suppose $Z_0 \neq \mathbb{P}^1 _k$. Since the filtration is one step we only need to show sheaves at degree zero have t-decompositions, all other shifted sheaves have obvious t-decompositions. The functor $\Gamma_{Z_0}(-)$ gives a t-decomposition of sheaves at degree zero.
\end{proof}

Next, we give an explicit description of \tensor copreaisles of $\deriveb$ in terms of simple torsion sheaves and line bundles.

\begin{prop}
Any proper thick \tensor copreaisle of\/ $\deriveb$ is one of the following forms:
\begin{enumerate}\romanenumi
\item \label{29i} $\langle k(x)^* [i] \mid x \in \phi (i) \rangle ^{\geq 0};$\\
where $\phi$ is a type-1 Thomason filtration of\/ $\mathbb{P}^1_k$.
\item \label{29ii} $\langle \CO (n) [i_0], k(x)^*[i] \mid \forall n \in \Z \text{ and } x \in \phi (i) \rangle ^{\geq 0}$\\
where $\phi$ is a type-2 Thomason filtration of $\mathbb{P}^1_k$ and $i_0$ a fixed integer.
\end{enumerate}
\end{prop}

\begin{proof}
By the proof of Lemma~\ref{preaisle and copreaisle}, we know that every \tensor copreaisle of $\deriveb$ is of the form $\cat A^*$ where $\cat A $ is a \tensor preaisle. Now using the description given in Proposition~\ref{Main} we conclude our result.
\end{proof}

The trivial \tensor copreaisles $\deriveb$ and $0$ give rise to tensor weight structures on $\deriveb$. In contrast to the case of t-structures (see Proposition~\ref{aisle}), the next result shows that, there are no other tensor weight structures on $\deriveb$.

\begin{prop}
The trivial weight structures are the only tensor weight structures on $\deriveb$.
\end{prop}

\begin{proof}
Suppose $\cat A$ is a non-zero \tensor copreaisle of $\deriveb$ which induces a weight structure. We first claim that $\cat A$ can not be a copreaisle containing only torsion sheaves. Indeed, since $\cat A$ is non-zero it contains some indecomposable torsion sheaf (up to shift), say $T_x[i]$. Without loss of generality we can assume $i = 0$, that is, $T_x \in \cat A$. By our assumption $\cat A$ contains only torsion sheaves hence $\scr{L}[1] \notin \cat A$, and since $\Ext ^1(T_x, \scr{L}) \neq 0$, we also have $\scr{L}[1] \notin \cat A ^{\perp}$. Suppose $\delta : E \rar \scr{L}[1]$ is a map with $E \in \cat A$, then by Lemma~\ref{delta lemma}$\MK$\eqref{25i} $\Hom (T_x, \cone(\delta)) \neq 0$ which implies $\cone(\delta) \notin \cat A^{\perp}$. This shows $\mathscr {L}[1]$ does not have a weight decomposition. Hence, $\cat A$ can not be a copreaisle containing only torsion sheaves.

If $\mathscr{L}[i]$ is in $\cat A$ for some $\mathscr{L}$, then for any line bundle $\mathscr{M}$, the tensor property of $\cat A$ implies $\mathscr{M}[i] = \scrHom(\check{\mathscr{M}}\otimes{\mathscr{L}}, \mathscr{L}[i])$ is in $\cat A$. Now, there are two cases:
\begin{enumerate}
\item \label{p30i} either there is an integer $i$ such that for any line bundle $\scr{L}$, $\mathscr{L}[i] \in \cat A$ and $\mathscr{L}[i+1] \notin \cat A$, or
\item \label{p30ii} there is no such $i$, and $\cat A$ contains all line bundles and their shifts.
\end{enumerate}

\begin{proof}[Case~\meqref{p30i}]
Suppose there is an integer $i$, then without loss of generality we can assume $i = 0$. By our assumption, for any line bundle $\mathscr{L}$ we have $\mathscr{L} [1] \notin \cat A$. Since $\cat A$ contains all the line bundles, in particular, contains twists of $\scr{L}$ and $\Ext ^1(\scr{L}(2), \scr{L}) \neq 0$ implies $\Hom(\scr{L}(2), \scr{L}[1]) \neq 0$, so $\scr{L}[1] \notin \cat A^{\perp} $. Again by our assumption on $\cat A$, for any $E \in \cat A$ the cohomology sheaf $H^{-1}(E)$ is a torsion sheaf. For any map $\delta : E
\rar \scr{L}[1]$ with $E \in \cat A$, Lemma~\ref{delta lemma}$\MK$\eqref{25ii} implies $\cone(\delta) \notin
\cat A^{\perp}$. This shows that $\mathscr{L} [1] $ does not have a weight decomposition. This is a contradiction.
\let\qed\relax
\end{proof}

\begin{proof}[Case~\meqref{p30ii}]
Now, suppose $\cat A$ contains all line bundles and their shifts. In particular, it contains $\CO (-1)$, $\CO (-2)$ and all their shifts. Now, for any indecomposable torsion sheaf of degree $d$ say $ T_x$ supported on a closed point $x \in \mathbb{P}^1_k$, consider the following distinguished triangle coming from the corresponding short exact sequence in $\Coh \mathbb{P}^1 _k$,
\[
\CO (-2)^{\oplus d} \longrightarrow \CO (-1) ^{\oplus d} \longrightarrow T_x
\longrightarrow \CO (-2)[1].
\]

As $\cat A$ is closed under extensions, $T_x $ is in $\cat A$. This proves $\cat A$ contains all the torsion sheaves and their shifts. Therefore, $\cat A$ must be equal to $\deriveb$.
\end{proof}
\let\qed\relax
\end{proof}

\begin{rema}
The above proposition says when $X = \mathbb{P}^1 _k$, the bounded derived category $\mathbf{D}^b(\Coh X)$ has no non-trivial tensor weight structure. However, when $X = \spec R$ for a regular Noetherian ring $R$, the derived category $\mathbf{D}^b(\Coh X)$ has tensor weight structures. In this case, $\mathbf{D}^b(\mathrm{mod}$-$R)$ is equivalent to the homotopy category $\mathbf{K} ^b(\proj$-$R)$, and the brutal truncations provide non-trivial tensor weight structures.
\end{rema}

\subsection*{Acknowledgement}

The authors are grateful for the excellent work environment and the assistance of the support staff of HRI, Prayagraj. The second author acknowledges the financial aid provided by the INFOSYS scholarship. We thank the referee for the valuable suggestions and comments which greatly improved the exposition.

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