Proof. Let $A, B$ be objects in $\mathscr {A}_f$ and let $\varphi \in \mathop {{\rm Hom}}\nolimits _X(A,B)$. Consider the kernel $K$, the cokernel $C$, and the image $I$ of $\varphi$. As $Rf_*A \simeq 0 \simeq Rf_* B$, the long exact sequences of higher derived functors for $f_*$ applied to short exact sequences on $X$

\[ 0 \to K \to A \to I\to 0,\quad 0 \to I \to B \to C \to 0, \]

together with vanishing of $R^if_*$, for $i>1$, imply that first $I$, hence $K$ and $C$ lie in $\mathscr {A}_f$. That $\mathscr {A}_f$ is closed under extensions is obvious.

Proof. As the $t$-structure on ${{\mathcal {C}}_f}_{{\rm qc}}$ is defined as the restriction of the standard $t$-structure on ${\mathcal {D}}_{{\rm qc}}(X)$, the functor $\iota _{f*} \colon {{\mathcal {C}}_f}_{{\rm qc}} \to {\mathcal {D}}_{\rm qc}(X)$ is exact, in particular, it commutes with the truncation functors $\tau _{\geqslant i}$. This implies isomorphism of functors

\[ \tau^X_{\geqslant -1} \iota_{f*} \simeq \iota_{f*} \tau^{\mathscr{A}_f}_{\geqslant -1}. \]

It follows that $\tau ^X_{\geqslant -1} \iota _{f*} \iota _f^* {\mathcal {M}}$ is isomorphic to $\iota _{f*}\circ {}^{-1}\iota _f^* {\mathcal {M}}$, where

\[ {}^{-1} \iota_f^* := {\mathcal{H}}^{-1}_{\mathscr{A}_f} \circ \iota_f^* \colon \mathop{{}^{-1}{\rm Per}}\nolimits(X/Y) \to \mathscr{A}_f[1] \]

is the left adjoint functor to the inclusion $\mathscr {A}_f[1] \to \mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$. By Lemma 2.5, ${\mathcal {P}}\in \mathscr {A}_f$ is a projective generator.

Proof. Let $E$ be an object in ${\mathcal {A}}_0$. Then $i_*E \in {\mathcal {A}}$, hence $0=\mathop {{\rm Ext}}\nolimits ^1_{{\mathcal {D}}}({\mathcal {M}}, i_*E) \simeq \mathop {{\rm Ext}}\nolimits ^1_{{\mathcal {D}}_0}(i^*{\mathcal {M}}, E)$. As functor $i^*$ is left adjoint to a $t$-exact functor, it is right $t$-exact, i.e. $i^* {\mathcal {M}} \in {\mathcal {D}}_0^{\leqslant 0}$. Thus, we have an exact triangle

(13)\begin{equation} \tau_{\leqslant -1} i^* {\mathcal{M}} \to i^* {\mathcal{M}} \to {\mathcal{P}} \to \tau_{\leqslant -1}i^* {\mathcal{M}}[1]. \end{equation}

As, for degree reasons, $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}_0}(\tau _{\leqslant -1}i^*M, E) =0$, by applying $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}_0}(-,E)$ to (13), we get that $\mathop {{\rm Ext}}\nolimits ^1_{{\mathcal {D}}_0}({\mathcal {P}}, E)=0$, i.e. ${\mathcal {P}}\in {\mathcal {A}}_0$ is projective.

If ${\mathcal {M}}\in {\mathcal {A}}$ is a projective generator and $i_*$ is fully faithful, then, for any non-zero $E\in {\mathcal {A}}_0$, object $i_*E$ is non-zero, hence $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}}({\mathcal {M}}, i_*E) \simeq \mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}_0}(i^*{\mathcal {M}}, E)\neq 0$. It follows from the long exact sequence obtained by applying $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}_0}(-,E)$ to (13) that $\mathop {{\rm Hom}}\nolimits _{{\mathcal {D}}_0}({\mathcal {P}}, E) \neq 0$, i.e. ${\mathcal {P}}$ is a projective generator for ${\mathcal {A}}_0$.

Proof. The derived projection formula $Rf_*Lf^* E \simeq E$ and Leray spectral sequence $R^pf_* L^qf^*(E) \Rightarrow {\mathcal {H}}_X^{p-q} E$ imply $R^1f_*f^* E \simeq 0$, $f_*L^1f^* E \simeq 0$, $Rf_* L^if^* E \simeq 0$, for any $i>1$. As $R^1f_*L^1f^* E$ is supported on $f({\rm Ex}\, f)$, the assumption that $E$ has no torsion supported on $f(\mathop {{\rm Ex}}\nolimits f)$ and sequence (20) imply that $R^1f_*L^1f^*E$ is zero and $E \simeq f_*f^*E$.

Proof. Sequence (20) implies that morphism $\alpha \colon f_* E' \to f_*f^*f_* E'$ is surjective. Morphism $\alpha$ is the inclusion of a direct summand, because its composition with the canonical map $f_*f^*f_* E'\to f_* E'$ is the identity. Hence, $\alpha$ is an isomorphism. Exact sequence (20) implies that $R^1f_*L^1f^*f_* E' = 0$. The rest follows from Lemma 2.8.

Proof. Morphism $g$ is proper by the valuative criterion. Replace $Z$ by the (closed) image of $g$, if necessary. Then $h$ becomes a proper morphism, cf. [Reference GrothendieckGro61, Corollaire 5.4.3]. As the dimension of fibers for $f$ is bounded by one, so is the dimension of fibers for $h$.

As $Rf_*(E)$ is a sheaf on $Y$, spectral sequence $R^qh_* R^sg_* E \Rightarrow R^{q+s}f_* E$ implies that $Rh_*R^1g_*(E) = 0$. Sheaf $R^1g_*(E)$ is supported in the locus of points $z \in Z$ such that the fiber of morphism $g$ over $z$ is one-dimensional. As the null category of a finite morphism is zero, morphism $h\colon Z\to Y$ restricted to the support of $R^1g_*(E)$ must have fibers of dimension one. Let $y \in Y$ be a point in $h({\rm Supp}\, R^1g_*(E))$. Then the fiber of $f$ over $y$ is two-dimensional, which contradicts the assumptions. Thus, $R^1g_*(E)$ is zero.

For any $E \in \mathscr {A}_f$, the spectral sequence $R^ph_*R^qg_* E \Rightarrow R^{p+q}f_* E = 0$ degenerates. It follows that $g_* E \in \mathscr {A}_h$. As $R^1g_*(E) = 0$, functor $g_* \colon \mathscr {A}_f \to \mathscr {A}_h$ is exact.

Proof. The construction of $X\times _Y Z$ and the statement are local on $Y$, hence we may assume that $Y=\mathop {{\rm Spec}}\nolimits (R)$ is affine. Moreover, the statement is local on $Z$, so we may further reduce to the case when $Z = \mathop {{\rm Spec}}\nolimits (A)$ is affine. Hence, $g\colon Z\to Y$ is an affine morphism, which implies that $\pi _X$ is affine.

The commutativity of (21) implies $R(f\pi _X)_*\pi _X^* E \simeq R(g\pi _Z)_*\pi _X^* E$. Morphisms $g$ and $\pi _X$ are affine, hence we have an isomorphism $R^lf_* (\pi _{X*}\pi _X^* E) \simeq g_*R^l\pi _{Z*}(\pi _X^* E)$, for any $l\geqslant 0$. Morphism $g$ is affine, thus $g_*R^l\pi _{Z*}(\pi _X^*E)$ is zero if and only if $R^l\pi _{Z*}(\pi _X^* E)$ is also zero. Hence, we need to show that $R^lf_* (\pi _{X*}\pi _X^* E) = 0$, for $l\geqslant l_0$.

Morphism $\pi _X$ is affine, hence $\pi _{X*}\pi _X^* E \simeq E \otimes \pi _{X*}({\mathcal {O}}_{X\times _Y Z})$. The base-change morphism (cf. (47)) $f^*g_* {\mathcal {O}}_{Z}\to \pi _{X*} {\mathcal {O}}_{X\times _Y Z}$ is an isomorphism. Indeed, this can be checked locally on $X$: if $X = \mathop {{\rm Spec}}\nolimits (B)$, then both sheaves $\pi _{X*} {\mathcal {O}}_{X\times _Y Z}$ and $f^*g_* {\mathcal {O}}_Z$ correspond to $B$-module $B\otimes _R A$. Thus,

\[ \pi_{X*}\pi_X^* E \simeq E \otimes f^*g_* {\mathcal{O}}_Z \simeq {\mathcal{H}}^0_X(E \otimes^L Lf^* g_* {\mathcal{O}}_Z). \]

Derived projection formula $Rf_*(E \otimes ^L Lf^*g_* {\mathcal {O}}_Z) \simeq Rf_*(E) \otimes ^L g_*{\mathcal {O}}_Z$ implies that $Rf_* (E \otimes ^L Lf^* g_* {\mathcal {O}}_Z) \in {\mathcal {D}}(Y)^{< l_0}$. Morphism $f$ is proper and with dimension of fibers bounded by one, hence $R^pf_*(F)=0$, for $p>1$ and for any sheaf $F$. It follows that spectral sequence

\[ R^pf_*{\mathcal{H}}_X^q(E \otimes^L Lf^*g_* {\mathcal{O}}_Z) \Rightarrow R^{p+q}f_*(E \otimes^L Lf^*g_* {\mathcal{O}}_Z) \]

degenerates. Therefore, $R^lf_*(\pi _{X*}\pi _X^* E) \simeq R^l f_* {\mathcal {H}}_X^0(E \otimes Lf^*g_* {\mathcal {O}}_Z) = 0$, for $l\geqslant l_0$.

Proof. First, we check that $({\mathcal {O}}_{\Gamma ^t},{\mathcal {O}}_{\Gamma }, \eta, \varepsilon )$ is a 2-categorical adjunction, i.e. that (C.1) are equal to the identity morphisms.

As ${\mathcal {O}}_{\Gamma } \simeq (\mathop {{\rm Id}}\nolimits _D\times i)_*{\mathcal {O}}_{\Delta ^D}$, the composite ${\mathcal {O}}_{\Gamma }\xrightarrow {\eta \ast {\mathcal {O}}_{\Gamma } } {\mathcal {O}}_{\Gamma } \ast {\mathcal {O}}_{\Gamma ^t} \ast {\mathcal {O}}_{\Gamma } \xrightarrow { {\mathcal {O}}_{\Gamma }\ast \varepsilon } {\mathcal {O}}_{\Gamma }$ reads

\[ (\mathop{{\rm Id}}\nolimits_D\times i)_*{\mathcal{O}}_{\Delta^D}\xrightarrow{\eta \ast {\mathcal{O}}_{\Gamma} } (\mathop{{\rm Id}}\nolimits_D\times i)_*(\mathop{{\rm Id}}\nolimits_D\times i)^*(\mathop{{\rm Id}}\nolimits_D\times i)_*{\mathcal{O}}_{\Delta^D} \xrightarrow{{\mathcal{O}}_{\Gamma}\ast \varepsilon } (\mathop{{\rm Id}}\nolimits_D\times i)_*{\mathcal{O}}_{\Delta^D}. \]

Object $(\mathop {{\rm Id}}\nolimits _D\times i)_*{\mathcal {O}}_{\Delta ^D}$ is supported on $D\times D$, hence the first morphism, restriction to $D\times D$, is the identity on the zeroth cohomology. The second morphism is the truncation at zeroth cohomology, hence $({\mathcal {O}}_\Gamma \ast \varepsilon )\circ (\eta \ast {\mathcal {O}}_{\Gamma } ) = \mathop {{\rm Id}}\nolimits _{{\mathcal {O}}_{\Gamma }}$.

As ${\mathcal {O}}_{\Gamma ^t} \simeq (i\times \mathop {{\rm Id}}\nolimits _D)_*{\mathcal {O}}_{\Delta ^D}$, the composite ${\mathcal {O}}_{\Gamma ^t}\xrightarrow { {\mathcal {O}}_{\Gamma ^t}\ast \eta } {\mathcal {O}}_{\Gamma ^t} \ast {\mathcal {O}}_{\Gamma } \ast {\mathcal {O}}_{\Gamma ^t} \xrightarrow {\varepsilon \ast {\mathcal {O}}_{\Gamma ^t} } {\mathcal {O}}_{\Gamma ^t}$ reads

\[ (i \times \mathop{{\rm Id}}\nolimits_D)_*{\mathcal{O}}_{\Delta^D}\xrightarrow{{\mathcal{O}}_{\Gamma^t}\ast \eta } (i \times \mathop{{\rm Id}}\nolimits_D)_*(i \times \mathop{{\rm Id}}\nolimits_D)^*(i \times \mathop{{\rm Id}}\nolimits_D)_*{\mathcal{O}}_{\Delta^D} \xrightarrow{\varepsilon \ast {\mathcal{O}}_{\Gamma^t} } (i\times \mathop{{\rm Id}}\nolimits_D)_*{\mathcal{O}}_{\Delta^D}. \]

The arguments as above show that $(\varepsilon \ast {\mathcal {O}}_{\Gamma ^t} ) \circ ( {\mathcal {O}}_{\Gamma ^t}\ast \eta )$ is the identity on ${\mathcal {O}}_{\Gamma ^t}$.

Now we calculate the twist and cotwist of the 2-categorical adjunction $({\mathcal {O}}_{\Gamma ^t},{\mathcal {O}}_{\Gamma }, \eta,\varepsilon )$. Morphism $\eta$ in (23) is induced by the morphism ${\mathcal {O}}_{X}\to {\mathcal {O}}_D$ with kernel ${\mathcal {I}}_D$. Hence,

\[ {\rm diag}_*^X {\mathcal{I}}_D \to {\rm diag}_*^X{\mathcal{O}}_{X} \xrightarrow{\eta} {\rm diag}_*^X{\mathcal{O}}_D \to {\rm diag}_*^X {\mathcal{I}}_D[1] \]

is an exact triangle in ${\mathcal {D}}_{{\rm qc}}(X\times X)$. Thus, the cotwist equals ${\rm diag}_*^X{\mathcal {I}}_D$. It is an equivalence whose inverse is ${\rm diag}_*^X{\mathcal {I}}_D^{-1}$. The corresponding functorial exact triangle is (26).

Morphism $\varepsilon$ in (25) is induced by the truncation to the zeroth cohomology of the object $(i\times \mathop {{\rm Id}}\nolimits _D)^*(i\times \mathop {{\rm Id}}\nolimits _D)_*{\mathcal {O}}_{\Delta ^D}$. As we noted before, $(i\times \mathop {{\rm Id}}\nolimits _D)^*(i\times \mathop {{\rm Id}}\nolimits _D)_*{\mathcal {O}}_{\Delta ^D}$ has two non-zero cohomology sheaves and the truncation to the zeroth cohomology yields an exact triangle

\[ {\mathcal{O}}_{\Delta^D}\otimes {\mathcal{I}}_{D\times D}|_{D\times D}[1] \to (i\times\mathop{{\rm Id}}\nolimits_D)^*(i\times\mathop{{\rm Id}}\nolimits_D)_*{\mathcal{O}}_{\Delta^D} \to {\mathcal{O}}_{\Delta^D}\to {\mathcal{O}}_{\Delta^D}\otimes {\mathcal{I}}_{D\times D}|_{D\times D}[2]. \]

The twist

\[ {\mathcal {O}}_{\Delta ^D} \otimes {\mathcal {I}}_{D\times D}|_{D\times D}[1] \simeq {\rm diag}^D_*{\mathcal {I}}_D|_D[1] \]

is an equivalence whose inverse is ${\rm diag}^D_*{\mathcal {I}}_D^{-1}|_D[-1]$. The corresponding functorial exact triangle is (27).

Proof. Let $\iota _i \colon D_i \to X$ denote the embedding of $D_i$ into $X$. We denote by $h$ the composite $h=if\iota _i \colon D_i \to \mathcal {Y}$. Morphism $h$ is finite, hence $h_*$ is exact and, by Grothendieck duality (A.1), we have

(28)\begin{equation} {\mathcal{E}} xt^i_{{\mathcal{Y}}}(Rh_* {\mathcal{O}}_{D_i}, {\mathcal{O}}_{{\mathcal{Y}}}) \simeq Rh_* {\mathcal{E}} xt^i_{D_i}({\mathcal{O}}_{D_i}, h^!{\mathcal{O}}_{{\mathcal{Y}}}) \simeq Rh_* {\mathcal{H}}_{D_i}^i(h^! {\mathcal{O}}_{{\mathcal{Y}}}). \end{equation}

Divisor $D_i$ is Cartier in a Gorenstein scheme, hence it is Gorenstein itself. As ${\mathcal {Y}}$ is smooth and it is the spectrum of a complete ring, we have $\omega _{{\mathcal {Y}}}^{\boldsymbol {\cdot }} \simeq {\mathcal {O}}_{{\mathcal {Y}}}[n+1]$. It follows that $h^! {\mathcal {O}}_{{\mathcal {Y}}}=h^!\omega _{{\mathcal {Y}}}^{\boldsymbol {\cdot }}[-n-1]=\omega _{D_i}[-2]$ is a sheaf in homological degree $2$. As $h_*$ is exact, $Rh_* {\mathcal {H}}_{D_i}^i(h^! {\mathcal {O}}_{{\mathcal {Y}}}) \simeq h_* {\mathcal {H}}_{D_i}^i(h^! {\mathcal {O}}_{{\mathcal {Y}}})$. Hence, by (28), we have

\[ {\mathcal{E}} xt^i_{\mathcal{Y}}(h_* {\mathcal{O}}_{D_i}, {\mathcal{O}}_{\mathcal{Y}}) = 0, \quad {\rm for}\ i\neq 2. \]

As $\mathcal {Y}$ is affine, $\mathop {{\rm Ext}}\nolimits _{{\mathcal {Y}}}^i(h_*{\mathcal {O}}_{D_i}, {\mathcal {O}}_{\mathcal {Y}}) = 0$, for $i\neq 2$. Hence, the projective dimension of $h_* {\mathcal {O}}_{D_i}$ as an ${\mathcal {O}}_{{\mathcal {Y}}}$ module is two. In other words, $h_* {\mathcal {O}}_{D_i}$ has a locally free resolution

\[ 0 \to {\mathcal{E}}_{-2} \to {\mathcal{E}}_{-1} \to {\mathcal{E}}_0 \to h_* {\mathcal{O}}_{D_i} \to 0. \]

Denote by ${\mathcal {E}}_{{\boldsymbol {\cdot }}}$ the complex $0 \to {\mathcal {E}}_{-2} \to {\mathcal {E}}_{-1} \to {\mathcal {E}}_0 \to 0$. The cohomology of complex $i_*i^* {\mathcal {E}}_{{\boldsymbol {\cdot }}}$ is ${\rm Tor}^{{\mathcal {Y}}}(h_* {\mathcal {O}}_{D_i},{\mathcal {O}}_Y)$. As $Y \subset \mathcal {Y}$ is a Cartier divisor, ${\mathcal {O}}_Y$ has a resolution of length two on ${\mathcal {Y}}$, i.e. the zeroth and first Tor groups only can be non-zero. As functor $i_*$ is exact, we conclude that the morphism $i^*{\mathcal {E}}_{-2} \to i^* {\mathcal {E}}_{-1}$ is injective.

Morphism $f$ is birational, hence the higher derived functors of $f^*$ are torsion. In particular, the kernel of $f^*i^* {\mathcal {E}}_{-2} \to f^*i^* {\mathcal {E}}_{-1}$ is torsion, meaning zero because $f^*i^* {\mathcal {E}}_{-2}$ is locally free. This shows that morphism $f^*i^* {\mathcal {E}}_{-2} \to f^*i^* {\mathcal {E}}_{-1}$ is also injective, thus complex $\widetilde {{\mathcal {E}}}_{\boldsymbol {\cdot }} := f^*i^* {\mathcal {E}}_{{\boldsymbol {\cdot }}}$ has zero cohomology except for ${\mathcal {H}}^{0}_X(\widetilde {{\mathcal {E}}}_{{\boldsymbol {\cdot }}})$ and ${\mathcal {H}}^{-1}_X(\widetilde {{\mathcal {E}}}_{{\boldsymbol {\cdot }}})$.

Let $E\subset X$ be the exceptional locus of $f$. Assume $L^1f^*f_* {\mathcal {O}}_{D_i} \neq 0$. Its support is contained in $E$. Moreover, because $L^1f^*f_* {\mathcal {O}}_{D_i} \in \mathscr {A}_f$ (see Lemma 2.9), the support of this sheaf should contain at least one component of the curve in the fiber over the closed point of $Y$ (see Theorem D.1 for the structure of the fiber). Therefore, there exists a closed point $x \in E \setminus D_i$ in the support of $L^1f^*f_* {\mathcal {O}}_{D_i}$.

By Theorem 3.1 for divisor $Y$ in ${\mathcal {Y}}$, object $i^* {\mathcal {E}}_{{\boldsymbol {\cdot }}} \simeq Li^*i_*f_* {\mathcal {O}}_{D_i}$ fits into an exact triangle

\[ f_* {\mathcal{O}}_{D_i}[1] \to i^* {\mathcal{E}}_{{\boldsymbol{\cdot}}} \to f_* {\mathcal{O}}_{D_i} \to f_* {\mathcal{O}}_{D_i}[2]. \]

By applying functor $L f^*$ to this triangle, we obtain an exact sequence

\[ f^* f_* {\mathcal{O}}_{D_i} \to \mathcal{H}_X^{-1}(\widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}}) \to L^1f^*f_* {\mathcal{O}}_{D_i}\to 0. \]

Sheaf $f^*f_*{\mathcal {O}}_{D_i}$ is supported on the preimage of $f(D_i) \subset Y$, i.e. on the set $E \cup D_i$. Let us restrict our attention to an open affine neighbourhood $\widetilde {i} \colon \widetilde {X}\hookrightarrow X$ of point $x$. The support of both sheaves $f^*f_* {\mathcal {O}}_{D_i}$ and $L^1f^*f_*{\mathcal {O}}_{D_i}$ is contained in $E$. Therefore, the assumptions on the exceptional locus for $f$ imply that ${\mathcal {H}}_X^{-1}(\widetilde {{\mathcal {E}}})$ is a non-zero sheaf with support of some codimension $l>1$.

Let $j\colon \widetilde {X}\to \mathcal {X}$ be a closed embedding into a smooth variety of dimension $m$. We have $\omega _{\widetilde {X}} \simeq \widetilde {i}^!f^! {\mathcal {O}}_Y \simeq \widetilde {i}^! {\mathcal {O}}_X \simeq {\mathcal {O}}_{\widetilde {X}}$, hence Grothendieck duality (A.1) gives

\[ {\mathcal{E}} xt^{{\boldsymbol{\cdot}}}_{\mathcal{X}}(j_* \widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}}, \omega_{\mathcal{X}}) \simeq j_* {\mathcal{E}} xt^{{\boldsymbol{\cdot}}}_{\widetilde{X}}(\widetilde{{\mathcal{E}}}_{\boldsymbol{\cdot}},{\mathcal{O}}_{\widetilde{X}}[n-m]). \]

Complex $\widetilde {{\mathcal {E}}}_{{\boldsymbol {\cdot }}}$ consists of three locally free sheaves, thus ${\mathcal {E}} xt^k_X(\widetilde {{\mathcal {E}}}_{{\boldsymbol {\cdot }}}, {\mathcal {O}}_X)$ is zero, for $k>2$. It follows that ${\mathcal {E}} xt^k_{\mathcal {X}}(j_* \widetilde {{\mathcal {E}}}_{\boldsymbol {\cdot }}, \omega _{\mathcal {X}}) = 0$, for $k>m-n+2$. By applying functor ${\mathcal {H}} om^{{\boldsymbol {\cdot }}}_{\mathcal {X}}(-, \omega _{\mathcal {X}})$ to the exact triangle

\[ j_*{\mathcal{H}}^{-1}_X(\widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}})[1] \to j_*\widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}} \to j_*{\mathcal{H}}^0_X(\widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}}) \to j_*{\mathcal{H}}^{-1}_X(\widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}})[2], \]

we obtain an isomorphism

(29)\begin{equation} {\mathcal{E}} xt^k_{\mathcal{X}}(j_*{\mathcal{H}}^{-1}_X(\widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}}), \omega_{\mathcal{X}}) \xrightarrow{\simeq} {\mathcal{E}} xt^{k+2}_{\mathcal{X}}(j_*{\mathcal{H}}^0_X(\widetilde{{\mathcal{E}}}_{{\boldsymbol{\cdot}}}), \omega_{\mathcal{X}}), \end{equation}

for any $k\geqslant m-n+2$. Sheaf $j_* {\mathcal {H}}^{-1}_X(\widetilde {{\mathcal {E}}}_{{\boldsymbol {\cdot }}})$ is non-zero with support of codimension $l+m-n \geqslant m-n+2$. It follows that sheaf ${\mathcal {E}} xt^{l+m-n}_{\mathcal {X}}(j_*{\mathcal {H}}^{-1}_X(\widetilde {{\mathcal {E}}}_{{\boldsymbol {\cdot }}}), \omega _{\mathcal {X}})$ is non-zero with support of codimension $l+m-n$, cf. [Reference Huybrechts and LehnHL10, Proposition 1.1.6].

On the other hand, the codimension of the support of ${\mathcal {E}} xt^{l+m-n+2}_{\mathcal {X}}(j_*{\mathcal {H}}^0_X(\widetilde {{\mathcal {E}}}_{{\boldsymbol {\cdot }}}), \omega _{\mathcal {X}})$ is greater than or equal to $l+m-n+2$, which contradicts (29).

Proof. The sheaf ${\mathcal {M}}_i$ is defined as an extension

\[ 0 \to {\mathcal{O}}_X^{r_i-1} \to {\mathcal{M}}_i \xrightarrow{q} {\mathcal{O}}_X(D_i) \to 0. \]

For any divisor $D_i$, one can find a linearly equivalent divisor $D'_i$ such that $D_i \cap D'_i$ is empty. Indeed, let $D'_i$ be a divisor as in Remark 2.7 linearly equivalent to $D_i$ which intersects $C_i$ in a point $x'_i$, not equal to $x_i = D_i \cap C_i$. Then $D_i\cap D'_i$ contains no closed points (as all closed points of $X$ lie on $C$), hence empty. It follows that ${\mathcal {O}}_{D_i}(D_i) \simeq {\mathcal {O}}_{D_i}(D'_i) \simeq {\mathcal {O}}_{D_i}$ and sequence

\[ 0 \to {\mathcal{O}}_X \to {\mathcal{O}}_X(D_i) \xrightarrow{r} {\mathcal{O}}_{D_i} \to 0 \]

is exact. As $\mathop {{\rm Ext}}\nolimits ^1_X({\mathcal {O}}_X, {\mathcal {O}}_X) \simeq 0$, the snake lemma for the two top rows of the following diagram gives the following commutative diagram in which rows and columns are exact sequences.

It gives exact sequence (30). By considering local homomorphisms into the structure sheaf we obtain sequence (31).

Proof. The statement is local in $Y$, therefore we might assume that $Y = \mathop {{\rm Spec}}\nolimits R$ is a spectrum of a complete Noetherian local ring with hypersurface singularities. Then ${\mathcal {M}}$ is a direct sum of copies of ${\mathcal {M}}_i$ and ${\mathcal {O}}_X$. Thus, it suffices to verify the lemma for ${\mathcal {M}}_i$, as clearly $L^1f^*{\mathcal {O}}_Y = 0$.

As $R^1f_* {\mathcal {O}}_X \simeq 0$, applying $Rf_*$ to sequence (30) gives an exact sequence

\[ 0 \to {\mathcal{O}}_Y^{r_i} \to f_* {\mathcal{M}}_i \to f_* {\mathcal{O}}_{D_i} \to 0. \]

It implies an exact sequence

\[ 0 \to L^1f^*f_* {\mathcal{M}}_i \to L^1f^*f_* {\mathcal{O}}_{D_i} \to {\mathcal{O}}_X^{r_i}. \]

By Lemma 3.2, the sheaf $L^1f^*f_* {\mathcal {O}}_{D_i}$ is zero, hence the result.

Proof. First, we show that $f^*f_* {\mathcal {N}}$ is torsion-free. This can be done locally, i.e. we may assume that $Y$ is a spectrum of a complete Noetherian local ring. By [Reference Van den BerghVdB04, Lemma 3.5.2] sheaf ${\mathcal {N}}$ is a direct sum of copies of ${\mathcal {O}}_X$ and ${\mathcal {N}}_i$, where ${\mathcal {N}}_i$ are vector bundles dual to ${\mathcal {M}}_i$.

By [Reference Van den BerghVdB04, Lemma 3.1.2], the sheaf $R^1f_* {\mathcal {N}}_i$ is zero. By applying $f_*$ to sequence (31), we obtain an exact sequence on $Y$

(33)\begin{equation} 0 \to f_* {\mathcal{N}}_i \to {\mathcal{O}}_Y^{r_i} \to f_* {\mathcal{O}}_{D_i} \to 0. \end{equation}

As $f_*{\mathcal {O}}_{D_i}$ is a torsion sheaf, $f_*{\mathcal {N}}_i$ is non-zero. By applying $f^*$ to sequence (33), we obtain

\[ 0 \to L^1f^*f_* {\mathcal{O}}_{D_i} \to f^*f_* {\mathcal{N}}_i \to {\mathcal{O}}_X^{r_i}. \]

It implies that the torsion of $f^*f_* {\mathcal {N}}_i$ is $L^1f^*f_* {\mathcal {O}}_{D_i}$, which is zero by Lemma 3.2. Hence, sheaf $f^*f_* {\mathcal {N}}_i$ is torsion free and the counit morphism $\varepsilon _{{\mathcal {N}}}\colon f^*f_* {\mathcal {N}}_i \to {\mathcal {N}}_i$ is an embedding ($\varepsilon _{{\mathcal {N}}}$ is non-zero as under the $f^*\dashv f_*$ adjunction it corresponds to the identity endomorphism of $f_*{\mathcal {N}}_i$).

The cokernel of the morphism $f^*f_* {\mathcal {N}} \to {\mathcal {N}}$ is, by the definition of $\iota _f^*$, isomorphic to ${\mathcal {H}}_X^0(\iota _f^* {\mathcal {N}})$, hence $\mathcal {Q}$ is an object in $\mathscr {A}_f$. Now let $E$ be any object in $\mathscr {A}_f$. Then $E$, considered as an object in ${\mathcal {D}}^b(X)$, lies in $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$, hence $\mathop {{\rm Ext}}\nolimits ^1_X({\mathcal {N}}, E) \simeq 0$. Moreover, $f^* \vdash f_*$ adjunction implies that $\mathop {{\rm Hom}}\nolimits _X(f^*f_* {\mathcal {N}}, E) \simeq 0$. Thus, by applying higher derived functors of $\mathop {{\rm Hom}}\nolimits _X(-,E)$ to sequence (32), we obtain $\mathop {{\rm Ext}}\nolimits ^1_X(\mathcal {Q}, E) \simeq 0$, which proves that $\mathcal {Q} \in \mathscr {A}_f$ is indeed a projective object.

Proof. Lemma 3.8 implies that $f^*f_* {\mathcal {M}}$ is an object in $\mathop {{\rm Coh}}\nolimits (X) \cap {}^{-1}{\rm Per}(X/Y)$, hence ${\mathcal {T}}_{-1}(f^*f_* {\mathcal {M}}) = f^*f_* {\mathcal {M}}$. Moreover, $L^1f^*f_* {\mathcal {M}} = 0$ by Lemma 3.4. We conclude by sequence (34).

Proof. We have ${\rm Coh}(X) \cap {}^{0}{\rm Per}(X/Y) = {\mathcal {T}}_{0}$ and ${\rm Coh}(X) \cap {}^{-1}{\rm Per}(X/Y) = {\mathcal {T}}_{-1}$, for categories ${\mathcal {T}}_0$ and ${\mathcal {T}}_{-1}$ defined in (8) and (10), respectively.

Let $E$ be a coherent sheaf on $Y$. By Lemma 2.9, sheaf $R^1f_*f^* E$ is zero, hence $f^*E$ is an object in $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$. From adjunction

\[ \mathop{{\rm Hom}}\nolimits_X(f^* E, {\mathcal{A}}_f) \simeq \mathop{{\rm Hom}}\nolimits_Y(E, f_* {\mathcal{A}}_f) \simeq 0, \]

it follows that $f^*E$ lies also in $\mathop {{}^{-1}{\rm Per}}\nolimits (X/Y)$.

Now we assume further that $E$ is locally free and $Y$ is affine. Let $E'$ be any object in $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$, for $p=-1, 0$. Then

\[ \mathop{{\rm Ext}}\nolimits^1_X(f^*E, E') \simeq \mathop{{\rm Ext}}\nolimits^1_X(Lf^* E, E') \simeq \mathop{{\rm Ext}}\nolimits^1_Y(E, Rf_* E'). \]

Because $Rf_*$ is exact for the perverse $t$-structures on ${\mathcal {D}}^b(X)$, object $Rf_* E'$ is a coherent sheaf on $Y$. As $E$ is locally free and $Y$ is affine, $\mathop {{\rm Ext}}\nolimits ^1_Y(E, Rf_* E') \simeq 0$, i.e. $f^*E$ is projective in $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$.

Proof. For a torsion-free sheaf $F$ on $X$, its push-forward $f_* F$ is also torsion-free. Thus, $f_* F$ will be reflexive if we check that it is normal.

Let $G$ be a torsion-free sheaf on $Y$ which is not normal. There exist open sets $j \colon U \hookrightarrow V$ with the complement of $U$ in codimension greater than one in $V$ such that canonical morphism $G \to j_*j^* G$ is not an isomorphism. Then we have a non-trivial extension

\[ 0 \to G \to j_*j^*G \to Q \to 0. \]

The question is local, so we can assume $Y$ to be affine. Sheaf $Q$ is supported in codimension greater than one, hence there exists a closed subscheme $Z \subset Y$ of codimension greater than one such that sequence

\[ 0 \to K \to {\mathcal{O}}_{Z}^{\oplus k} \to Q \to 0 \]

is exact. As $G$ is torsion-free, $\mathop {{\rm Hom}}\nolimits (K, G)$ is zero. Hence, vanishing of $\mathop {{\rm Ext}}\nolimits ^1_Y({\mathcal {O}}_Z, G)$ implies that $\mathop {{\rm Ext}}\nolimits ^1_Y(Q, G)$ is also zero. Thus, in order to show that a torsion-free sheaf $G$ is normal it suffices to check that $\mathop {{\rm Ext}}\nolimits ^1_Y({\mathcal {O}}_Z, G)$ vanishes, for any closed $Z \subset Y$ of codimension greater than one.

Group $\mathop {{\rm Ext}}\nolimits ^1_Y({\mathcal {O}}_Z, f_* F)$ is zero if and only if morphism $\mathop {{\rm Hom}}\nolimits ({\mathcal {O}}_Y, f_* F) \to \mathop {{\rm Hom}}\nolimits (I_Z, f_* F)$ given by short exact sequence

\[ 0 \to I_Z \to {\mathcal{O}}_Y \to {\mathcal{O}}_Z \to 0 \]

is an isomorphism. By adjunction, $\mathop {{\rm Hom}}\nolimits _Y(I_Z, f_* F)$ is isomorphic to $\mathop {{\rm Hom}}\nolimits _X(f^* I_Z, F)$. Exceptional locus of $f$ has codimension greater than one, hence $f^* I_Z$ is isomorphic to ${\mathcal {O}}_X$ in codimension one. As $F$ is reflexive, this implies $\mathop {{\rm Hom}}\nolimits _X(f^*I_Z, F)= \mathop {{\rm Hom}}\nolimits _X({\mathcal {O}}_X, F)$.

Thus, functor $f_*$ maps reflexive sheaves on $X$ to reflexive sheaves on $Y$. Clearly, the same is true about $(f^*(-))^{\vee \vee }$.

Now let $F$ be an object in $\textrm{Ref}\,(X)$. Adjunction implies a morphism $\alpha \colon (f^*f_*F)^{\vee \vee } \to F^{\vee \vee } \simeq F$. Both sheaves are reflexive and $\alpha$ is an isomorphism outside exceptional locus of $f$, which has codimension greater than one. Therefore, $\alpha$ is an isomorphism.

For $G \in \textrm{Ref}\,(Y)$, reflexification $f^*G \to (f^*G)^{\vee \vee }$ together with adjunction $G\to f_*f^*G$ give $\beta \colon G \to f_*((f^*G)^{\vee \vee })$. Again, morphism $\beta$ is an isomorphism in codimension one between reflexive sheaves, hence an isomorphism.

Finally, a morphism between reflexive sheaves is determined by its restriction to any open set with the complement of codimension greater than one. Thus, functors $f_*$ and $(f^*(-))^{\vee \vee }$ give isomorphisms on the groups of morphisms between objects in $\textrm{Ref}\,(X)$ and $\textrm{Ref}\,(Y)$.

Proof. By [Reference Van den BerghVdB04, Proposition 3.2.6], categories $\mathscr {P}_{-1}$ and $\mathscr {P}_0$ are subcategories of $\textrm{Ref}\,(X)$. By [Reference Van den BerghVdB04, Proposition 4.3.1], subcategories $f_* \mathscr {P}_{-1}$ and $f^+_* \mathscr {P}_0^+$ of $\textrm{Ref}\,(Y)$ are equivalent. By Lemma 3.11, the subcategory $(f^{+*}f_* \mathscr {P}_{-1})^{\vee \vee }$ of $\textrm{Ref}\,(X^+)$ corresponds to $\mathscr {P}_{-1}$ under equivalences $\textrm{Ref}\,(X)\simeq \textrm{Ref}\,(Y) \simeq \textrm{Ref}\,(X^+)$, hence $\mathscr {P}_0^+ \simeq (f^{+*}f_* \mathscr {P}_{-1})^{\vee \vee }$.

Exchanging the roles of $f$ and $f^+$, we get equivalence of $\mathscr {P}_{-1}^+$ with $\mathscr {P}_0$.

Proof. As $X$ and $Y$ are Gorenstein, $\omega _X^{\boldsymbol {\cdot }}$ and $\omega _Y^{\boldsymbol {\cdot }}$ are, up to shift, line bundles. In particular, $\omega _Y^{\boldsymbol {\cdot }}$ is a perfect complex, hence, by Lemma A.1, $f^!(\omega _Y^{\boldsymbol {\cdot }}) \simeq f^!({\mathcal {O}}_Y) \otimes Lf^*(\omega _Y^{\boldsymbol {\cdot }})$. Isomorphism $\omega _X^{\boldsymbol {\cdot }} \simeq f^!(\omega _Y^{\boldsymbol {\cdot }}) \simeq f^!({\mathcal {O}}_Y)\otimes Lf^*(\omega _Y)^{\boldsymbol {\cdot }}$ implies $f^!({\mathcal {O}}_Y) \simeq \omega _X^{\boldsymbol {\cdot }} \otimes (Lf^*(\omega _Y^{\boldsymbol {\cdot }}))^{-1}$. Hence, $Lf^*(\omega _Y^{\boldsymbol {\cdot }}) \simeq \omega _X^{\boldsymbol {\cdot }}$ if and only $f^!({\mathcal {O}}_Y) \simeq {\mathcal {O}}_X$.

Proof. Let ${\mathcal {M}}$ be an object in $\mathscr {P}_{-1}$. As ${\mathcal {Y}}$ is affine, the length of the locally free resolution for $g_* {\mathcal {M}}$ is $l+1$, for the maximal $l$ such that ${\mathcal {E}} xt^l_{{\mathcal {Y}}}(g_* {\mathcal {M}}, {\mathcal {O}}_{{\mathcal {Y}}})$ is non-zero. Thus, the question is local on ${\mathcal {Y}}$ and, by restricting to a smaller affine open subset, we can assume that $\omega ^{{\boldsymbol {\cdot }}}_{{\mathcal {Y}}} \simeq {\mathcal {O}}_{{\mathcal {Y}}}[n+1]$ and $\omega ^{{\boldsymbol {\cdot }}}_{Y} \simeq {\mathcal {O}}_{Y}[n]$. Then $\omega _X^{{\boldsymbol {\cdot }}} \simeq {\mathcal {O}}_X[n]$, because $f$ is crepant. Let ${\mathcal {N}} = R {\mathcal {H}} om_X({\mathcal {M}}, {\mathcal {O}}_X)={\mathcal {H}} om_X({\mathcal {M}}, {\mathcal {O}}_X)$. It is an object in $\mathscr {P}_0$, see [Reference Van den BerghVdB04, Proposition 3.2.6]. As ${\mathcal {N}}$ is an object in $\mathop {{}^{0}{\rm Per}}\nolimits (X/Y)$, we have $Rg_* {\mathcal {N}} \simeq g_* {\mathcal {N}}$. Then we have, by Grothendieck duality (A.1),

\[ R{\mathcal{H}} om_{{\mathcal{Y}}}(g_* {\mathcal{M}}, {\mathcal{O}}_{{\mathcal{Y}}})\simeq R{\mathcal{H}} om_{{\mathcal{Y}}}(Rg_* {\mathcal{M}}, \omega_{{\mathcal{Y}}})[-1] \simeq Rg_* R {\mathcal{H}} om_X({\mathcal{M}}, \omega_X)[-1]\simeq g_* {\mathcal{N}}[-1]. \]

Hence, $g_* {\mathcal {M}}$ is of projective dimension one. Analogously, for ${\mathcal {N}} \in \mathscr {P}_0$, its dual ${\mathcal {M}} = R{\mathcal {H}} om_X({\mathcal {N}},{\mathcal {O}}_X)$ lies in $\mathscr {P}_{-1}$ and $R{\mathcal {H}} om(g_* {\mathcal {N}}, {\mathcal {O}}_{{\mathcal {Y}}}) \simeq g_* {\mathcal {M}}[-1]$.

Proof. As $R^1g_*{\mathcal {R}} =0$, the zero cohomology of $Lg^*Rg_*{\mathcal {R}}$ is the sheaf $g^*g_*{\mathcal {R}}$. The adjunction counit for $i\colon Y \to {\mathcal {Y}}$ gives an isomorphism $g^*g_*\xrightarrow {\simeq } f^*f_*$, hence ${\mathcal {H}}^0_X(Lg^*Rg_*{\mathcal {R}}) = f^*f_*{\mathcal {R}}$.

As $g_*{\mathcal {R}}$ admits a locally free resolution (see Lemma 3.14):

\[ 0 \to {\mathcal{E}}_{-1} \to {\mathcal{E}}_0 \to g_* {\mathcal{R}} \to 0, \]

the sheaves ${\mathcal {H}}^i_X(Lg^*Rg_*{\mathcal {R}})$ are zero, for $i\leqslant -2$. Moreover, $L^1g^*g_*{\mathcal {R}}$ is reflexive, as the kernel of a morphism of locally free sheaves [Reference HartshorneHar80, Proposition 1.1].

Applying $Lf^*$ to exact triangle $f_* {\mathcal {R}}[1] \to Li^*g_* {\mathcal {R}} \to f_* {\mathcal {R}} \to f_*{\mathcal {R}}[2]$ (given by Theorem 3.1 for $Y\subset {\mathcal {Y}}$) implies the exact sequence

\[ L^2f^*f_*{\mathcal{R}}\to f^*f_*{\mathcal{R}}\to L^1g^*g_*{\mathcal{R}}\to L^1f^*f_*{\mathcal{R}}. \]

As $f$ is an isomorphism in codimension one, the sheaves $L^2f^*f_*{\mathcal {R}}$ and $L^1f^*f_*{\mathcal {R}}$ have the support of codimension at least two. As the universal map $f^*f_*{\mathcal {R}}\to (f^*f_*{\mathcal {R}})^{\vee \vee }$ is also an isomorphism in codimension one, the induced map $(f^*f_*{\mathcal {R}})^{\vee \vee } \to L^1g^*g_*{\mathcal {R}}$ is an isomorphism in codimension one. Hence, it is an isomorphism, see Lemma 3.10. It follows that $L^1g^*g_*{\mathcal {R}} \simeq (f^*f_*{\mathcal {R}})^{\vee \vee }$. As ${\mathcal {R}}$ is locally free [Reference Van den BerghVdB04, Proposition 3.2.6], $(f^*f_*{\mathcal {R}})^{\vee \vee }\simeq {\mathcal {R}}$ by Lemma 3.11.

By Lemma 3.8, $f^*f_*{\mathcal {R}}$ is an object of $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$. As the same holds for ${\mathcal {R}} \in \mathscr {P}_{p} \subset \mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$, the exact triangle ${\mathcal {R}}[1] \to Lg^*g_*{\mathcal {R}} \to f^*f_*{\mathcal {R}}$ is also the decomposition of $Lg^*g_*{\mathcal {R}}$ in the $t$-structure with heart $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$.

Proof. By applying $Lf^*$ to functorial exact triangle (27), we obtain a morphism $Lf^*E \to Lf^*E[2]$ with cone $Lf^*Li^*i_*E[1]$. As ${\mathcal {Y}}$ is smooth, ${\mathcal {D}}^b({\mathcal {Y}})$ coincides with the category of perfect complexes on ${\mathcal {Y}}$. It follows that the functor $Lf^*Li^*i_*$ takes ${\mathcal {D}}^b(Y)$ to ${\mathcal {D}}^b(X)$ (even to $\mathop {{\rm Perf}}\nolimits (X)$), hence, for any $E\in {\mathcal {D}}^b(Y)$, there exists $l\in \mathbb {Z}$ such that ${\mathcal {H}}^i(Lf^*Li^*i_*E) = 0$, for $i < l$. Then ${\mathcal {H}}^i Lf^*E \simeq {\mathcal {H}}^{i-2} Lf^* E$, for any $i< l$.

Proof. Composing resolution (37) with sequence (38) gives the projective resolution for ${\mathcal {P}}_{{\mathcal {M}}}[1]$ Similarly, resolution (41) and sequence (32) give projective resolution for ${\mathcal {P}}_{{\mathcal {N}}}$.

Proof. By [Reference Van den BerghVdB04, Proposition 3.2.6], objects in $\mathscr {P}_{p}$ are vector bundles on $X$. Hence, ${\mathcal {E}} xt^i({\mathcal {M}}_1, {\mathcal {M}}_2) \simeq 0$, for $i > 0$. As morphism $f$ has fibers of dimension bounded by one, the local-to-global spectral sequence implies that $\mathop {{\rm Ext}}\nolimits ^i({\mathcal {M}}_1, {\mathcal {M}}_2)\simeq H^0(Y, R^if_* {\mathcal {H}} om({\mathcal {M}}_1, {\mathcal {M}}_2))$ vanishes, for $i> 1$. Group $\mathop {{\rm Ext}}\nolimits ^1_X({\mathcal {M}}_1,{\mathcal {M}}_2)$ is isomorphic to $\mathop {{\rm Ext}}\nolimits ^1_{\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)}({\mathcal {M}}_1, {\mathcal {M}}_2)$, hence also zero, as ${\mathcal {M}}_1$ is an object of $\mathscr {P}_p$.

Proof. As objects in $\mathscr {P}_{p}$ are coherent sheaves on $X$, we have a functor $\Theta \colon {\rm Hot}^{-,b}(\mathscr {P}_{p})\to {\mathcal {D}}^b(X)$. There exists $N_0$ such that $\mathop {{\rm Ext}}\nolimits ^q_{X}(P, C) = 0$, for any $P \in \mathscr {P}_{p}$ and $C \in \mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$ and $q\geqslant N_0$. Indeed, $C$ has non-zero cohomology sheaves in degrees $-1$ and $0$ only, and objects in $\mathscr {P}_p$ are locally free sheaves on $X$ (see [Reference Van den BerghVdB04, Proposition 3.2.6]). Then Proposition E.4 implies that cohomology groups of complex $\prod _{j-i = p} \mathop {{\rm Hom}}\nolimits _X(A^i, B^j)$ are isomorphic to $\mathop {{\rm Hom}}\nolimits _X(A^{\boldsymbol {\cdot }}, B^{\boldsymbol {\cdot }})$, for any $A^{\boldsymbol {\cdot }}, B^{\boldsymbol {\cdot }} \in \mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_p)$ (see Lemma 4.3). Hence, $\mathop {{\rm Hom}}\nolimits ^{\boldsymbol {\cdot }}_X(A^{\boldsymbol {\cdot }}, B^{\boldsymbol {\cdot }}) \simeq \mathop {{\rm Hom}}\nolimits _{\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_p)}(A^{\boldsymbol {\cdot }}, B^{\boldsymbol {\cdot }})$, i.e. $\Theta$ is fully faithful. It is essentially surjective, because $\mathop {{}^{p}{\rm Per}}\nolimits (X/Y)$ is the heart of a bounded $t$-structure.

Proof. Lemma 3.11 implies that functor

\[ T_o := (f^*f^+_*(-))^{\vee \vee} \colon \mathscr{P}^+_0 \to \mathscr{P}_{-1} \]

is the inverse of $\Sigma _o$. Let us extend it term-wise to a functor

\[ T \colon {\rm Hot}^{-,b}(\mathscr{P}_0^+) \to {\rm Hot}^{-}(\mathscr{P}_{-1}). \]

Proposition 3.12 implies that $\Sigma$ and $T$ are inverse equivalences between $\mathscr {P}_{-1}$ and $\mathscr {P}_0^+$. Hence, they are also inverse equivalences between $\mathop {{\rm Hot}}\nolimits ^-(\mathscr {P}_{-1})$ and $\mathop {{\rm Hot}}\nolimits ^-(\mathscr {P}_0)$. By Proposition 4.4, we have ${\mathcal {D}}^b(X) \simeq {\rm Hot}^{-,b}(\mathscr {P}_{-1})$ and ${\mathcal {D}}^b(X^+) \simeq {\rm Hot}^{-,b}(\mathscr {P}_0^+)$. Thus, it suffices to show that $\Sigma$ takes $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{-1})$ to $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_0^+)$ and $T$ takes $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{0}^+)$ to $\mathop {{\rm Hot}}\nolimits ^{-,b}(\mathscr {P}_{-1})$.