Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all objects that are direct summands of finite direct sums of copies of objects from $\mathcal C$. Let $X$ be a variety. Let $f: Y \to X$ and $g : Y'\to X$ be two resolution of singularities. Then, there exists a smooth variety $Y''$ and birational morphisms $f': Y''\to Y$ and $g': Y''\to Y'$ such that $f\circ f'=g\circ g'$ .

Consider the derived push-forward $Rf_*: D(\text{QCoh } Y)\to D(\text{QCoh } X)$ and $R(f\circ f')_*: D(\text{QCoh } Y'')\to D(\text{QCoh } X)$.

Do the following equalities hold true:

(1) $\text{add}(Rf_* F : F \in D(\text{QCoh } Y))=\text{add}(R(f\circ g)_* G : G \in D(\text{QCoh } Y''))$ ?

(2) $\text{add}(Rf_* F : F \in D(\text{Coh } Y))=\text{add}(R(f\circ g)_* G : G \in D(\text{Coh } Y''))$ ?

Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all objects that are direct summands of finite direct sums of copies of objects from $\mathcal C$. Let $X$ be a variety. Let $f: Y \to X$ and $g : Y'\to X$ be two resolution of singularities. Then, there exists a smooth variety $Y''$ and birational morphisms $f': Y''\to Y$ and $g': Y''\to Y'$ such that $f\circ f'=g\circ g'$ .

Consider the derived push-forward $Rf_*: D(\text{QCoh } Y)\to D(\text{QCoh } X)$ and $R(f\circ f')_*: D(\text{QCoh } Y'')\to D(\text{QCoh } X)$.

Do the following equalities hold true:

(1) $\text{add}(Rf_* F : F \in D(\text{QCoh } Y))=\text{add}(R(f\circ f')_* G : G \in D(\text{QCoh } Y''))$ ?

(2) $\text{add}(Rf_* F : F \in D(\text{Coh } Y))=\text{add}(R(f\circ f')_* G : G \in D(\text{Coh } Y''))$ ?

Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all objects that are direct summands of finite direct sums of copies of objects from $\mathcal C$. Let $X$ be a variety. Let $f: Y \to X$ and $g : Y'\to X$ be two resolution of singularities. Then, there exists a smooth variety $Y''$ and birational morphisms $f': Y''\to Y$ and $g': Y''\to Y'$.

Consider the derived push-forward $Rf_*: D(\text{QCoh } Y)\to D(\text{QCoh } X)$ and $R(f\circ f')_*: D(\text{QCoh } Y'')\to D(\text{QCoh } X)$.

Do the following equalities hold true:

(1) $\text{add}(Rf_* F : F \in D(\text{QCoh } Y))=\text{add}(R(f\circ g)_* G : G \in D(\text{QCoh } Y''))$ ?

(2) $\text{add}(Rf_* F : F \in D(\text{Coh } Y))=\text{add}(R(f\circ g)_* G : G \in D(\text{Coh } Y''))$ ?

Let $\mathcal C$ be a full subcategory (closed under isomorphism also) of an additive category $\mathcal A$. Then, $\text{add}(\mathcal C)$ is the full subcategory of $\mathcal A$ consisting of all objects that are direct summands of finite direct sums of copies of objects from $\mathcal C$. Let $X$ be a variety. Let $f: Y \to X$ and $g : Y'\to X$ be two resolution of singularities. Then, there exists a smooth variety $Y''$ and birational morphisms $f': Y''\to Y$ and $g': Y''\to Y'$ such that $f\circ f'=g\circ g'$ .

Consider the derived push-forward $Rf_*: D(\text{QCoh } Y)\to D(\text{QCoh } X)$ and $R(f\circ f')_*: D(\text{QCoh } Y'')\to D(\text{QCoh } X)$.

Do the following equalities hold true:

(1) $\text{add}(Rf_* F : F \in D(\text{QCoh } Y))=\text{add}(R(f\circ g)_* G : G \in D(\text{QCoh } Y''))$ ?

(2) $\text{add}(Rf_* F : F \in D(\text{Coh } Y))=\text{add}(R(f\circ g)_* G : G \in D(\text{Coh } Y''))$ ?