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This is an answer to a question of Karl in the comments to my first answer to this question.

[EDIT: The following is a minimally simplified version of Proposition 3.3 of Hasett-Kovács04.]

Theorem. Let $X$ be a noetherian scheme, $r\in\mathbb N$, $Z\subseteq X$ a subscheme such that ${\rm codim}_XZ\geq r$, and $\mathscr F$ a coherent $\mathscr O_X$-module such that ${\rm supp}\,\mathscr F=X$ and $\mathscr F_x$ is $S_r$ for every $x\in Z$. Then $$ \mathscr H^i_Z(X,\mathscr F)=0\quad\text{for $i=0,\ldots,r-1$}. $$

Proof. Let $x\in Z$ and notice that we have the following equality of functors: $$ H^0_x = H^0_x\circ \mathscr H^0_Z $$ which induces a Grothendieck spectral sequence $$ E^{p,q}_2= H^p_x \circ \mathscr H^q_Z \Rightarrow H^{p+q}_x. $$ Now prove the statement using induction on $i$.

Suppose $\exists\,\sigma\in\mathscr H^0_Z(X,\mathcal F)$, $\sigma\neq 0$. Let $x\in Z$ be the general point of an irreducible component of ${\rm supp}\,\sigma$. Then $H^0_x(X, \mathscr H^0_Z(X,\mathscr F))\neq 0$ and hence $H^0_x(X,\mathscr F)\neq 0$. But this contradicts the assumption that $\mathscr F_x$ is $S_r$.

Now suppose that we already know that $$ \mathcal H^i_Z(X,\mathscr F)=0\quad\text{for $i=0,\ldots,k-1$} $$ for some $k<r$ and assume that $\mathscr H^k_Z(X,\mathscr F)\neq 0$. By the same argument as above we find a point such that $E^{0,k}_2=H^0_x(X,\mathscr H^k_Z(X,\mathscr F))\neq 0$. Since it is an $E^{0,k}$ term there are no differentials (including later pages of the spectral sequence) mapping to this term and all subsequent differentials mapping from this term map to something of the form $E^{p,q}$ with $0\leq q\leq k-1$. However those latter kind are zero by the inductive hypothesis. Therefore this implies that then $H^k_x(X,\mathscr F)\neq 0$ which is again a contradiction to the assumption that $\mathscr F_x$ is $S_r$. Q.E.D.

See also this MO answer

This is an answer to a question of Karl in the comments to my first answer to this question.

[EDIT: The following is a minimally simplified version of Proposition 3.3 of Hasett-Kovács04.]

Theorem. Let $X$ be a noetherian scheme, $r\in\mathbb N$, $Z\subseteq X$ a subscheme such that ${\rm codim}_XZ\geq r$, and $\mathscr F$ a coherent $\mathscr O_X$-module such that ${\rm supp}\,\mathscr F=X$ and $\mathscr F_x$ is $S_r$ for every $x\in Z$. Then $$ \mathscr H^i_Z(X,\mathscr F)=0\quad\text{for $i=0,\ldots,r-1$}. $$

Proof. Let $x\in Z$ and notice that we have the following equality of functors: $$ H^0_x = H^0_x\circ \mathscr H^0_Z $$ which induces a Grothendieck spectral sequence $$ E^{p,q}_2= H^p_x \circ \mathscr H^q_Z \Rightarrow H^{p+q}_x. $$ Now prove the statement using induction on $i$.

Suppose $\exists\,\sigma\in\mathscr H^0_Z(X,\mathcal F)$, $\sigma\neq 0$. Let $x\in Z$ be the general point of an irreducible component of ${\rm supp}\,\sigma$. Then $H^0_x(X, \mathscr H^0_Z(X,\mathscr F))\neq 0$ and hence $H^0_x(X,\mathscr F)\neq 0$. But this contradicts the assumption that $\mathscr F_x$ is $S_r$.

Now suppose that we already know that $$ \mathcal H^i_Z(X,\mathscr F)=0\quad\text{for $i=0,\ldots,k-1$} $$ for some $k<r$ and assume that $\mathscr H^k_Z(X,\mathscr F)\neq 0$. By the same argument as above we find a point such that $E^{0,k}_2=H^0_x(X,\mathscr H^k_Z(X,\mathscr F))\neq 0$. Since it is an $E^{0,k}$ term there are no differentials (including later pages of the spectral sequence) mapping to this term and all subsequent differentials mapping from this term map to something of the form $E^{p,q}$ with $0\leq q\leq k-1$. However those latter kind are zero by the inductive hypothesis. Therefore this implies that then $H^k_x(X,\mathscr F)\neq 0$ which is again a contradiction to the assumption that $\mathscr F_x$ is $S_r$. Q.E.D.

See also this MO answer

This is an answer to a question of Karl in the comments to my first answer to this question.

[EDIT: The following is a minimally simplified version of Proposition 3.3 of Hasett-Kovács04.]

Theorem. Let $X$ be a noetherian scheme, $r\in\mathbb N$, $Z\subseteq X$ a subscheme such that ${\rm codim}_XZ\geq r$, and $\mathscr F$ a coherent $\mathscr O_X$-module such that ${\rm supp}\\,\mathscr F=X$ and $\mathscr F_x$ is $S_r$ for every $x\in Z$. Then $$ \mathscr H^i_Z(X,\mathscr F)=0\quad\text{for $i=0,\ldots,r-1$}. $$

Proof. Let $x\in Z$ and notice that we have the following equality of functors: $$ H^0_x = H^0_x\circ \mathscr H^0_Z $$ which induces a Grothendieck spectral sequence $$ E^{p,q}_2= H^p_x \circ \mathscr H^q_Z \Rightarrow H^{p+q}_x. $$ Now prove the statement using induction on $i$.

Suppose $\exists\\,\sigma\in\mathscr H^0_Z(X,\mathcal F)$, $\sigma\neq 0$. Let $x\in Z$ be the general point of an irreducible component of ${\rm supp}\\,\sigma$. Then $H^0_x(X, \mathscr H^0_Z(X,\mathscr F))\neq 0$ and hence $H^0_x(X,\mathscr F)\neq 0$. But this contradicts the assumption that $\mathscr F_x$ is $S_r$.

Now suppose that we already know that $$ \mathcal H^i_Z(X,\mathscr F)=0\quad\text{for $i=0,\ldots,k-1$} $$ for some $k<r$ and assume that $\mathscr H^k_Z(X,\mathscr F)\neq 0$. By the same argument as above we find a point such that $E^{0,k}_2=H^0_x(X,\mathscr H^k_Z(X,\mathscr F))\neq 0$. Since it is an $E^{0,k}$ term there are no differentials (including later pages of the spectral sequence) mapping to this term and all subsequent differentials mapping from this term map to something of the form $E^{p,q}$ with $0\leq q\leq k-1$. However those latter kind are zero by the inductive hypothesis. Therefore this implies that then $H^k_x(X,\mathscr F)\neq 0$ which is again a contradiction to the assumption that $\mathscr F_x$ is $S_r$. Q.E.D.

See also this MO answer

This is an answer to a question of Karl in the comments to my first answer to this question.

[EDIT: The following is a minimally simplified version of Proposition 3.3 of Hasett-Kovács04.]

Theorem. Let $X$ be a noetherian scheme, $r\in\mathbb N$, $Z\subseteq X$ a subscheme such that ${\rm codim}_XZ\geq r$, and $\mathscr F$ a coherent $\mathscr O_X$-module such that ${\rm supp}\,\mathscr F=X$ and $\mathscr F_x$ is $S_r$ for every $x\in Z$. Then $$ \mathscr H^i_Z(X,\mathscr F)=0\quad\text{for $i=0,\ldots,r-1$}. $$

Proof. Let $x\in Z$ and notice that we have the following equality of functors: $$ H^0_x = H^0_x\circ \mathscr H^0_Z $$ which induces a Grothendieck spectral sequence $$ E^{p,q}_2= H^p_x \circ \mathscr H^q_Z \Rightarrow H^{p+q}_x. $$ Now prove the statement using induction on $i$.

Suppose $\exists\,\sigma\in\mathscr H^0_Z(X,\mathcal F)$, $\sigma\neq 0$. Let $x\in Z$ be the general point of an irreducible component of ${\rm supp}\,\sigma$. Then $H^0_x(X, \mathscr H^0_Z(X,\mathscr F))\neq 0$ and hence $H^0_x(X,\mathscr F)\neq 0$. But this contradicts the assumption that $\mathscr F_x$ is $S_r$.

Now suppose that we already know that $$ \mathcal H^i_Z(X,\mathscr F)=0\quad\text{for $i=0,\ldots,k-1$} $$ for some $k<r$ and assume that $\mathscr H^k_Z(X,\mathscr F)\neq 0$. By the same argument as above we find a point such that $E^{0,k}_2=H^0_x(X,\mathscr H^k_Z(X,\mathscr F))\neq 0$. Since it is an $E^{0,k}$ term there are no differentials (including later pages of the spectral sequence) mapping to this term and all subsequent differentials mapping from this term map to something of the form $E^{p,q}$ with $0\leq q\leq k-1$. However those latter kind are zero by the inductive hypothesis. Therefore this implies that then $H^k_x(X,\mathscr F)\neq 0$ which is again a contradiction to the assumption that $\mathscr F_x$ is $S_r$. Q.E.D.

See also this MO answer

This is an answer to a question of Karl in the comments to my first answer to this question.

[EDIT: The following is a minimally simplified version of Proposition 3.3 of Hasett-Kovács04.]

Theorem. Let $X$ be a noetherian scheme, $r\in\mathbb N$, $Z\subseteq X$ a subscheme such that ${\rm codim}_XZ\geq r$, and $\mathscr F$ a coherent $\mathscr O_X$-module such that ${\rm supp}\\,\mathscr F=X$ and $\mathscr F_x$ is $S_r$ for every $x\in Z$. Then $$ \mathscr H^i_Z(X,\mathscr F)=0\quad\text{for $i=0,\ldots,r-1$}. $$

Proof. Let $x\in Z$ and notice that we have the following equality of functors: $$ H^0_x = H^0_x\circ \mathscr H^0_Z $$ which induces a Grothendieck spectral sequence $$ E^{p,q}_2= H^p_x \circ \mathscr H^q_Z \Rightarrow H^{p+q}_x. $$ Now prove the statement using induction on $i$.

Suppose $\exists\\,\sigma\in\mathscr H^0_Z(X,\mathcal F)$, $\sigma\neq 0$. Let $x\in Z$ be the general point of an irreducible component of ${\rm supp}\\,\sigma$. Then $H^0_x(X, \mathscr H^0_Z(X,\mathscr F))\neq 0$ and hence $H^0_x(X,\mathscr F)\neq 0$. But this contradicts the assumption that $\mathscr F_x$ is $S_r$.

Now suppose that we already know that $$ \mathcal H^i_Z(X,\mathscr F)=0\quad\text{for $i=0,\ldots,k-1$} $$ for some $k<r$ and assume that $\mathscr H^k_Z(X,\mathscr F)\neq 0$. By the same argument as above we find a point such that $E^{0,k}_2=H^0_x(X,\mathscr H^k_Z(X,\mathscr F))\neq 0$. Since it is an $E^{0,k}$ term there are no differentials (including later pages of the spectral sequence) mapping to this term and all subsequent differentials mapping from this term map to something of the form $E^{p,q}$ with $0\leq q\leq k-1$. However those latter kind are zero by the inductive hypothesis. Therefore this implies that then $H^k_x(X,\mathscr F)\neq 0$ which is again a contradiction to the assumption that $\mathscr F_x$ is $S_r$. Q.E.D.

This is an answer to a question of Karl in the comments to my first answer to this question.

[EDIT: The following is a minimally simplified version of Proposition 3.3 of Hasett-Kovács04.]

Theorem. Let $X$ be a noetherian scheme, $r\in\mathbb N$, $Z\subseteq X$ a subscheme such that ${\rm codim}_XZ\geq r$, and $\mathscr F$ a coherent $\mathscr O_X$-module such that ${\rm supp}\\,\mathscr F=X$ and $\mathscr F_x$ is $S_r$ for every $x\in Z$. Then $$ \mathscr H^i_Z(X,\mathscr F)=0\quad\text{for $i=0,\ldots,r-1$}. $$

Proof. Let $x\in Z$ and notice that we have the following equality of functors: $$ H^0_x = H^0_x\circ \mathscr H^0_Z $$ which induces a Grothendieck spectral sequence $$ E^{p,q}_2= H^p_x \circ \mathscr H^q_Z \Rightarrow H^{p+q}_x. $$ Now prove the statement using induction on $i$.

Suppose $\exists\\,\sigma\in\mathscr H^0_Z(X,\mathcal F)$, $\sigma\neq 0$. Let $x\in Z$ be the general point of an irreducible component of ${\rm supp}\\,\sigma$. Then $H^0_x(X, \mathscr H^0_Z(X,\mathscr F))\neq 0$ and hence $H^0_x(X,\mathscr F)\neq 0$. But this contradicts the assumption that $\mathscr F_x$ is $S_r$.

Now suppose that we already know that $$ \mathcal H^i_Z(X,\mathscr F)=0\quad\text{for $i=0,\ldots,k-1$} $$ for some $k<r$ and assume that $\mathscr H^k_Z(X,\mathscr F)\neq 0$. By the same argument as above we find a point such that $E^{0,k}_2=H^0_x(X,\mathscr H^k_Z(X,\mathscr F))\neq 0$. Since it is an $E^{0,k}$ term there are no differentials (including later pages of the spectral sequence) mapping to this term and all subsequent differentials mapping from this term map to something of the form $E^{p,q}$ with $0\leq q\leq k-1$. However those latter kind are zero by the inductive hypothesis. Therefore this implies that then $H^k_x(X,\mathscr F)\neq 0$ which is again a contradiction to the assumption that $\mathscr F_x$ is $S_r$. Q.E.D.

See also this MO answer