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Let $S$ be an abelian or K3 surface, $H$ an ample class on $S$, $v\in H^{ev}(S,\mathbb{Z})$ and $M$ the moduli space of $H$-stable sheaves on $S$ with invariants fixed by $v$. Let $\mathcal{E}$ be a universal sheaf on $M\times S$. I indicate with $\pi_{ij}$ the projection from $M\times S\times M$ to the $i^{th}$ and $j^{th}$ factors. My goal is to show that the relative Ext sheaf $$ \mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$$ is a line bundle supported on the diagonal $\Delta\subset M\times M$. I had solved this problem in the following way. First of all let's recall the (one of the corollary) Base Change Theorem as stated in Mumford, Abelian Varieties:

let $f:X\to Y$ be a proper morphism of noetherian schemes and $\mathcal{F}$ a coherent sheaf on $X$. If $\dim H^p(X_y,\mathcal{F}_y)$ is constant for every $y\in Y$, then $R^pf_*\mathcal{F}$ is locally free and $R^pf_*\mathcal{F}\otimes_{\mathcal{O}_Y}k(y)=H^p(X_y,\mathcal{F}_y)$, where $k(y)$ is the field at $y$.

Let's apply this with $f=\pi_{13}$ and $\mathcal{F}=\mathcal{H}om(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$. Since $M$ parametrizes stable sheaves on $S$, the latter sheaves is supported on the diagonal. On the other hand, since both $M$ and $S$ are symplectic, Serre duality assures us that also $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$ is supported on the diagonal. If $(m,m)=([F],[F])\in\Delta$, where $[F]$ is the class of a stable sheaf on $S$, then $$\dim H^2(X_{(m,m)},\mathcal{F}_{(m,m)})=\dim H^2(S,\mathcal{H}om_{\mathcal{O}_S}(F,F))=$$$$=\dim Ext^2(F,F)=\dim End(F)^{\vee}=1$$ where we used Serre duality and the fact that a stable sheaf is simple. Hence the BCT implies that $R^2\pi_{13,*}\mathcal{H}om(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})=\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$ is locally free of rank $1$.

Anyway, I have recently looked at the Mukai paper, 'On the moduli space of bundle on K3 surfaces I', Proposition 4.10, where there is Mukai's proof of this statement. It turns out that his proof is quite different: he uses a lot of facts about the diagonal (e.g. $\Delta$ is locally a complete intersection and the ideal of sheaves $\mathcal{I}_{\Delta}$ annihilates $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$). Of course I trust Mukai more than myself and in fact I can see that the remark that the diagonal is a locally complete intersection is quite important and I needed it in my argument. Nevertheless, I cannot see other points in which my argument fails.

Can anybody point out some errors in my argument and suggest how to fix them?

Thank you very much.

EDIT: as Bernie noted, I cannot use the classical Base Change Theorem in this case and I need the ext-formulation (see the referement in the comments under Bernie's answer). Up to this modification, the conclusion looks 'formally' right. Anyway, why does Mukai need that $\mathcal{I}_{\Delta}$ annihilates $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$, while I don't need this (or at least, I cannot see where I am using it, if I am using it)?

Let $S$ be an abelian or K3 surface, $H$ an ample class on $S$, $v\in H^{ev}(S,\mathbb{Z})$ and $M$ the moduli space of $H$-stable sheaves on $S$ with invariants fixed by $v$. Let $\mathcal{E}$ be a universal sheaf on $M\times S$. I indicate with $\pi_{ij}$ the projection from $M\times S\times M$ to the $i^{th}$ and $j^{th}$ factors. My goal is to show that the relative Ext sheaf $$ \mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$$ is a line bundle supported on the diagonal $\Delta\subset M\times M$. I had solved this problem in the following way. First of all let's recall the (one of the corollary) Base Change Theorem as stated in Mumford, Abelian Varieties:

let $f:X\to Y$ be a proper morphism of noetherian schemes and $\mathcal{F}$ a coherent sheaf on $X$. If $\dim H^p(X_y,\mathcal{F}_y)$ is constant for every $y\in Y$, then $R^pf_*\mathcal{F}$ is locally free and $R^pf_*\mathcal{F}\otimes_{\mathcal{O}_Y}k(y)=H^p(X_y,\mathcal{F}_y)$, where $k(y)$ is the field at $y$.

Let's apply this with $f=\pi_{13}$ and $\mathcal{F}=\mathcal{H}om(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$. Since $M$ parametrizes stable sheaves on $S$, the latter sheaves is supported on the diagonal. On the other hand, since both $M$ and $S$ are symplectic, Serre duality assures us that also $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$ is supported on the diagonal. If $(m,m)=([F],[F])\in\Delta$, where $[F]$ is the class of a stable sheaf on $S$, then $$\dim H^2(X_{(m,m)},\mathcal{F}_{(m,m)})=\dim H^2(S,\mathcal{H}om_{\mathcal{O}_S}(F,F))=$$$$=\dim Ext^2(F,F)=\dim End(F)^{\vee}=1$$ where we used Serre duality and the fact that a stable sheaf is simple. Hence the BCT implies that $R^2\pi_{13,*}\mathcal{H}om(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})=\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$ is locally free of rank $1$.

Anyway, I have recently looked at the Mukai paper, 'On the moduli space of bundle on K3 surfaces I', Proposition 4.10, where there is Mukai's proof of this statement. It turns out that his proof is quite different: he uses a fact about the diagonal, e.g. the ideal of sheaves $\mathcal{I}_{\Delta}$ annihilates $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$. Of course I trust Mukai more than myself. Nevertheless, I cannot see why and where I need that remark in my argument.

Can anybody point out some errors in my argument and suggest how to fix them?

Thank you very much.

EDIT: as Bernie noted, I cannot use the classical Base Change Theorem in this case and I need the ext-formulation (see the referement in the comments under Bernie's answer). Up to this modification, the conclusion looks 'formally' right. Anyway, why does Mukai need that $\mathcal{I}_{\Delta}$ annihilates $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$?

Let $S$ be an abelian or K3 surface, $H$ an ample class on $S$, $v\in H^{ev}(S,\mathbb{Z})$ and $M$ the moduli space of $H$-stable sheaves on $S$ with invariants fixed by $v$. Let $\mathcal{E}$ be a universal sheaf on $M\times S$. I indicate with $\pi_{ij}$ the projection from $M\times S\times M$ to the $i^{th}$ and $j^{th}$ factors. My goal is to show that the relative Ext sheaf $$ \mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$$ is a line bundle supported on the diagonal $\Delta\subset M\times M$. I had solved this problem in the following way. First of all let's recall the (one of the corollary) Base Change Theorem as stated in Mumford, Abelian Varieties:

let $f:X\to Y$ be a proper morphism of noetherian schemes and $\mathcal{F}$ a coherent sheaf on $X$. If $\dim H^p(X_y,\mathcal{F}_y)$ is constant for every $y\in Y$, then $R^pf_*\mathcal{F}$ is locally free and $R^pf_*\mathcal{F}\otimes_{\mathcal{O}_Y}k(y)=H^p(X_y,\mathcal{F}_y)$, where $k(y)$ is the field at $y$.

Let's apply this with $f=\pi_{13}$ and $\mathcal{F}=\mathcal{H}om(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$. Since $M$ parametrizes stable sheaves on $S$, the latter sheaves is supported on the diagonal. On the other hand, since both $M$ and $S$ are symplectic, Serre duality assures us that also $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$ is supported on the diagonal. If $(m,m)=([F],[F])\in\Delta$, where $[F]$ is the class of a stable sheaf on $S$, then $$\dim H^2(X_{(m,m)},\mathcal{F}_{(m,m)})=\dim H^2(S,\mathcal{H}om_{\mathcal{O}_S}(F,F))=$$$$=\dim Ext^2(F,F)=\dim End(F)^{\vee}=1$$ where we used Serre duality and the fact that a stable sheaf is simple. Hence the BCT implies that $R^2\pi_{13,*}\mathcal{H}om(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})=\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$ is locally free of rank $1$.

Anyway, I have recently looked at the Mukai paper, 'On the moduli space of bundle on K3 surfaces I', Proposition 4.10, where there is Mukai's proof of this statement. It turns out that his proof is quite different: he uses a lot of facts about the diagonal (e.g. $\Delta$ is locally a complete intersection and the ideal of sheaves $\mathcal{I}_{\Delta}$ annihilates $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$). Of course I trust Mukai more than myself and in fact I can see that the remark that the diagonal is a locally complete intersection is quite important and I needed it in my argument. Nevertheless, I cannot see other points in which my argument fails.

Can anybody point out some errors in my argument and suggest how to fix them?

Thank you very much.

Let $S$ be an abelian or K3 surface, $H$ an ample class on $S$, $v\in H^{ev}(S,\mathbb{Z})$ and $M$ the moduli space of $H$-stable sheaves on $S$ with invariants fixed by $v$. Let $\mathcal{E}$ be a universal sheaf on $M\times S$. I indicate with $\pi_{ij}$ the projection from $M\times S\times M$ to the $i^{th}$ and $j^{th}$ factors. My goal is to show that the relative Ext sheaf $$ \mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$$ is a line bundle supported on the diagonal $\Delta\subset M\times M$. I had solved this problem in the following way. First of all let's recall the (one of the corollary) Base Change Theorem as stated in Mumford, Abelian Varieties:

let $f:X\to Y$ be a proper morphism of noetherian schemes and $\mathcal{F}$ a coherent sheaf on $X$. If $\dim H^p(X_y,\mathcal{F}_y)$ is constant for every $y\in Y$, then $R^pf_*\mathcal{F}$ is locally free and $R^pf_*\mathcal{F}\otimes_{\mathcal{O}_Y}k(y)=H^p(X_y,\mathcal{F}_y)$, where $k(y)$ is the field at $y$.

Let's apply this with $f=\pi_{13}$ and $\mathcal{F}=\mathcal{H}om(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$. Since $M$ parametrizes stable sheaves on $S$, the latter sheaves is supported on the diagonal. On the other hand, since both $M$ and $S$ are symplectic, Serre duality assures us that also $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$ is supported on the diagonal. If $(m,m)=([F],[F])\in\Delta$, where $[F]$ is the class of a stable sheaf on $S$, then $$\dim H^2(X_{(m,m)},\mathcal{F}_{(m,m)})=\dim H^2(S,\mathcal{H}om_{\mathcal{O}_S}(F,F))=$$$$=\dim Ext^2(F,F)=\dim End(F)^{\vee}=1$$ where we used Serre duality and the fact that a stable sheaf is simple. Hence the BCT implies that $R^2\pi_{13,*}\mathcal{H}om(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})=\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$ is locally free of rank $1$.

Anyway, I have recently looked at the Mukai paper, 'On the moduli space of bundle on K3 surfaces I', Proposition 4.10, where there is Mukai's proof of this statement. It turns out that his proof is quite different: he uses a lot of facts about the diagonal (e.g. $\Delta$ is locally a complete intersection and the ideal of sheaves $\mathcal{I}_{\Delta}$ annihilates $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$). Of course I trust Mukai more than myself and in fact I can see that the remark that the diagonal is a locally complete intersection is quite important and I needed it in my argument. Nevertheless, I cannot see other points in which my argument fails.

Can anybody point out some errors in my argument and suggest how to fix them?

Thank you very much.

EDIT: as Bernie noted, I cannot use the classical Base Change Theorem in this case and I need the ext-formulation (see the referement in the comments under Bernie's answer). Up to this modification, the conclusion looks 'formally' right. Anyway, why does Mukai need that $\mathcal{I}_{\Delta}$ annihilates $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$, while I don't need this (or at least, I cannot see where I am using it, if I am using it)?

Let $S$ be an abelian or K3 surface, $H$ an ample class on $S$, $v\in H^{ev}(S,\mathbb{Z})$ and $M$ the moduli space of $H$-stable sheaves on $S$ with invariants fixed by $v$. Let $\mathcal{E}$ be a universal sheaf on $M\times S$. I indicate with $\pi_{ij}$ the projection from $M\times S\times M$ to the $i^{th}$ and $j^{th}$ factors. My goal is to show that the relative Ext sheaf $$ \mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$$ is a line bundle supported on the diagonal $\Delta\subset M\times M$. I had solved this problem in the following way. First of all let's recall the (one of the corollary) Base Change Theorem as stated in Mumford, Abelian Varieties:

let $f:X\to Y$ be a proper morphism of noetherian schemes and $\mathcal{F}$ a coherent sheaf on $X$. If $\dim H^p(X_y,\mathcal{F}_y)$ is constant for every $y\in Y$, then $R^pf_*\mathcal{F}$ is locally free and $R^pf_*\mathcal{F}\otimes_{\mathcal{O}_Y}k(y)=H^p(X_y,\mathcal{F}_y)$, where $k(y)$ is the field at $y$.

Let's apply this with $f=\pi_{13}$ and $\mathcal{F}=\mathcal{H}om(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$. Since $M$ parametrizes stable sheaves on $S$, the latter sheaves is supported on the diagonal. On the other hand, since both $M$ and $S$ are symplectic, Serre duality assures us that also $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$ is supported on the diagonal. If $(m,m)=([F],[F])\in\Delta$, where $[F]$ is the class of a stable sheaf on $S$, then $$\dim H^2(X_{(m,m)},\mathcal{F}_{(m,m)})=\dim H^2(S,\mathcal{H}om_{\mathcal{O}_S}(F,F))=$$$$=\dim Ext^2(F,F)=\dim End(F)^{\vee}=1$$ where we used Serre duality and the fact that a stable sheaf is simple. Hence the BCT implies that $R^2\pi_{13}\mathcal{H}om(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})=\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$ is locally free of rank $1$.

Anyway, I have recently looked at the Mukai paper, 'On the moduli space of bundle on K3 surfaces I', Proposition 4.10, where there is Mukai's proof of this statement. It turns out that his proof is quite different: he uses a lot of facts about the diagonal (e.g. $\Delta$ is locally a complete intersection and the ideal of sheaves $\mathcal{I}_{\Delta}$ annihilates $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$). Of course I trust Mukai more than myself and in fact I can see that the remark that the diagonal is a locally complete intersection is quite important and I needed it in my argument. Nevertheless, I cannot see other points in which my argument fails.

Can anybody point out some errors in my argument and suggest how to fix them?

Thank you very much.

Let $S$ be an abelian or K3 surface, $H$ an ample class on $S$, $v\in H^{ev}(S,\mathbb{Z})$ and $M$ the moduli space of $H$-stable sheaves on $S$ with invariants fixed by $v$. Let $\mathcal{E}$ be a universal sheaf on $M\times S$. I indicate with $\pi_{ij}$ the projection from $M\times S\times M$ to the $i^{th}$ and $j^{th}$ factors. My goal is to show that the relative Ext sheaf $$ \mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$$ is a line bundle supported on the diagonal $\Delta\subset M\times M$. I had solved this problem in the following way. First of all let's recall the (one of the corollary) Base Change Theorem as stated in Mumford, Abelian Varieties:

let $f:X\to Y$ be a proper morphism of noetherian schemes and $\mathcal{F}$ a coherent sheaf on $X$. If $\dim H^p(X_y,\mathcal{F}_y)$ is constant for every $y\in Y$, then $R^pf_*\mathcal{F}$ is locally free and $R^pf_*\mathcal{F}\otimes_{\mathcal{O}_Y}k(y)=H^p(X_y,\mathcal{F}_y)$, where $k(y)$ is the field at $y$.

Let's apply this with $f=\pi_{13}$ and $\mathcal{F}=\mathcal{H}om(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$. Since $M$ parametrizes stable sheaves on $S$, the latter sheaves is supported on the diagonal. On the other hand, since both $M$ and $S$ are symplectic, Serre duality assures us that also $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$ is supported on the diagonal. If $(m,m)=([F],[F])\in\Delta$, where $[F]$ is the class of a stable sheaf on $S$, then $$\dim H^2(X_{(m,m)},\mathcal{F}_{(m,m)})=\dim H^2(S,\mathcal{H}om_{\mathcal{O}_S}(F,F))=$$$$=\dim Ext^2(F,F)=\dim End(F)^{\vee}=1$$ where we used Serre duality and the fact that a stable sheaf is simple. Hence the BCT implies that $R^2\pi_{13,*}\mathcal{H}om(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})=\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$ is locally free of rank $1$.

Anyway, I have recently looked at the Mukai paper, 'On the moduli space of bundle on K3 surfaces I', Proposition 4.10, where there is Mukai's proof of this statement. It turns out that his proof is quite different: he uses a lot of facts about the diagonal (e.g. $\Delta$ is locally a complete intersection and the ideal of sheaves $\mathcal{I}_{\Delta}$ annihilates $\mathcal{E}xt^2_{\pi_{13}}(\pi^*_{12}\mathcal{E},\pi^*_{23}\mathcal{E})$). Of course I trust Mukai more than myself and in fact I can see that the remark that the diagonal is a locally complete intersection is quite important and I needed it in my argument. Nevertheless, I cannot see other points in which my argument fails.

Can anybody point out some errors in my argument and suggest how to fix them?

Thank you very much.