Pull-back of push-out under the diagonal embedding Let $X$ be a variety, $\Delta:X\to X\times X$ be a diagonal embedding. How to compute $\Delta^*\Delta_*\mathcal{O}

Pull-back of push-out under the diagonal embedding Let $X$ be a variety, $\Delta:X\to X\times X$ be a diagonal embedding. How to compute $\Delta^\Delta_\mathcal{O}_X$?

Recall the defintion of pullback: Let $f:X\rightarrow Y$ and $G$ be a sheaf on $Y$. Then $f^*G:$$=f^{-1}{G} \otimes_{f^{-1}O_Y}O_X$.

Let $F$ be a sheaf on $X$. If $f$ is injective then we have $$f^{-1}f_*F(U)=\lim_{\rightarrow_{f(U)\subset V}}f_*F(V)=\lim_{\rightarrow_{U\subset f^{-1}(V)}}F(f^{-1}(V))=F(U).$$ This implies $f^{-1}f_*F=F$ so $f^* f_*F=F\otimes_{f^{-1}O_Y}O_X$.

Now take $f=\Delta$ and $Y=X\times X$. Then $\Delta^* \Delta_*O_X=O_X\otimes_{\Delta^{-1}O_{X\times X}}O_X$.

Pull-back of push-out under the diagonal embedding Let $X$ be a variety, $\Delta:X\to X\times X$ be a diagonal embedding. How to compute $\Delta^*\Delta_*\mathcal{O}_X$?

Pull-back of push-out under the diagonal embedding Let $X$ be a variety, $\Delta:X\to X\times X$ be a diagonal embedding. How to compute $\Delta^\Delta_\mathcal{O}_X$?