Pushing forward certain pull back sheaf

To my previous question:

Let $Y_1=Y_2=\mathbb{P}^1$, $Y=Y_1\times Y_2$, $p_i:Y\rightarrow Y_i$, $i=1,2$ be a canonnical projections. How to compute explicitly the sheaf $p_{2*}p_1^*(\mathcal{O}_{Y_1}(1))$?

Using Proposition 9.3 of Hartshorne, $$(p_2)_*p_1^*\mathcal{O}_{\mathbb{P}^1}(1)\simeq f^*f_*\mathcal{O}_{\mathbb{P}^1}(1)$$ where $f:\mathbb{P}^1\to\mbox{Spec}(k)$ is the structural morphism. Now $$f_*\mathcal{O}_{\mathbb{P}^1}(1)\simeq H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(1))\simeq k^2=\mathcal{O}_{\mbox{Spec}(k)}\oplus\mathcal{O}_{\mbox{Spec}(k)},$$ and $$f^*f_*\mathcal{O}_{\mathbb{P}^1}(1)\simeq H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(1))\otimes_k\mathcal{O}_{\mathbb{P}^1}\simeq\mathcal{O}_{\mathbb{P}^1}\oplus\mathcal{O}_{\mathbb{P}^1}$$ Of course this can also be done by writing out the definition of the pullback and pushforward, finding the morphism on every open set, and proving it is an isomorphism on stalks.
Edit: This can be easily generalized: If $\mathcal{L}$ is a line bundle on $X\times X$ for any variety $X$, then $(p_2)_*p_1^*\mathcal{L}\simeq H^0(X,\mathcal{L})\otimes_k\mathcal{O}_X$.
• Thanx. The last thing I didn't understand precisely is the meaning of $$f^*(H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(1)))\cong H^0(\mathbb{P}^1,\mathcal{O}_{\mathbb{P}^1}(1))\otimes_k\mathcal{O}_{\mathbb{P}^1}.$$ Is it true in general that for $f:X\rightarrow Y$ and $Y$ affine and for any quasi-coherent sheaf $F$ on $X$ $$f^*f_*F\cong H^0(X, F)^{\tilde{}}\otimes_{\mathcal{O}_Y}\mathcal{O}_X?$$ Jan 12, 2015 at 22:04
• This shouldn't be true in general. Here we use that the only non empty open set of $\mbox{Spec}(k)$ is $\mbox{Spec}(k)$, and so we get global sections. Jan 16, 2015 at 11:31