twist and product of projective spaces

Question

On $\mathbb P^n_k\times_k \mathbb P^m_k$, is it true that $T_{\mathbb P^n_k\times_k \mathbb P^m_k}\otimes \mathcal O_{\mathbb P^n_k\times_k \mathbb P^m_k}(d,e)\simeq p_1^*T_{\mathbb P^n_k}(d)\oplus p_2^*T_{\mathbb P^m_k}(e)$ ? ($n,m\geq 1$ and $p_i$ are the two projections).