Tate resolutions for Segre embeddings

We give an explicit description of the terms and differentials of the Tate resolution of sheaves arising from Segre embeddings of \(\mathbb{P}^a\times\mathbb{P}^b\). We prove that the maps in this Tate resolution are either coming from Sylvester-type maps, or from Bezout-type maps arising from the so-called toric Jacobian.