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We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence modules. We present bases of persistence modules, and give motivation as for the advantages of using them. Our focus is on developing efficient methods for the computation of homology of chains of persistence modules. Later we give a brief, self contained presentation of the Mayer-Vietoris spectral sequence. Then we study the Persistent Mayer-VietorisarXiv:1907.05228v2 fatcat:em6gkpef7rdpne56tqbhqqykrq