i will talk about joint work with raymond heitmann that gives a construction of big cohen-macaulay algebra in mixed characteristics following the recent breakthroughs on the direct summand conjecture by andr\'{e} and bhatt. in fact, we prove a weakly functorial version for certain surjective ring homomorphism that leads to the solution of the vanishing conjecture for maps of tor in mixed characteristic. our work also gives a simplified proof of the direct summand conjecture, and that direct summand of regular rings are cohen-macaulay.