We use a mass complete (log($M/M_{\odot}$) $\geqslant$ 9.6) sample of galaxies with accurate photometric redshifts in the COSMOS field to construct the density field and the cosmic web to $z$=1.2. The comic web extraction relies on the density field Hessian matrix and breaks the density field into clusters, filaments and the field. We provide the density field and cosmic web measures to the community. We show that at $z$ $\lesssim$ 0.8, the median star-formation rate (SFR) in the cosmic web gradually declines from the field to clusters and this decline is especially sharp for satellites ($\sim$ 1 dex vs. $\sim$ 0.5 dex for centrals). However, at $z$ $\gtrsim$ 0.8, the trend flattens out for the overall galaxy population and satellites. For star-forming galaxies only, the median SFR is constant at $z$ $\gtrsim$ 0.5 but declines by $\sim$ 0.3-0.4 dex from the field to clusters for satellites and centrals at $z$ $\lesssim$ 0.5. We argue that for satellites, the main role of the cosmic web environment is to control their star-forming fraction, whereas for centrals, it is mainly to control their overall SFR at $z$ $\lesssim$ 0.5 and to set their fraction at $z$ $\gtrsim$ 0.5. We suggest that most satellites experience a rapid quenching mechanism as they fall from the field into clusters through filaments, whereas centrals mostly undergo a slow environmental quenching at $z$ $\lesssim$ 0.5 and a fast mechanism at higher redshifts. Our preliminary results highlight the importance of the large-scale cosmic web on galaxy evolution.