  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.
