A resolved warm/dense gas Schmidt-Kennicutt relationship in a binary
  HyLIRG at $z=2.41$
  Hyperluminous infrared galaxies (HyLIRGs) provide crucial "long lever arm"
constraints on galaxy evolution. H-ATLAS $J084933.4+021443$, a $z=2.41$ binary
HyLIRG with at least two additional luminous companion galaxies, is thus an
optimal test-ground for studies of star formation and galaxy evolution during
"cosmic noon". We have used ALMA to obtain resolved imaging and kinematics of
atomic and molecular emission lines, and rest-frame $340$ to $1160$GHz
continuum emission, for the known luminous component galaxies in H-ATLAS
$J084933.4+021443$: W, T, M, C. All four component galaxies are spatially
($\sim 0 .\!\!^{''} 3$ or $2.5$kpc) resolved in CO J:7-6, [C$_\mathrm{I}$] 2-1,
H$_2$O and the millimetre (mm) to sub-mm continuum. Rotation-dominated gas
kinematics is confirmed in W and T. The significant extension to component T,
in gas and continuum, along its kinematic minor axis, is attributable to its
lensing magnification. Spatially resolved sub-mm spectral energy distributions
reveal that component W is well fit with greybody emission from dust at a
single temperature over the full extent of the galaxy, despite it containing a
powerful AGN, while component T requires an additional component of hotter
nuclear dust and additional sources of emission in the mm. We confirm that
[C$_\mathrm{I}$] 2-1 can be used as a rough tracer of warm/dense molecular gas
in extreme systems, though the [C$_\mathrm{I}$] 2-1/CO luminosity ratio
increases sub-linearly. We obtain an exquisite and unprecedented resolved
($2.5$-kpc-scale) "warm/dense molecular gas" Schmidt-Kennicutt (SK)
relationship for components W and T. Gas exhaustion times for all apertures in
W (T) are $1-4$Gyr ($0.5-2$Gyr). Both W and T follow a resolved "warm/dense
gas" SK relationship with power law $n\sim1.7$, significantly steeper than the
$n\sim1$ found previously via "cold" molecular gas in nearby "normal"
star-forming galaxies.
