We present high resolution ( 0.3 ^ { \prime \prime } ) Atacama Large Millimeter Array ( ALMA ) 870 \mu m imaging of five z \approx 1.5–4.5 X-ray detected AGN ( with luminosities of L _ { { 2 - 8 keV } } > 10 ^ { 42 } erg s ^ { -1 } ) .
These data provide a \gtrsim 20 \times improvement in spatial resolution over single-dish rest-frame FIR measurements .
The sub-millimetre emission is extended on scales of FWHM \approx 0.2 ^ { \prime \prime } –0.5 ^ { \prime \prime } , corresponding to physical sizes of 1–3 kpc ( median value of 1.8 kpc ) .
These sizes are comparable to the majority of z =1–5 sub-millimetre galaxies ( SMGs ) with equivalent ALMA measurements .
In combination with spectral energy distribution analyses , we attribute this rest-frame far-infrared ( FIR ) emission to dust heated by star formation .
The implied star-formation rate surface densities are \approx 20–200 M _ { \sun } yr ^ { -1 } kpc ^ { -2 } , which are consistent with SMGs of comparable FIR luminosities ( i.e. , L _ { IR } \approx [ 1–5 ] \times 10 ^ { 12 } L _ { \sun } ) .
Although limited by a small sample of AGN , which all have high FIR luminosities , our study suggests that the kpc-scale spatial distribution and surface density of star formation in high-redshift star-forming galaxies is the same irrespective of the presence of X-ray detected AGN .