Relating dust, gas, and the rate of star formation in M 31

Aims. We investigate the relationships between dust and gas, and study the star formation law in M 31. Methods. We have derived distributions of dust temperature and dust opacity across M 31 at 45$\arcsec$ resolution using the Spitzer data. With the opacity map and a standard dust model we de-reddened the Hα emission yielding the first Hα map of M 31 corrected for extinction. We compared the emissions from dust, Hα, HI, and H2 by means of radial distributions, pixel-to-pixel correlations, and wavelet cross-correlations. We calculated the star formation rate and star formation efficiency from the de-reddened Hα emission. Results. The dust temperature steeply decreases from 30 K near the center to 15 K at large radii. The mean dust optical depth at the Hα wavelength along the line of sight is about 0.7. The radial decrease in the dust-to-gas ratio is similar to that of the oxygen abundance. Extinction is nearly linearly correlated with the total gas surface density within limited radial intervals. On scales