  Analyses of type Ia supernovae (SNe Ia) have found puzzling correlations
between their standardised luminosities and host galaxy properties: SNe Ia in
high-mass, passive hosts appear brighter than those in lower-mass, star-forming
hosts. We examine the host galaxies of SNe Ia in the Dark Energy Survey
three-year spectroscopically-confirmed cosmological sample, obtaining
photometry in a series of "local" apertures centred on the SN, and for the
global host galaxy. We study the differences in these host galaxy properties,
such as stellar mass and rest-frame $U-R$ colours, and their correlations with
SN Ia parameters including Hubble residuals. We find all Hubble residual steps
to be $>3\sigma$ in significance, both for splitting at the traditional
environmental property sample median and for the step of maximum significance.
For stellar mass, we find a maximal local step of $0.098\pm0.018$ mag; $\sim
0.03$ mag greater than the largest global stellar mass step in our sample
($0.070 \pm 0.017$ mag). When splitting at the sample median, differences
between local and global $U-R$ steps are small, both $\sim 0.08$ mag, but are
more significant than the global stellar mass step ($0.057\pm0.017$ mag). We
split the data into sub-samples based on SN Ia light curve parameters: stretch
($x_1$) and colour ($c$), finding that redder objects ($c > 0$) have larger
Hubble residual steps, for both stellar mass and $U-R$, for both local and
global measurements, of $\sim0.14$ mag. Additionally, the bluer (star-forming)
local environments host a more homogeneous SN Ia sample, with local $U-R$
r.m.s. scatter as low as $0.084 \pm 0.017$ mag for blue ($c < 0$) SNe Ia in
locally blue $U-R$ environments.
