  Multicomponent nitrides are a hot research topic in the search of hard
coatings. The effect of substitutions on the phase stabilities, magnetic, and
elastic properties of $Al_{1-x-y}Cr_{x}Ti_{y}N$ $(0\leq x,y\leq1)$ was studied
using first principles calculations based on the density functional theory.
These calculations provide the lattice parameter, formation energy, mixing
enthalpy and elastic constants. The calculated values are in good agreement
with experiments and compare well with other theoretical results. A magnetic
transition from antiferromagnetic to ferromagnetic state occurs at
concentrations of B1-TiN higher than 60$\%$. The quaternary zone has a lower
aluminum solubility than the constituent ternary systems. The Poisson's ratio,
Shear and Young modulus were used as predictors of the hardness, indicating
that the higher hardness values are found on the transition line from B1 to B4.
The obtained results enable the design of new $Al_{1-x-y}Cr_{x}Ti_{y}N$-based
materials for coating applications.
