Computing the global and Hochschild dimensions of a free product of direct products of fields.

Let $k$ be a field (algebraically closed of characteristic 0, but I do not expect it to make a difference).

Consider the algebra $A=k^{n+1}\ast k^{m+1}$, where $\ast$ denotes the free product of associative algebras. Then $A$ can also be written as a quotient of a free algebra $A=k\langle x_1,\ldots,x_n,y_1,\ldots,y_m\rangle/I$, where $I$ is generated by the relations

1. $x_i^2=x_i$, $1\leqslant i\leqslant n;$
2. $y_j^2=y_j$, $1\leqslant j\leqslant m;$
3. $x_ix_j$, $1\leqslant i,j\leqslant n$, $i\neq j$;
4. $y_iy_j$, $1\leqslant i,j\leqslant m$, $i\neq j$.

How would one compute the Hochschild homology and cohomology of $A$? I know that $A$ has left global dimension 1 as a free product of algebras of global dimension 0.

What is the projective dimension of $A$ as an $A$-bimodule? I didn't find the bar resolution suitable for this computation and failed miserably at finding a simpler one.