Acyclic assembly with exact columns

Each nonzero column is $k\stackrel{1}{\to }k$ in degrees one and zero. Its degree-one kernel and degree-zero cokernel are zero, and all other columns and homology degrees are zero. Thus every column is acyclic, including its degree-zero homology. The first-quadrant support satisfies [F1], so there is no surviving edge complex and assembly gives an acyclic total complex.

For a direct check order total degree one as $C_{1,0}\oplus C_{0,1}$. The total complex is $k\stackrel{x\mapsto (x,0)}{\to }{k}^2\stackrel{(u,v)\mapsto v}{\to }k$ in degrees $2,1,0$. The first map is injective, the second is surjective, and the kernel of the second is $k\oplus 0$, exactly the first image. Hence $H_2=H_1=H_0=0$, with all other degrees already zero. This checks the two endpoints and the middle image-kernel equality independently of the assembly invocation, using explicit maps and no AC.