For an Artin algebra $A$ and an indecomposable non-projective module $M$ we must always have that $tau(M) cong nu Omega^2(M)$ iff $Ext_A^i(M,A)=0$ for $i=1,2$. ($nu$ being the Nakayama functor)

Making use of this to the indecomposable module $A$ as an $A^e$-module ($A^e$ being the enveloping algebra of A) and noting that $Ext_{A^e}^i(A,A^e)=Ext_A^i(D(A),A)$, we get that in case $Ext_A^i(D(A),A)=0$ for $i=1,2$, we have now that $tau(A) cong nu Omega^2(A)$ as $A^e$-modules.

Now tensoring this with an arbitrary $A$-module $M$, we get $tau(A) otimes_A M cong nu Omega^2(A) otimes_A M$.

Now we must always have that as much as projective summands $P$ that $tau(A) otimes_A M = tau(M) oplus P$ as $A$-modules.

Now I wonder if we are able to additionally write $nu Omega^2(A) otimes_A M$ in a pleasant type (avoiding the tensor product) to get a non-trivial international method for $tau(M)$?

I do know that $Omega^2(A) otimes_A M cong Omega^2(M) oplus P’$ for some projective module $P’$, however the Nakayama functor coming from the enveloping algebra appears to be extra difficult.
It cannot be true that $tau(M) cong nu Omega^2(M)$ typically aside from $A$ being selfinjective, since for instance $A$ may have finite international dimension after which the method is improper for $M$ having projective dimension equal to 1 typically.