The Nakayama conjecture on dominant dimension

Claims Corollary 1.2: for the symmetric pair $(A,M)$ of the Tachikawa counterexample and every field extension $K$ of $\mathbb F_2(q,H_1,H_2)$, the finite-dimensional algebra $\Gamma_K=\mathrm{End}(A_K\oplus M_K)^{op}$ has all injective terms projective but infinite self-injective dimension, has a nonzero simple $S$ with $\mathrm{Ext}^i(S,\Gamma_K)=0$ for all $i\ge0$, and has a projective-injective Wakamatsu tilting module that is not tilting. Hence the classical, generalized and strong Nakayama conjectures, the Auslander-Gorenstein conjecture and the Wakamatsu tilting conjecture all fail. Corollary 1.3: $\Gamma_K$ has infinite little and big finitistic dimension. It does NOT give characteristic-zero examples.

Claims Corollary 1.2: for the symmetric pair $(A,M)$ of the Tachikawa counterexample and every field extension $K$ of $\mathbb F_2(q,H_1,H_2)$, the finite-dimensional algebra $\Gamma_K=\mathrm{End}(A_K\oplus M_K)^{op}$ has all injective terms projective but infinite self-injective dimension, has a nonzero simple $S$ with $\mathrm{Ext}^i(S,\Gamma_K)=0$ for all $i\ge0$, and has a projective-injective Wakamatsu tilting module that is not tilting. Hence the classical, generalized and strong Nakayama conjectures, the Auslander-Gorenstein conjecture and the Wakamatsu tilting conjecture all fail. Corollary 1.3: $\Gamma_K$ has infinite little and big finitistic dimension. It does NOT give characteristic-zero examples.