The Ringel dual of a finite highest weight ideal

Ext-self-orthogonality and finite projective dimension. Each $T(A)$ has a standard flag and each $T(B)$ has a costandard flag. All positive $\mathrm{Ext}$ groups between these two kinds of flags vanish by the locally proved criterion of [F1], giving ${\mathrm{Ext}}^i(T(A),T(B))=0$ for every $i>0$. The actual projective sequence onto each standard has a standard-filtered kernel with strictly higher labels; starting at maximal labels, where the standard is projective, downward induction on the finite poset gives finite projective dimension for every standard. Long exact sequences along a finite standard flag then give finite projective dimension for every $T(A)$ and their finite sum $T_{\mathrm{\Gamma }}$. The second flag in the Ext argument is costandard, and no standard/standard Ext vanishing is assumed.

Costandard presentations by tiltings. Dual to the normalized standard inclusion $M(A)\hookrightarrow T(A)$, every costandard object admits a surjection from a tilting object with costandard-filtered kernel; induction on the costandard flag length gives such presentations for every costandard-filtered object.

Exactness and full faithfulness of $\mathrm{Hom}(T_{\mathrm{\Gamma }},-)$. On costandard flags the functor sends the presentations of step 2.1 to projective presentations of the images; two tilting presentations lift every map and identify its kernel, so the functor is exact on costandard-filtered objects and fully faithful. It sends tilting objects to projective objects because $\mathrm{Hom}(T_{\mathrm{\Gamma }},T(A))$ is a direct summand of the regular module.

The highest-weight structure of the dual. The quotient $\mathrm{Hom}(T_{\mathrm{\Gamma }},T(A))\to {\mathrm{\Delta }}_R(A)=\mathrm{Hom}(T_{\mathrm{\Gamma }},\mathrm{\nabla }(A))$ has a kernel filtered by standards with labels strictly higher in the reversed order, and its endomorphism ring is $\mathbb{C}$ with head labelled $A$; this follows from full faithfulness in step 3.1 and the maximal tilting support of [F1]. Hence the Ringel-dual category is highest weight with the reversed poset and the displayed standards, and its filtered image is $\mathcal{F}({\mathrm{\Delta }}_R)$, not $\mathcal{F}({\mathrm{\nabla }}_R)$.

Conclusion. $T_{\mathrm{\Gamma }}$ is a characteristic tilting generator with Ext-self-orthogonal summands and finite projective dimension, the Ringel-dual category has the reversed poset with standards ${\mathrm{\Delta }}_R(A)=\mathrm{Hom}(T_{\mathrm{\Gamma }},\mathrm{\nabla }(A))$, and $\mathrm{Hom}(T_{\mathrm{\Gamma }},-)$ is an exact fully faithful equivalence from costandard-filtered objects to standard-filtered dual objects, exchanging tiltings with projectives; the algebraic realization is proved in the next theorem.