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Many connections and dualities in representation theory can be explained
using quasi-hereditary covers in the sense of Rouquier. The concepts of
relative dominant and codominant dimension with respect to a module, introduced
recently by the first-named author, are important tools to evaluate and
classify quasi-hereditary covers.

In this paper, we prove that the relative dominant dimension of the regular
module of a quasi-hereditary algebra with a simple preserving duality with
respect to a summand $Q$ of a characteristic tilting module equals twice the
relative dominant dimension of a characteristic tilting module with respect to
$Q$.

To resolve the Temperley-Lieb algebras of infinite global dimension, we apply
this result to the class of Schur algebras $S(2, d)$ and $Q=V^{\otimes d}$ the
$d$-tensor power of the 2-dimensional module and we completely determine the
relative dominant dimension of the Schur algebra $S(2, d)$ with respect to
$V^{\otimes d}$. The $q$-analogues of these results are also obtained.

As a byproduct, we obtain a Hemmer-Nakano type result connecting the Ringel
duals of $q$-Schur algebras and Temperley-Lieb algebras. From the point of view
of Temperley-Lieb algebras, we obtain the first complete classification of
their connection to their quasi-hereditary covers formed by Ringel duals of
$q$-Schur algebras.

These results are compatible with the integral setup, and we use them to
deduce that the Ringel dual of a $q$-Schur algebra over the ring of Laurent
polynomials over the integers together with some projective module is the best
quasi-hereditary cover of the integral Temperley-Lieb algebra.