  Let $\sO$ be a discrete valuation ring with fraction field $K$ and residue
field $k$. A quasi-hereditary algebra $\wA$ over $\sO$ provides a bridge
between the representation theory of the quasi-hereditary algebra
$\wA_K:=K\otimes \wA$ over the field $K$ and the quasi-hereditary algebra
$A_k:=k\otimes_\sO\wA$ over $k$. In one important example, $\wA_K$--mod is a
full subcategory of the category of modules for a quantum enveloping algebra
while $\wA_k$--mod is a full subcategory of the category of modules for a
reductive group in positive characteristic. This paper considers first the
question of when the positively graded algebra $\gr \wA:= \bigoplus_{n\geq
0}(\wA\cap\rad^n\wA_K)/(\wA\cap\rad^{n+1}\wA_K)$ is quasi-hereditary. A main
result gives sufficient conditions that $\gr\wA$ be quasi-hereditary. The main
requirement is that each graded module $\gr\wDelta(\lambda)$ arising from a
$\wA$-standard (Weyl) module $\wDelta(\lambda)$ have an irreducible head. An
additional hypothesis requires that the graded algebra $\gr \wA_K$ be
quasi-hereditary, a property recently proved by us to hold in some important
cases involving quantum enveloping algebras. In the case where $\wA$ arises
from regular dominant weights for a quantum enveloping algebra at a primitive
$p$th root of unity for a prime $p>2h-2$ (where $h$ is the Coxeter number), a
second main result shows that $\gr\wA$ is quasi-hereditary. The proof depends
on previous work of the authors, including a continuation of the methods there
involving tightly graded subalgebras, and a development of a quantum
deformation theory over $\sO$, worthy of attention in its own right, extending
the work of Andersen-Jantzen-Soergel. As we point out, this work provides an
essential step in our work on $p$-filtrations of Weyl modules for reductive
algebraic groups over fields of positive characteristic.
