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Let $\sO$ be a discrete valuation ring with fraction field $K$ and residuefield $k$. A quasi-hereditary algebra $\wA$ over $\sO$ provides a bridgebetween 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 afull subcategory of the category of modules for a quantum enveloping algebrawhile $\wA_k$--mod is a full subcategory of the category of modules for areductive group in positive characteristic. This paper considers first thequestion of when the positively graded algebra $\gr \wA:= \bigoplus_{n\geq0}(\wA\cap\rad^n\wA_K)/(\wA\cap\rad^{n+1}\wA_K)$ is quasi-hereditary. A mainresult gives sufficient conditions that $\gr\wA$ be quasi-hereditary. The mainrequirement is that each graded module $\gr\wDelta(\lambda)$ arising from a$\wA$-standard (Weyl) module $\wDelta(\lambda)$ have an irreducible head. Anadditional hypothesis requires that the graded algebra $\gr \wA_K$ bequasi-hereditary, a property recently proved by us to hold in some importantcases involving quantum enveloping algebras. In the case where $\wA$ arisesfrom 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), asecond main result shows that $\gr\wA$ is quasi-hereditary. The proof dependson previous work of the authors, including a continuation of the methods thereinvolving tightly graded subalgebras, and a development of a quantumdeformation theory over $\sO$, worthy of attention in its own right, extendingthe work of Andersen-Jantzen-Soergel. As we point out, this work provides anessential step in our work on $p$-filtrations of Weyl modules for reductivealgebraic groups over fields of positive characteristic.

Forced gradings in integral quasi-hereditary algebras with applications to quantum groups