Global dimension of endomorphism rings and rings of differential operators of toric algebras

Eleonore Faber (Leeds)

Frank Adams 1,

In this talk we consider a normal toric algebra \(R\) over a field \(k\) of arbitrary characteristic. The module \(M\) of \((p^e)^{\operatorname{th}}\) roots of \(R\), where \(p\) and \(e\) are positive integers, is then the direct sum of so-called conic modules. With a combinatorial method we construct certain complexes of conic modules over \(R\) and explain how these yield projective resolutions of simple modules over the endomorphism ring \(\operatorname{End}_R(M)\). Thus we obtain a bound on the global dimension of \(\operatorname{End}_R(M)\), which shows that this endomorphism ring is a so-called noncommutative resolution of singularities (NCR) of \(R\) (or \(\operatorname{Spec}(R)\)). If the characteristic of \(k\) is \(p>0\), then this fact allows us to bound the global dimension of the ring of differential operators \(D(R)\). This is joint work with Greg Muller and Karen E. Smith.

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