Rings of differential operators on (k,s)-th Tjurina algebras of singularities

Abstract

Abstract In this paper, we give a description of differential operators on tensor products A ⊗ 𝕂 B {A\otimes_{\mathbb{K}}B} , where A and B are finitely generated 𝕂 {\mathbb{K}} -algebras. We prove that any differential operator on A ⊗ 𝕂 B {A\otimes_{\mathbb{K}}B} can be written as a finite sum of D 1 ⊗ D 2 {D_{1}\otimes D_{2}} , where D 1 {D_{1}} and D 2 {D_{2}} are differential operators on A and B, respectively. Moreover, we introduce a series of new invariants, the ( k , s ) {(k,s)} -th Tjurina algebra A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} for an isolated hypersurface singularity ( V , 𝟎 ) = ( V ⁢ ( f ) , 𝟎 ) ⊆ ( ℂ r , 𝟎 ) {(V,\boldsymbol{0})=(V(f),\boldsymbol{0})\subseteq(\mathbb{C}^{r},\boldsymbol{% 0})} . We formulate a sharp upper estimate for the dimension of the ℂ {\mathbb{C}} -vector space of differential operators on A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} of order at most 1, and we give lower and upper bounds for the dimension of the ℂ {\mathbb{C}} -vector space of differential operators on A ( k , s ) ⁢ ( V ) {A_{(k,s)}(V)} of order at most n.