Classification of isolated complete intersection singularities (Q1825914)

Let (V,0) and (W,0) be isolated complete intersection singularities and consider the \({\mathcal O}_ V\)-module \(A(V)=Ext^ 1_{{\mathcal O}_ V}(\Omega_ V,{\mathcal O}_ V)\) and the ideal \({\mathcal I}(V)=Ann_{{\mathcal O}_ V}(A(V))\) resp. A(W) and \({\mathcal I}(W)\). Main result: Suppose there exist a \({\mathbb{C}}_ n\)-algebra isomorphism \(h:\quad {\mathcal O}_ V/{\mathcal I}(V)\to {\mathcal O}_ W/{\mathcal I}(W)\) and an abelian group isomorphism \(\psi:\quad A(V)\to A(W)\) such that \(\psi (a\cdot m)=h(a)\cdot \psi (m)\) for all \(m\in A(V)\) and \(a\in {\mathcal O}_ V/{\mathcal I}(V)\). Then \({\mathcal O}_ V\to {\mathcal O}_ W\) as \({\mathbb{C}}\)-algebras. This result can be regarded as a generalization of a result of Lê and Ramanujan (which says that the moduli algebra determines the topological type of an isolated hypersurface singularity) and of a result of Mather and Yau (which establishes the theorem for hypersurface singularities).