Deforming syzygies of liftable modules and generalised Knörrer functors (Q2474808)

Let \(k\) be a field, \(A,C\) Henselizations of local \(k\)-algebras essentially of finite type and \(M\) a finite \(A\)-module. For a noetherian local Henselian \(k\)-algebra \((S,m_S)\) consider the Henselization of \(A\otimes_kS\) in the ideal \(A\otimes_km_S\). A deformation of \(M\) to \(S\) is an \(A_S\)-module \(M_S\), flat over \(S\), together with an \(A_S\)-linear map \(\pi:M_S\rightarrow M\) inducing an isomorphism \(M_S\otimes_Sk \rightarrow M\). Two deformations are equivalent if they are isomorphic over \(M\). Let \(Def_M^A:Hens_k\rightarrow Sets\) be the functor associating to \(S\) the set of equivalent classes of deformations \(M_S\) of \(M\) to \(S\). Let \(I\subset C\) be an ideal generated by a regular sequence of length \(n\). Suppose that \(C/I\cong A\) and the induced map \(C/I^2\rightarrow A\) has a section. Then there exists an isomorphism of \(Def_M^A\) on a naturally defined subfunctor of \(Def_{\Omega^n_M}^C\), \(\Omega^n_M\) being the \(n\)-syzygy of \(M\) over \(C\). As a consequence, if the Henselization \(A\) of \(k[X]/(f)\) in \(X=(X_1,\ldots,X_n)\), \(f\in k[X]\), is geometrically wild then the Henselization \(C\) of \(k[X,Y]/(F)\) in \(X, Y=(Y_1,\ldots,Y_q)\), \(F=f+Y_1^{n_1}+\ldots+Y_q^{n_q}\) with \(n_i>2\), is geometrically wild too. If \(A\cong P/(f)\) for a regular element \(f\in P\), \(M\) is a maximal Cohen-Macaulay \(A\)-module and \(C\) is the Henselization of \(P[Y_1,Y_2]/(f+Y_1Y_2)\) then the Knörrer functor \(H\) from the \textit{H. Knörrer} Periodicity Theorem [Invent. Math. 88, 153--164 (1987; Zbl 0617.14033)], induces an isomorphism \(Def_M^A\cong Def_{H(M)}^C\). This an extension of a previous result obtained by the author in his master's thesis and later by \textit{G. Pfister} and the reviewer [Math. Z. 223, 309--332 (1996; Zbl 0870.13003)].