On Aluffi's problem and blowup algebras of certain modules (Q343579)

Let \(\varphi: R \to A\) be a surjective homomorphism of noetherian commutative rings, \(\mathfrak{a} \subset A\) an ideal, and \(I=\phi^{-1}(\mathfrak{a}) \subseteq R\). Introduced by \textit{P. Aluffi} [Tohoku Math. J., II. Ser. 56, No. 4, 593--619 (2004; Zbl 1061.14006)], the quasi-symmetric algebra of \(\mathfrak{a}\) with respect to \(\varphi\) is defined by NEWLINE\[NEWLINE\text{qSym}_\varphi(\mathfrak a)=\text{Sym}_A(\mathfrak a) \otimes_{\text{Sym}_R(I)} \mathcal{R}_R(I),NEWLINE\]NEWLINE where \(\text{Sym}_A(\mathfrak a) \) is the symmetric algebra of the \(A\)-ideal \(\mathfrak a\) and \(\mathcal{R}_R(I)\) is the Rees algebra of the \(R\)-ideal \(I\).NEWLINENEWLINEIn this paper, the author defines a generalization of this construction for modules. Let \(R\) be a noetherian ring, \(M\) a finitely generated \(R\)-module, \(J\) an ideal of \(R\) such that \(JM=(0)\), and \(\pi : N \to M\) an epimorphism where \(N\) is a finitely generated \(R\)-module of generic, constant rank. The quasi-symmetric algebra of \(M\) with respect to \((\pi, J)\) is defined to be NEWLINE\[NEWLINE\text{qSym}_{(\pi,J)}(M)=\text{Sym}_{R/J}(M) \otimes_{\text{Sym}_R(N)} \mathcal{R}_R(N).NEWLINE\]NEWLINENEWLINENEWLINEThis construction recovers the original concept of Aluffi. Indeed, with the notation introduced above, if one takes \(\varphi\mid_ {I}: I \to \mathfrak{a}\) to be the surjective homomorphism induced by \(\varphi\) and \(J = \text{Ker} \varphi\), then NEWLINE\[NEWLINE\text{qSym}_\varphi(\mathfrak a)=\text{qSym}_{(\varphi\mid_I, J)}(\mathfrak a).NEWLINE\]NEWLINENEWLINENEWLINEA similar, but not identical, generalization has been introduced by \textit{Z. Ramos} and \textit{A. Simis} [J. Algebra 467, 155--182 (2016; Zbl 1354.13010)]; the author discusses the differences between the two approaches, particularly with respect to torsion modules.NEWLINENEWLINEIn the case when \(X=V(F)\) is a quasi-homogeneous algebraic hypersurface in \(\mathbb{A}_k^n\), the author provides an explicit computation of the quasi-symmetric algebra of the module of derivations \(\text{Der}_k(\mathcal{O}_X))\).