Free resolutions for multiple point spaces

This paper relates the multiple point spaces in the source and target of a corank 1 map-germ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${(\mathbb {C}^n, 0)\to(\mathbb {C}^{n+1}, 0)}$$\end{document} . Let f be such a map-germ, and, for 1 ≤ k ≤ multiplicity( f ), let Dk( f ) be its k’th multiple point scheme – the closure of the set of ordered k-tuples of pairwise distinct points sharing the same image. There are natural projections Dk+1( f ) → Dk( f ), determined by forgetting one member of the (k + 1)-tuple. We prove that the matrix of a presentation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}_{D^{k+1}(f)}}$$\end{document} over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}_{D^k(f)}}$$\end{document} appears as a certain submatrix of the matrix of a suitable presentation of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}_{\mathbb {C}^n,0}}$$\end{document} over \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {O}_{\mathbb {C}^{n+1},0}}$$\end{document} . This does not happen for germs of corank > 1.