Restricting homology to hypersurfaces. (English) Zbl 07907778
Carlson, Jon F. (ed.) et al., Geometric and topological aspects of the representation theory of finite groups. PIMS summer school and workshop, Vancouver, Canada, July 27 – August 5, 2016. Cham: Springer; Vancouver: Pacific Institute for the Mathematical Sciences. Springer Proc. Math. Stat. 242, 1-23 (2018).
Summary: This paper concerns the homological properties of a module \(M\) over a ring \(R\) relative to a presentation \(R\cong P/I\), where \(P\) is local ring. It is proved that the Betti sequence of \(M\) with respect to \(P/(f)\) for a regular element \(f\) in \(I\) depends only on the class of \(f\) in \(I/\mathfrak{n}I\), where \(\mathfrak{n}\) is the maximal ideal of \(P\). Applications to the theory of supports sets in local algebra and in the modular representation theory of elementary abelian groups are presented.
For the entire collection see [Zbl 1410.20001].
For the entire collection see [Zbl 1410.20001].
MSC:
| 13D07 | Homological functors on modules of commutative rings (Tor, Ext, etc.) |
| 16E45 | Differential graded algebras and applications (associative algebraic aspects) |
| 13D02 | Syzygies, resolutions, complexes and commutative rings |
| 13D40 | Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series |
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