Abstract
Let \((A,\mathfrak{m}, \mathbb{k})\) denote a local Noetherian ring and \(\mathfrak{q}\) an ideal such that \(\ell_{A}(M/\mathfrak{q}M) <\infty\) for a finitely generated \(A\)-module \(M\). Let \(\underline{a} = a_{1},\ldots,a_{d}\) denote a system of parameters of \(M\) such that \(a_{i} \in \mathfrak{q}^{c_{i}}\setminus \mathfrak{q}^{c_{i}+1}\) for \(i = 1,\ldots,d\). It follows that \(\chi:= e_{0}(\underline{a};M) - c \cdot e_{0}(\mathfrak{q};M) \geq 0\), where \(c = c_{1} \cdot \ldots \cdot c_{d}\). The main results of the report are a discussion when \(\chi = 0\) resp. to describe the value of \(\chi\) in some particular cases. Applications concern results on the multiplicity \(e_{0}(\underline{a};M)\) and applications to Bezout numbers.
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Acknowledgements
The authors are grateful to the reviewer for bibliographical comments and suggestions. Furthermore, the first named author is thankful to DAAD and HEC, Pakistan for the support of his PhD research under grant number 91524811 and 112-21480-2PS1-015 (50021731) respectively.
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Khadam, M.A., Schenzel, P. (2017). About Multiplicities and Applications to Bezout Numbers. In: Conca, A., Gubeladze, J., Römer, T. (eds) Homological and Computational Methods in Commutative Algebra. Springer INdAM Series, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-319-61943-9_13
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DOI: https://doi.org/10.1007/978-3-319-61943-9_13
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