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The theory of ordinal length

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Abstract

Gulliksen, following Bass’s observations, extended the notion of length to an ordinal-valued invariant defined on the class of finitely generated modules over a Noetherian ring. We show how to calculate this combinatorial invariant by means of the fundamental cycle of the module, thus linking the lattice of submodules to homological properties of the module. Using this, we equip each module with its canonical topology. From ordinal length, other ordinal-valued invariants can be derived, such as filtration rank.

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Notes

  1. Although very plausible, I do not know whether the valence of the filtration rank is equal to \(l^{\text {pr}}(M)\).

  2. See, for instance, (Eisenbud (1995), p. 102); not to be confused with the multiplicity of a module at a primary ideal given in terms of its Hilbert function.

  3. Our terminology differs slightly from the literature, where, at least for local rings, the condition is formulated over the completion.

  4. This is Eisenbud’s nomenclature in (Eisenbud (1995), p. 93), whereas Dress (in the original paper Dress 1993) would require in addition that all primes in \(\mathcal P\) are non-embedded. Since everything is determined by the nature of \(\mathrm{Ass }(M)\), there seems to be no need in distinguishing between the two.

  5. The adverb ‘pretty’ can mean either ‘fairly’ or ‘very’, and it is the latter usage (i.e., more than clean) that is intended here, whereas in informal speech ‘pretty clean’ would rather mean the former (i.e., less than clean).

  6. I am grateful to S. Arslan for providing me the semi-group data.

  7. Here ‘ci’ stands for ‘complete intersection’, although this terms is usually only reserved for ideals of this type in regular rings.

  8. As alluded to above, the counterintuitive notation normally adopted for the resulting ordinal is \({\mathrm{\mathtt {b} }}{\mathrm{\mathtt {a} }}\).

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  1. Department of Mathematics, The CUNY Graduate Center, 365 5th Avenue, New York, NY, 10016, USA

    Hans Schoutens

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Schoutens, H. The theory of ordinal length. Beitr Algebra Geom 57, 67–118 (2016). https://doi.org/10.1007/s13366-014-0229-z

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