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A Brief History of Order Ideals

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Abstract

The notion of order ideal is no doubt implicit in a paper by Serre in 1958 on free summands of projective modules. A formal definition is given by Bass. However, any algebraist contemplating the question, “on what locus of prime ideals in Spec(R) does an element e in a module E generate a free summand?”, has in fact encountered the concept of an order ideal. In the account on order ideals and their applications in this paper, it is our intent to elaborate on four basic theorems - as we see them - that give insight into the height properties of these ideals. We do this both from a historical view as well as a view of their utility.

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Notes

  1. 1.

    Throughout the remainder of this section when discussing kth syzygy modules E of finite projective dimension we assume that the ring R satisfies the Serre condition (S k ). (For definition see [22, p. 3].)

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Acknowledgements

We first met David Eisenbud in the late 1960s at University of Chicago. David and the first author were graduate students and the second author was a post-doctoral fellow. Even at this young age David was a catalyst in seminars and a central force for ideas at afternoon tea. It was patently clear that he would become a leader in mathematical research and the greater mathematical community. To each of the authors he was an inspiration—especially in their early work. We conclude by recounting memorable conversations between David Eisenbud, Mike Stillman, and the first author. The essence of these conversations served as motivation for our joint work on syzygies.

In the fall of 1976 the first author was talking with Mike Stillman about computing syzygies of a collection of homogeneous polynomials generating an ideal and realized that if they bounded the degree they were looking at they had a simple set of linear equations over a field. They could solve those and build a complex and check the Buchsbaum–Eisenbud criteria for exactness [10]. This was possible to do by hand as the complexes were quite small and the entries were not too complicated. They noticed that if a column in the kth matrix map had entries in the ideal generated by fewer than k of the variables which might as well be x 1, …, x k − 1, then that entry was a linear combination of entries of lower degrees. The first author mentioned this to David and he said one should be able to prove this by computing \(\mathrm{Tor}_{k}(R/I,R/(x_{1},\ldots,x_{k-1}))\). It is zero since R modulo the x’s has projective dimension k − 1 but would be nonzero if the syzygy were a minimal generator. Of course it isn’t too hard to see the same proof would show that the order ideal of a minimal generator of a kth syzygy couldn’t be in the annihilator of a finitely generated module of projective dimension less than k (see Theorem 4∗, Sect. 1). Thus a Cohen–Macaulay module over R modulo the order ideal of a minimal generator of a kth syzygy would be useful. Alas it took the authors some time to realize that. Happily Mike went to Harvard and met David Bayer. They learned about Grobner bases and created the first version of Macaulay. The authors then used Macaulay to compute lots of resolutions. Eventually they understood how to use the above ideas to formulate a version of the improved new intersection theorem that applied to order ideals of minimal generators which led to the proof of their syzygy theorem.

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  1. Department of Mathematics, University of Illinois at Urbana Champaign, Urbana, IL, 61801, USA

    E. Graham Evans & Phillip Griffith

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  1. E. Graham Evans
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  2. Phillip Griffith
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Correspondence to Phillip Griffith .

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  1. Cornell University, 547 Malott Hall, Ithaca, 14853, New York, USA

    Irena Peeva

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Evans, E.G., Griffith, P. (2013). A Brief History of Order Ideals. In: Peeva, I. (eds) Commutative Algebra. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5292-8_12

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