Irena Swanson

Irena Swanson's Papers

  1. Joint reductions, tight closure and the Briancon-Skoda theorem, J. Algebra, 147 (1992), 128-136.
  2. Mixed multiplicities, joint reductions, and a theorem of Rees, J. London Math. Soc., 48 (1993), 1-14.
  3. A note on analytic spread, Comm. in Alg., 22(2) (1994), 407-411.
  4. Primary decompositions of powers of ideals, ``Proceedings of Mt. Holyoke Conference on Commutative Algebra: Syzygies, Multiplicities and Birational Algebra", Contemporary Mathematics, Volume 159, 1994, pp. 367-371.
  5. Joint reductions, tight closure and the Briancon-Skoda theorem, II, J. Algebra, 170 (1994), 567-583.
  6. Cores of ideals in two dimensional regular local rings (with C. Huneke), Michigan Math. J., 42 (1995), 193-208.
  7. Integral closure of ideals in excellent local rings, (with D. Delfino), J. Algebra, 187 (1997), 422-445. Ray Heitmann pointed out that Theorem 2.7 in the published version is wrong. Fortunately, the main results of the paper are still true. We give new proofs in this version: Integral closure of ideals in excellent local rings (new). If you just want to look at the erratum, here it is: (Erratum). J. Algebra 274 (2004), 422-428.
  8. Ideals contracted from 1-dimensional overrings with an application to the primary decomposition of ideals (with W. Heinzer), Proc. Amer. Math. Soc., 125 (1997), 387-392.
  9. Powers of ideals: primary decompositions, Artin-Rees lemma and regularity, Math. Annalen, 307 (1997), 299-313. April 2005: Francesc Planas pointed out a gap in Theorem 4.1, the result on the Artin-Rees lemma for powers of ideals: there are in fact infinitely many primes at which one should localize as J varies, so it is not clear that there is a global upper bound. In fact, if Theorem 4.1 is true, then there is an easy proof that every pair of finitely generated modules over a Noetherian ring has the uniform Artin-Rees property. The correct statement of Theorem 4.1 should be: Let R be a Noetherian local ring and I an ideal. Then there exists an integer k such that for all n, all m < kn, and all ideals J, the intersection of J^m and I^n is contained in J^{m-kn} I^n. The result on primary decomposition is unaffected. (The new proof of 4.1 needs the passage to the non-local Rees algebra S, but it suffices to prove the theorem in S only for ideals J extended from R and for the principal ideal I = (t^{-1})S, whence it suffices to prove the theorem in S localized at the complement of the unique homogeneous maximal ideal.)
  10. Linear bounds on growth of associated primes for monomial ideals (with K. E. Smith), Comm. in Algebra, 25 (1997), 3071-3079.
  11. Linear equivalence of topologies, Math. Zeitschrift, 234 (2000), 755-775.
  12. Zeros of differentials along ideals, appendix to R. H\"ubl's paper Derivations and the Integral Closure of Ideals, Proc. Amer. Math. Soc., 127 (1999), 3503-3511.
  13. Permanental ideals, (with R. C. Laubenbacher), J. Symbolic Comput., 30 (2000), 195-205.
  14. Discrete valuations centered on local domains (with R. Huebl), 1998, J. Pure Appl. Algebra, 161 (2001), 145-166. Added in December 2008, after discussion with Shuzo Izumi and Reinhold Huebl: the Izumi-Rees theorem on comparability of two m-valuations in a Noetherian local ring (R,m) holds if R is analytically irreducible, no need to assume in addition that R be excellent. Namely, first pass to the completion, which is an excellent Noetherian local domain, and the m-valuations in R extend naturally to the completion.
  15. Jacobian ideals of trilinear forms: an application of 1-genericity (with Anna Guerrieri), J. Algebra, 226 (2000), 410-435.
  16. Normal cones of monomial primes, (with R. Huebl), Math. of Computation 72 (2002), 459-475.
  17. The Zarankiewicz problem via Chow forms (with Marko Petkovsek ) and Jamie Pommersheim), in ``Computational Commutative Algebra and Combinatorics", Advanced Studies in Pure Mathematics, 33, editor T. Hibi, Mathematical Society of Japan, Tokyo, 2002, 203-212.

    The following 4 papers all arose from the attempt to answer a question of Bayer, Huneke and Stillman of how or whether the doubly exponential ideal membership property of the Mayr-Mayer ideals is reflected in their primary decompositions.
  18. The first Mayr-Meyer ideal, in ``Proceedings of the Fourth International Conference on Commutative Ring Theory and Applications", Fez, Morocco, June 7 - 12. This paper analyzes the primary decomposition structure of the first Mayr-Meyer ideal and shows that a specific membership problem's complexity does not depend on the existence of embedded primes or on the unreducedness.
  19. The minimal components of the Mayr-Meyer ideals, J. Algebra 267 (2003), 127-155. This paper analyzes the minimal primes and their components of the Mayr-Meyer ideals J(n,d), for n, d at least 2. The numbers of minimal primes is n(d')^2 + 20, where d' is the largest factor of d which is relatively prime with the characteristic of the field. [Correction on 8 Nov 2009: d' = d if the characteristic is 0. This is also incorporated in the paper.] It is shown that the the doubly exponential ideal membership property of the Mayr-Meyer ideals is due to the embedded primes.
  20. On the embedded primes of the Mayr-Meyer ideals, J. Algebra 275 (2004), 143-190. This paper analyzes the embedded primes of the Mayr-Meyer ideals J(n,d), for n, d at least 2. The main technique is the usage of short exact sequences to find the associated primes. This method in general produces possible but not necessarily associated primes. Removal of redundancies gets progressively harder. It is proved that J(n,d) definitely has O(nd^3) embedded primes. A recursive procedure shows that an upper bound on the number of embedded primes is doubly exponential in n. In the process a new family of ideals is found which exhibits the same doubly exponential ideal membership property as the Mayr-Meyer family of ideals. I copied ">1" from page 163 (in the published paper) incorrectly onto page 189 (where it appears asa ">0"), and this made me count the number of possibly embedded primes slightly off as O(d^3n). The paper that is posted here has the corrected count. I thank Thomas Kahle for making me count.
  21. (Only for the really really really determined!!!) For some previous attempts at finding the primary decomposition of the Mayr-Meyer ideals J(n,d), you may click here, but be warned that you will see 55 unedited pages of hard-to-read and incomplete attempts.

  22. Ten lectures on tight closure. These are the notes from the mini course I gave at IPM (Institute for Studies in Theoretical Physics and Mathematics) in Tehran, Iran, in January 2002. It consists of approximately 65 pages of lectures and 5 pages of tight closure references. The latest version was posted on 7 July 2004: improvements were suggested by Janet Striuli and Graham Leuschke. I also changed spacing so more gets printed per page now. The sections are: 1. The basics; 2. Briancon-Skoda theorem, rings of invariants, ...; 3. The localization problem; 4. Tight closure for modules; 5. Application to symbolic and ordinary powers of ideals; 6. Test elements and the persistence of tight closure; 7. More on test elements, or what is needed in Section 5; 8. Tight closure in characteristic 0; 9. A bit on the Hilbert-Kunz function; 10. Summary of research in tight closure.
  23. Infinitely many associated primes of Frobenius powers and local cohomology permanent preprint,2002. Katzman gave an example (several years ago) of an ideal in a two-dimensional ring of positive prime characteristic p whose Frobenius powers have infinitely many associated primes. The ring in Katzman's example is not an integral domain. This paper gives a modification of Katzman's example to produce a two-generated ideal in a two-dimensional Noetherian integral domain of characteristic 2 for which the set of associated primes of all the Frobenius powers is infinite. A further modification yields a four-dimensional Noetherian integral domain and a five-dimensional Noetherian local integral domain for which an explicit second local cohomology module has infinitely many associated primes.
  24. The previous paper is superseded by the much better paper: Associated primes of local cohomology modules and of Frobenius powers (with Anurag Singh), International Mathematics Research Notices 33 (2004) 1703-1733.
  25. On the ideals of minors of matrices of linear forms (with Anna Guerrieri), in ``Proceedings of the Special Session on Commutative Algebra and Its Interaction with Algebraic Geometry and Conference on Commutative Algebra and Algebraic Geometry". We analyze the ideals of 2 x 2 minors of a generic Hankel matrix. We provide a combinatorial criterion for when these ideals are prime and what their components are.
  26. Notes on the behavior of the Ratliff-Rush filtration (with Maria Evelina Rossi), in ``Proceedings of the Special Session on Commutative Algebra and Its Interaction with Algebraic Geometry and Conference on Commutative Algebra and Algebraic Geometry". An erratum: on page 1, line 3 from bottom up: the assumption I : a = I should be replaced by the assumption (a) : I = (a). In example 1.8, add the observation that for that ideal I, I = I^2 : I is strictly contained in I^3 : I^2.
  27. Computing instanton numbers of curve singularities (with Elizabeth Gasparim), Journal of Symbolic Computation 40 (2005), 965--978. The Macaulay2 code of this algorithm can be found in instanton.m2.
  28. Computations with Frobenius powers (with Susan Hermiller), Journal of Experimental Mathematics 14 (2005), 161-173. We thank Aldo Conca and Enrico Sbarra for pointing out a problem with a previous version.
  29. Symbolic powers of radical ideals (with Aihua Li), Rocky Mountain J. of Math. 36 (2006), 997-1009.
  30. On free integral extensions generated by one element (with Orlando Villamayor), in 'Commutative Algebra with a focus on geometric and homological aspects', Proceedings of Sevilla, June 18-21, 2003 and Lisbon, June 23-27, 2003. Marcel Dekker's Lecture Notes in Pure and Applied Mathematics Series. Editors Alberto Corso, Philippe Gimenez, Maria Vaz Pinto, Santiago Zarzuela. Chapman-Hall 2005, pp. 239-257.
  31. Primary decompositions, an expanded version of my expository talks at the International Conference on Commutative Algebra and Combinatorics, Allahabad, India, December 2003. Editor W. Bruns et al, No. 2, 2006, pp. 117-155. 42 pages. (Latest version posted on 10 December 2021, after Justin Chen pointed out a typo in Exercise 4.30.)
  32. Integral Closure of Ideals, Rings, and Modules, book, (with Craig Huneke), Cambridge University Press, Cambridge, 2006.
  33. Permanental ideals of Hankel matrices (with Elena Grieco and Anna Guerrieri), Abh. Math. Sem. Univ. Hamburg 77 (2007), 39-58. 22 pages.
  34. Multi-graded Hilbert functions, mixed multiplicities, expository chapter in Syzygies and Hilbert Functions, edited by I. Peeva. Lecture Notes in Pure and Applied Mathematics series by CRC, (2007), 267--280.
  35. Adjoints of ideals (with R. Huebl), Michigan Math. J. 57 (2008), 447--462.
  36. The Goto numbers of parameter ideals (with W. Heinzer), J. Algebra 321 (2009), 152--166.
  37. An algorithm for computing the integral closure (with Anurag Singh), preprint 2008.
  38. Every numerical semigroup is one over d of infinitely many symmetric numerical semigroups, and if d is at least 3, there are infinitely many pseudo-symmetric numerical semigroups with this property. In Commutative Algebra and its Applications, coedited with Marco Fontana, Salah-Eddine Kabbaj, and Bruce Olberding, published by de Gruyter, 2009.
    Click here to link to the book information at de Gruyter.
  39. Rees valuations, expository paper. In Commutative Algebra, coedited with Marco Fontana, Salah-Eddine Kabbaj, and Bruce Olberding, published by Springer, 2011. Click here to link to the book information at Springer. Pedro Lima pointed out in March 2020 that J" is not needed in the proof of Proposition 2.2.
  40. My lecture notes for the School on Local Rings and Local Study of Algebraic Varieties, ICTP, Trieste 31 May--4 June 2010 in pdf format: Integral closure of ideals and rings (pdf). This school and conference was in honor of Tito Valla, and click here (pdf) for the song/poem in his honor that we sang at the end of my last talk. This is approximately (mp3) what the singing should have been like. The whistling accompaniment is by Adam Boocher.
  41. An article on mathematics and quilting (first three sections only), in particular, on piecing semiregular tessellations. Semiregular tessellations are explained at a high-school level, some results are proved, some mathematical activities are proposed for learners of less and more advanced levels, and then quick and new piecing methods are developed for making quilts. This is now a chapter in the book Crafting by Concepts (link to A K Peters), edited by sarah-marie belcastro and Carolyn Yackel and published by A K Peters in 2011.
  42. Searching for Cutkosky's example, (with Francesca Di Giovannantonio and Anna Guerrieri), Rocky Mountain J. of Math. 44 (2014), 865--876.
  43. Minimal primes of ideals arising from conditional independence statements, (with Amelia Taylor), preprint 2013. J. Algebra 392 (2013), 299--314.
  44. 2 x 2 permanental ideals of hypermatrices, (with Julia Porcino). Comm. Alg. 43 (2015) special issue of Communications in Algebra in honor of Marco Fontana, 84--101.
  45. Hilbert-Kunz functions of 2 x 2 determinantal rings, (with Lance Miller), preprint 2012. Illinois J. of Math. 57 (2013), 251--277.
  46. Frobenius numbers of numerical semigroups generated by three consecutive squares or cubes, (with M. Lepilov, J. O'Rourke), accepted for publication in Semigroup Forum, 2014.
  47. Explicit Hilbert-Kunz functions of 2 x 2 determinantal rings, (with Marcus Robinson), accepted for publication in the Pacific Journal of Mathematics, 2014. Pacific Journal of Mathematics 275 (2015), 433--442.
  48. Integral closure, expository paper and open questions, in the book Commutative Algebra, Recent Advances in Commutative Rings, Integer-Valued Polynomials, and Polynomial Functions, edited by M. Fontana, S. Frisch, and S. Glaz. Springer, 2014. 331--351.
  49. Three lectures on primary decompositions, binomial ideals, and algebraic statistics, EACA's Second International School On Computer Algebra and Applications, June 2013, Valladolid, Spain. Chapter with E. Saenz-de-Cabezon in the book Computations and Combinatorics in Commutative Algebra, EACA School, Valladolid, 2013. Editors A. M. Bigatti, P. Gimenez and E. Saenz-de-Cabezon, Springer, 2017, pages 41--75. This pdf is an early version: Three lectures (Computation of primary decompositions, Expanded lectures on binomial ideals, Primary decomposition in algebraic statistics.
  50. Commutative algebra provides a big surprise for Craig Huneke's birthday, Notices of the American Mathematical Society 64 (2017), 256--259. Online at AMS Notices.
  51. Many associated primes of powers of prime ideals, with Jesse Kim. Published in the Journal of Pure and Applied Algebra, 2019.
  52. Tensor-multinomial sums of ideals: primary decompositions and persistence of associated primes, with Robert M. Walker. Published in Proc. Amer. Math. Soc. 147 (2019), 5081--5082.
  53. Predicted decay ideals, with Sarah Jo Weinstein. Published online in Communications in Algebra in October 2019.

Irena Swanson's Books: written or co-edited