Irena Swanson
Irena Swanson's Papers

Joint reductions, tight closure and the BrianconSkoda theorem,
J. Algebra, 147 (1992), 128136.

Mixed multiplicities, joint reductions, and a theorem of Rees,
J. London Math. Soc., 48 (1993), 114.

A note on analytic spread,
Comm. in Alg., 22(2) (1994), 407411.

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. 367371.

Joint reductions, tight closure and the BrianconSkoda theorem, II,
J. Algebra, 170 (1994), 567583.

Cores of ideals in two dimensional regular local rings
(with C. Huneke),
Michigan Math. J., 42 (1995), 193208.

Integral closure of ideals in excellent local rings,
(with D. Delfino),
J. Algebra, 187 (1997), 422445.
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), 422428.

Ideals contracted from 1dimensional overrings with an application
to the primary decomposition of ideals
(with W. Heinzer),
Proc. Amer. Math. Soc., 125 (1997), 387392.

Powers of ideals: primary decompositions, ArtinRees lemma and regularity,
Math. Annalen, 307 (1997), 299313.
April 2005:
Francesc Planas pointed out a gap in Theorem 4.1,
the result on the ArtinRees 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 ArtinRees 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^{mkn} I^n.
The result on primary decomposition is unaffected.
(The new proof of 4.1 needs the passage to the nonlocal 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.)

Linear bounds on growth of associated primes for monomial ideals
(with K. E. Smith),
Comm. in Algebra, 25 (1997), 30713079.

Linear equivalence of topologies,
Math. Zeitschrift, 234 (2000), 755775.

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), 35033511.

Permanental ideals,
(with R. C. Laubenbacher),
J. Symbolic Comput., 30 (2000), 195205.

Discrete valuations centered on local domains
(with R. Huebl),
1998,
J. Pure Appl. Algebra, 161 (2001), 145166.
Added in December 2008, after discussion with Shuzo Izumi and Reinhold Huebl:
the IzumiRees theorem on comparability of two mvaluations
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 mvaluations in R extend naturally to the completion.

Jacobian ideals of trilinear forms:
an application of 1genericity
(with Anna Guerrieri),
J. Algebra, 226 (2000), 410435.

Normal cones of monomial primes,
(with R. Huebl),
Math. of Computation 72 (2002), 459475.

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, 203212.
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 MayrMayer ideals
is reflected in their primary decompositions.

The first MayrMeyer 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
MayrMeyer ideal
and shows that a specific membership problem's complexity does not
depend on the existence of embedded primes or on the unreducedness.

The minimal components of the MayrMeyer ideals,
J. Algebra 267 (2003), 127155.
This paper analyzes the minimal primes and their components
of the MayrMeyer 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 MayrMeyer ideals is due to the embedded primes.

On the embedded primes of the MayrMeyer ideals,
J. Algebra 275 (2004), 143190.
This paper analyzes the embedded primes
of the MayrMeyer 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 MayrMeyer 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.

(Only for the really really really determined!!!)
For some previous attempts at finding the primary decomposition of
the MayrMeyer ideals J(n,d),
you may click here,
but be warned that you will
see 55 unedited pages of hardtoread and incomplete attempts.

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. BrianconSkoda 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 HilbertKunz function;
10. Summary of research in tight closure.

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 twodimensional 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 twogenerated ideal in a twodimensional 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 fourdimensional Noetherian integral domain
and a fivedimensional Noetherian local integral domain
for which an explicit second local cohomology module has infinitely many
associated primes.

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) 17031733.

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.

Notes on the behavior of the RatliffRush 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.

Computing instanton numbers of curve singularities
(with Elizabeth Gasparim),
Journal of Symbolic Computation 40 (2005), 965978.
The Macaulay2 code of this algorithm can be found in
instanton.m2.

Computations with Frobenius powers
(with Susan Hermiller),
Journal of Experimental Mathematics 14 (2005), 161173.
We thank
Aldo Conca
and
Enrico Sbarra
for pointing out a problem with a previous version.

Symbolic powers of radical ideals
(with
Aihua Li),
Rocky Mountain J. of Math. 36 (2006), 9971009.

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 1821, 2003 and
Lisbon, June 2327, 2003.
Marcel Dekker's Lecture Notes
in Pure and Applied Mathematics Series.
Editors Alberto Corso, Philippe Gimenez, Maria Vaz Pinto, Santiago Zarzuela.
ChapmanHall 2005,
pp. 239257.

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. 117155.
42 pages.
(Latest version posted on 10 December 2021,
after Justin Chen pointed out a typo in Exercise 4.30.)

Integral Closure of Ideals,
Rings, and Modules,
book,
(with Craig Huneke),
Cambridge University Press,
Cambridge, 2006.

Permanental ideals of Hankel matrices
(with Elena Grieco and
Anna Guerrieri),
Abh. Math. Sem. Univ. Hamburg 77 (2007), 3958.
22 pages.

Multigraded 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), 267280.

Adjoints of ideals
(with R. Huebl),
Michigan Math. J. 57 (2008), 447462.

The Goto numbers of parameter ideals
(with W. Heinzer),
J. Algebra 321 (2009), 152166.

An algorithm for computing the integral closure
(with Anurag Singh),
preprint 2008.

Every numerical semigroup is one over d of infinitely many
symmetric numerical semigroups,
and if d is at least 3,
there are infinitely many pseudosymmetric numerical semigroups
with this property.
In
Commutative Algebra and its Applications,
coedited with
Marco Fontana,
SalahEddine Kabbaj,
and
Bruce Olberding,
published by
de Gruyter, 2009.
Click here to link to the book information at de Gruyter.

Rees valuations,
expository paper.
In
Commutative Algebra,
coedited with
Marco Fontana,
SalahEddine 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.

My lecture notes for the
School on Local Rings and Local Study of Algebraic Varieties,
ICTP, Trieste 31 May4 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.

An article on mathematics and quilting
(first three sections only),
in particular,
on piecing semiregular tessellations.
Semiregular tessellations are explained at a highschool 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 sarahmarie belcastro
and Carolyn Yackel
and published by A K Peters in 2011.

Searching for Cutkosky's example,
(with Francesca Di Giovannantonio and
Anna Guerrieri),
Rocky Mountain J. of Math. 44 (2014), 865876.

Minimal primes of ideals arising
from conditional independence statements,
(with Amelia Taylor),
preprint 2013.
J. Algebra 392 (2013), 299314.

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,
84101.

HilbertKunz functions of 2 x 2 determinantal rings,
(with Lance Miller),
preprint 2012.
Illinois J. of Math. 57 (2013), 251277.

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.

Explicit HilbertKunz 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), 433442.

Integral closure,
expository paper and open questions,
in the book Commutative Algebra, Recent Advances in Commutative Rings,
IntegerValued Polynomials,
and Polynomial Functions,
edited by M. Fontana, S. Frisch, and S. Glaz.
Springer, 2014.
331351.

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. SaenzdeCabezon
in the book Computations and Combinatorics in Commutative Algebra,
EACA School, Valladolid, 2013.
Editors A. M. Bigatti, P. Gimenez and E. SaenzdeCabezon,
Springer, 2017, pages 4175.
This pdf is an early version:
Three lectures (Computation of primary
decompositions, Expanded lectures on binomial ideals,
Primary decomposition in algebraic statistics.

Commutative algebra provides a big surprise for Craig Huneke's birthday,
Notices of the American Mathematical Society 64 (2017), 256259.
Online at AMS Notices.

Many associated primes of powers of prime ideals,
with Jesse Kim.
Published in the Journal of Pure and Applied Algebra, 2019.

Tensormultinomial sums of ideals:
primary decompositions and persistence of associated primes,
with
Robert M. Walker.
Published in Proc. Amer. Math. Soc. 147 (2019), 50815082.

Predicted decay ideals,
with Sarah Jo Weinstein.
Published online in Communications in Algebra in October 2019.
Irena Swanson's Books: written or coedited

Cowrote
Integral Closure of Ideals, Rings, and Modules,
with
Craig Huneke,
published by
Cambridge University Press, Cambridge, 2006.
This is a graduatelevel textbook,
and it is also meant to be a reference for researchers.
Click here to link to the book information at Cambridge University Press,
and click here
for chapter titles, errata, and the online version.
 Coedited:
Commutative Algebra and its Applications,
coedited with
Marco Fontana,
SalahEddine Kabbaj,
and
Bruce Olberding,
published by
de Gruyter, 2009.
Contributors: D. D. Anderson, M. Zafrullah;
D. F. Anderson, A. Badawi, D. E. Dobbs, J. Shapiro;
A. Badawi;
C. Bakkari, N. Mahdou;
V. Barucci;
D. Bennis, N. Mahdou;
S. Bouchiba;
S. Bouchiba, S. Kabbaj;
P. Cesarz, S. T. Chapman, S. McAdam, G. J. Schaeffer;
J.L. Chabert;
F. Couchot;
M. D'Anna, C. A. Finocchiaro, M. Fontana;
D. E. Dobbs;
D. E. Dobbs, G. Picavet;
S. El Baghdadi;
L. El Fadil;
J. Elliott;
Y. Fares;
F. HalterKoch;
S. Hizem;
A. Jhilal, N. Mahdou;
S. Kabbaj, A. Mimouni;
N. Mahdou, K. Ouarghi;
R. Matsuda;
S. B. Mulay;
B. Olberding;
L. Salce;
S. SatherWagstaff;
I. Swanson.
Click here to link to the book information at de Gruyter.

Coedited a collection of expository papers:
Commutative Algebra,
coedited with
Marco Fontana,
SalahEddine Kabbaj,
and
Bruce Olberding,
published by
Springer, 2011.
Click here to link to the book information at Springer.
Contributors: D. D. Anderson;
D. F. Anderson, M. C. Axtell, J. A. Sticklers, Jr.;
S. Bazzoni, S.E. Kabbaj;
H. Brenner;
L. W. Christensen, H.B. Foxby, H. Holm;
M. Fontana, M. Zafrullah,
H. Haghighi, M. Tousi, S. Yassemi;
F. HalterKoch;
W. Heinzer, L. J. Ratliff, Jr., D. E. Rush;
F.V. Kuhlmann;
K. A. Loper;
B. Olberding;
L. Salce;
H. Schoutens;
I. Swanson;
M. Vitulli.

Ten lectures on tight closure, IPM Lecture Note Series, 3,
Tehran, 2002.
93 pages.

NEW
Introduction to Analysis with Complex Numbers,
456 pages,
published by World Scientific Publishing, 2021.
This is a selfcontained book that covers the standard topics in introductory
analysis and that in addition constructs the natural, rational, real and
complex numbers, and also handles complexvalued functions, sequences, and series.
The book teaches how to write proofs. Fundamental proofwriting logic is covered in Chapter 1 and is repeated and enhanced in two appendices. Many examples of proofs appear with words in a different font for what should be going on in the proof writer's head.
The book contains many examples and exercises to solidify the understanding. The material is presented rigorously with proofs and with many workedout examples. Exercises are varied, many involve proofs, and some provide additional learning materials.
Errata.

(Unpublished)
Summer Graduate Workshop at MSRI, Berkeley, 6 June17 June 2011
Coorganizing and lecturing with
Daniel Erman
then from Stanford University,
and
Amelia Taylor, then from from Colorado College.
The workshop will involve a combination of theory and symbolic computation in commutative algebra. The lectures are intended to introduce three active areas of research: BoijSoederberg theory, algebraic statistics, and integral closure. The lectures will be accompanied with tutorials on the computer algebra system Macaulay 2.
My first introductory lecture was on resolutions
and I got to the proof of the Hilbert's Syzygy Theorem
(at least of the graded modules).
You can get here the slide version of the resolutions talk.
My lectures two and three were on the EisenbudSturmfels paper
on binomial ideals.
You can get here my notes on binomial ideals (updated in June 2013,
in Valladolid, Spain).
I did not present lattices and characters neither in my lectures
nor in these notes;
I think the omission made the lectures more easily understable to beginners.

(Unpublished)
Reed College, Spring 2011,
MATH 411 Topics in Advanced Analysis: Functional Analysis.
Banach, Hilbert spaces;
compact, Fredholm, selfadjoint operators;
spectral theory; Lp spaces.
I am an algebraist,
but I love (teaching) analysis.
These
pdf lecture notes
are work in progress.
Many examples are worked out,
including counterexamples to the conclusions of big theorems
if various hypotheses are removed.
Feedback appreciated.

My lecture notes for the
School on Local Rings and Local Study of Algebraic Varieties,
ICTP, Trieste 31 May4 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.
Here is Gemma Colome's recording of it.
To hear
what the singing could have been like,
click here(mp3).
The whistling accompaniment is by
Adam Boocher.

(Unpublished)
My lecture notes on
homological algebra (pdf),
delivered at University of Rome III in MarchMay 2010,
with improvements for lectures at University of Graz in the Winter Semester
2018.

(Unpublished)
Reed College, Spring 2009, MATH 332 Abstract Algebra:
An elementary treatment of the algebraic structures of groups,
rings, fields, and/or algebras.
(These
pdf lecture notes
are work in progress.
Feedback appreciated.
In class I scatter some nonstandard topics only on Tuesdays,
but in the pdf notes they are all in one place.)