## The Michigan Mathematical Journal

### Counting the Ideals of Given Codimension of the Algebra of Laurent Polynomials in Two Variables

#### Abstract

We establish an explicit formula for the number $C_{n}(q)$ of ideals of codimension (colength) $n$ of the algebra $\mathbb{F}_{q}[x,y,x^{-1},y^{-1}]$ of Laurent polynomials in two variables over a finite field $\mathbb{F}_{q}$ of cardinality $q$. This number is a palindromic polynomial of degree $2n$ in $q$. Moreover, $C_{n}(q)=(q-1)^{2}P_{n}(q)$, where $P_{n}(q)$ is another palindromic polynomial; the latter is a $q$-analogue of the sum of divisors of $n$, which happens to be the number of subgroups of $\mathbb{Z}^{2}$ of index $n$.

#### Article information

Source
Michigan Math. J., Volume 67, Issue 4 (2018), 715-741.

Dates
Received: 23 November 2016
Revised: 2 February 2018
First available in Project Euclid: 16 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.mmj/1529114453

Digital Object Identifier
doi:10.1307/mmj/1529114453

Mathematical Reviews number (MathSciNet)
MR3877434

Zentralblatt MATH identifier
07056366

#### Citation

Kassel, Christian; Reutenauer, Christophe. Counting the Ideals of Given Codimension of the Algebra of Laurent Polynomials in Two Variables. Michigan Math. J. 67 (2018), no. 4, 715--741. doi:10.1307/mmj/1529114453. https://projecteuclid.org/euclid.mmj/1529114453

#### References

• [1] G. Andrews and F. G. Garvan, Dyson’s crank of a partition, Bull. Amer. Math. Soc. (N.S.) 18 (1988), no. 2, 167–171.
• [2] T. M. Apostol, Introduction to analytic number theory, Undergrad. Texts Math., Springer-Verlag, New York–Heidelberg, 1976.
• [3] R. Bacher and C. Reutenauer, The number of right ideals of given codimension over a finite field, Noncommutative birational geometry, representations and combinatorics, Contemp. Math., 592, pp. 1–18, Amer. Math. Soc., Providence, RI, 2013.
• [4] R. Bacher and C. Reutenauer, Number of right ideals and a $q$-analogue of indecomposable permutations, Canad. J. Math. 68 (2016), no. 3, 481–503.
• [5] A. Białynicki-Birula, Some theorems on actions of algebraic groups, Ann. of Math. (2) 98 (1973), 480–497.
• [6] A. Białynicki-Birula, Some properties of the decompositions of algebraic varieties determined by actions of a torus, Bull. Acad. Pol. Sci., Sér. Sci. Math. Astron. Phys. 24 (1976), no. 9, 667–674.
• [7] N. Bourbaki, Algèbre commutative, Hermann, Paris, 1961, English translation: Commutative Algebra, Chapters 1–7, Springer-Verlag, Berlin, 1989.
• [8] A. Conca and G. Valla, Canonical Hilbert–Burch matrices for ideals of $k[x,y]$, Michigan Math. J. 57 (2008), 157–172.
• [9] M. A. de Cataldo, T. Hausel, and L. Migliorini, Exchange between perverse and weight filtration for the Hilbert schemes of points of two surfaces, J. Singul. 7 (2013), 23–38.
• [10] D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, Grad. Texts in Math., 150, Springer-Verlag, New York, 1995.
• [11] G. Ellingsrud and S. A. Strømme, On the homology of the Hilbert scheme of points in the plane, Invent. Math. 87 (1987), no. 2, 343–352.
• [12] N. J. Fine, Basic hypergeometric series and applications, Math. Surveys Monogr., 27, Amer. Math. Soc., Providence, RI, 1988.
• [13] J. Fogarty, Algebraic families on an algebraic surface, Amer. J. Math. 90 (1968), 511–521.
• [14] F. G. Garvan, New combinatorial interpretations of Ramanujan’s partition congruences mod $5$, $7$ and $11$, Trans. Amer. Math. Soc. 305 (1988), no. 1, 47–77.
• [15] L. Göttsche and W. Soergel, Perverse sheaves and the cohomology of Hilbert schemes of smooth algebraic surfaces, Math. Ann. 296 (1993), no. 2, 235–245.
• [16] A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique. IV. Les schémas de Hilbert, Séminaire Bourbaki, 6, Exp. No. 221, pp. 249–276, W. A. Benjamin, New York–Amsterdam, 1966.
• [17] G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, 3rd edition, Clarendon Press, Oxford, 1954.
• [18] T. Hausel, E. Letellier, and F. Rodriguez-Villegas, Topology of character varieties and representation of quivers, C. R. Math. Acad. Sci. Paris 348 (2010), no. 3–4, 131–135.
• [19] T. Hausel, E. Letellier, and F. Rodriguez-Villegas, Arithmetic harmonic analysis on character and quiver varieties, Duke Math. J. 160 (2011), no. 2, 323–400.
• [20] T. Hausel, E. Letellier, and F. Rodriguez-Villegas, Arithmetic harmonic analysis on character and quiver varieties II, Adv. Math. 234 (2013), 85–128.
• [21] T. Hausel and F. Rodriguez-Villegas, Mixed Hodge polynomials of character varieties, Invent. Math. 174 (2008), no. 3, 555–624, With an appendix by Nicholas M. Katz.
• [22] C. Kassel and C. Reutenauer, Complete determination of the zeta function of the Hilbert scheme of $n$ points on a two-dimensional torus, Ramanujan J. (published online: 18 May 2018), doi:10.1007/s11139-018-0011-1, arXiv:1610.07793.
• [23] S. Mozgovoy and M. Reineke, Arithmetic of character varieties of free groups, Internat. J. Math. 26 (2015), no. 12, 1550100.
• [24] M. Reineke, Cohomology of noncommutative Hilbert schemes, Algebr. Represent. Theory 8 (2005), no. 4, 541–561.
• [25] The On-Line Encyclopedia of Integer Sequences, published electronically at http://oeis.org.