Graded poincare series of monomial rings

My master's thesis is on calculations of the graded Poincare-Betti series of monomial rings. More specifically, I look at the quotient ring R=k[x1,...,xn]/(M) for some monomial ideal M, and for this ring, the augmentation R→k→0 can be extended to a free resolution F→R→k→0 (in many ways). The Poincare-Betti series is defined as the series P(z)=Sum dimk Tor(k,k)izi where Tor(k,k)i is the homology modules Hi(F⊗k) for the given free resolution F.

It can be shown that the Poincare-Betti series does not depend on the resolution chosen - indeed, the Tor(k,k) does not depend no the chosen resolution either. Furthermore, that the generating series P(z) is a rational function in z, and that the denominator polynomial can be given by an explicit formula.

A lot of this information should, and will in the text of the thesis, be attributed to the proper authors. I will here only note that the explicit formula characterizing the Poincare-Betti series is due to Alexander Berglund 2004.

Theory, and master's thesis

You can download and read a PDF containing a recent draft of my thesis.

Program code download and information

A very preliminary very first version of the Poincare-Betti calculator may be found here. Indeed, this is where subsequent releases will be made.

Note that the calculator is only verified against Pari version 2.2.7. We have discovered that when linked against Pari 2.2.9, the homology calculationsgo awry. Caveat Utiliator.

Poincare calculator, 0.1
poincare-0.1.tar.gz First release
Poincare calculator, 0.2
poincare-0.2.tar.gz Support for prime characteristics added
Poincare calculator, 0.3
poincare-0.3.tar.gz A high degree of functionality, including support for native Weyman-Fröberg and stable i/o handling.
Poincare calculator, 0.4
poincare-0.4.tar.gz Last release before 1.0. Basically functional.
Poincare calculator, 1.0
poincare-1.0.tar.gz Released to the thesis defense
Instructions for building and using are located within the tarball.

Atomic lattice lists download

In the course of the thesis work, a list of all lattices on 4 and 5 atoms as well as a list of all lattices on 6 atoms with more than 44 elements has been generated. These may be found for download here

4 atoms (1962 bytes)
5 atoms (152k)
6 atoms(35M)

The listing of lattices on 6 atoms can also be retrieved in portions:

64 elements (789 bytes)
63 elements (798 bytes)
62 elements (837 bytes)
61 elements (910 bytes)
60 elements (1089 bytes)
59 elements (1423 bytes)
58 elements (2257 bytes)
57 elements (3640 bytes)
56 elements (6726 bytes)
55 elements (16k)
54 elements (28k)
53 elements (60k)
52 elements (108k)
51 elements (208k)
50 elements (396k)
49 elements (736k)
48 elements (1.3M)
47 elements (2.6M)
46 elements (6.0M)
45 elements (12M)
44 elements (18M)


Mikael Johansson