A Macaulay2 Tutorial with InvariantRing

Macaulay2 is a software system created by Dan Grayson and Mike Stillman devoted to supporting research in algebraic geometry and commutative algebra. Macaulay2 includes core algorithms for computing Gröbner bases and graded or multi-graded free resolutions of modules over quotient rings of graded or multi-graded polynomial rings with a monomial ordering. [...] Macaulay2 can compute Betti numbers, Ext, cohomology of coherent sheaves on projective varieties, primary decomposition of ideals, integral closure of rings, and more.

It has your favorite rings of coefficients coefficients.

We can define polynomial rings with ease ...

... and define ideals.

We can define the quotient ring

which is the coordinate ring of \(SL_4\text{.}\) We can localize the ring at a prime ideal.

Macaulay2 is by design good at resolutions and Gröbner bases. Let’s start with resolutions. For a polynomial ring, gens R returns the List of generators, while vars R returns the generators as a one-row Matrix.

Given a Matrix, we can take its cokernel.

Now that \(M\) is an \(R\)-Module, Macaulay2 knows how to produce a free resolution.

... and since \(C\) is a ChainComplex, we can access its differentials.

We want to see that \(C\) is indeed a chain complex. Indeed, the differentials composed twice would be

Next, let’s take a look at Gröbner bases. (What is .. A Gröbner basis?

¹

math.berkeley.edu/~bernd/what-is.pdf

) A Gröbner basis is a set of multivariate polynomials that has desirable algorithmic properties. Every set of polynomials can be transformed into a Gröbner basis. This process generalizes three familiar techniques: Gaussian elimination for solving linear systems of equations, the Euclidean algorithm for computing the greatest common divisor of two univariate polynomials, and the Simplex Algorithm for linear programming.