Internal duality for resolution of rings (Q1291100)

Let \(n\) be a positive integer, \(N=1+n+n^2\), and let \(K\) be a field containing a primitive \(N\)-th root of unity, say \(\omega\). Denote by \(G\) the cyclic group generated by \(\text{diag}(\omega,\omega^n,\omega^{n^2})\in SL_3(K)\). The main result of the paper is that the algebra \(B_n=K[x_1,x_2,x_3]^G\) of polynomial invariants exhibits internal duality. This means that for each \(i\) there exists an integer \(h_i\) such that the equalities \(\beta_{i,j}=\beta_{i,h_i-j}\) hold for the internal Betti numbers of \(B_n\) (here \(\beta_{i,j}\) denotes the number of generators of degree \(j\) in the \(i\)-th member of the minimal free resolution of \(B_n\)). To prove this the authors construct an explicit minimal Hilbert syzygy resolution for \(B_n\), and also for some other families of monomial rings.