Finitely presented graded Lie algebras and homomorphisms of local rings
For a (connected) graded algebra \(N=\oplus_{i\geq 0}N_ i\), its Hilbert series is \(N(z)=\sum_{i\geq 0}(\dim N_ i)z^ i\). Examples are known of finitely-presented N with transcendental N(z). In particular, there are graded Lie algebras G whose enveloping algebras U(G) are finitely- presented graded Hopf algebras of global dimension 3 with U(G)(z) transcendental. The author's first main theorem is that if N is finitely- presented by n generators \(a_ 1,...,a_ n\) (of degrees \(\geq 1)\) and r relations, then there is a finitely-presented U(G) of global dimension 3 with \[ U(G)(z)=(2-N(z))^{-1}(1-\sum^{n}_{i=1}z^{\deg a_ i})^{-2}. \] U(G) is given explicitly by 3n generators and \(2n^ 2+r\) relations. This shows that restriction to finitely-presented graded Hopf algebras of global dimension 3 does not essentially restrict the construction of transcendental Hilbert series. The other main result concerns the construction of transcendental Poincaré series \(P_ R(z)=\sum_{i\geq 0}\dim (Ext^ i_ R(k,k))\) for a commutative Noetherian local ring R with residue field k. It is known that \(P_ R(z)\) can be transcendental, but it was an open question if R is Golod attached. This means that there is a regular local ring \(S_ 0\) and a sequence \(S_ 0\to S_ 1\to...\to S_ t=R\) of Golod maps. A Golod map \(f: S\to R\) is an epimorphism of local rings with \(\pi^*(f): \pi^*(R)\to \pi^*(S)\) surjective with kernel a free Lie algebra, where \(\pi^*(R)\) is the graded Lie algebra with \(U(\pi^*(R))=Ext^*_ R(k,k)\) as graded Hopf algebra. The author proves that if N is finitely presented by n generators of degree 1 and r relations of degree 2, then there is a regular local ring \(S_ 0\) of embedding dimension 3n, and three Golod maps \(S_ 0\to S_ 1\to S_ 2\to R\) with \[ P_ R(z)=z/[(1+z)(1-nz)^ 2(2-N(z)-1+3nz- (2n^ 2+r)z^ 2)]. \] Using this, the author explicitly constructs a Golod attached local ring with transcendental Poincaré series.
scientific article; zbMATH DE number 3918519
scientific article