Some new Betti numbers of ideals generated by n+1 generic forms in n variables
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2025
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| author | Fröberg, Ralf |
| author_facet | Fröberg, Ralf |
| contents | Very little is known on the Hilbert series of graded algebras $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_r)$, $r>n$, $g_i$ generic form of degree $e_i$, in general. One instance when the series is known, is for $n+1$ forms in $n$ variables, \cite{St}. Of course even less is known about Betti numbers. There are some general results on the Betti table by Pardue and Richert in \cite{Pa-Ri,Pa-Ri1}, and by Diem in \cite{Di}. Then there are results on Betti numbers in the case $n+1$ relations in $n$ variables, described below, by Migliore and Mirò-Roig in \cite{Mi-Mi}, and more partial results in the general case by the same authors in \cite{Mi-Mi1}. In this paper we consider the same case as in \cite{Mi-Mi}, $n+1$ forms in $n$ variables. Our results can be described as follows. We can determine all graded Betti numbers of $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_{n+1})$, $g_i$ generic, at least if $\sum_{i=1}^{n+1}°(g_i)-n$ is even, often in more cases. Thus, given {\em any} set $\{ e_1,\ldots,e_n\}$, $e_i\ge2$ for all $i$, such that $°(g_i)=e_i$, $i=1,\ldots,n$, we get many numbers $D_j$, so that we can determine all graded Betti numbers of $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_{n+1})$, $°(g_i)=e_i$, $1\le i\le n$, $°(g_{n+1})=D_j$. The main ingredients of the proof is a theorem by Pardue and Richert, \cite{Pa-Ri,Pa-Ri1}, and later by Diem,\cite{Di}, and a new short proof of a theorem on Hilbert series of artinian complete intersections by Reid, Roberts, and Roitman, \cite{R-R-R}. We also give examples of algebras with many so called "ghost terms" in the minimal resolution. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_16155 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Some new Betti numbers of ideals generated by n+1 generic forms in n variables Fröberg, Ralf Commutative Algebra 13D02, 13D40, 13C13 Very little is known on the Hilbert series of graded algebras $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_r)$, $r>n$, $g_i$ generic form of degree $e_i$, in general. One instance when the series is known, is for $n+1$ forms in $n$ variables, \cite{St}. Of course even less is known about Betti numbers. There are some general results on the Betti table by Pardue and Richert in \cite{Pa-Ri,Pa-Ri1}, and by Diem in \cite{Di}. Then there are results on Betti numbers in the case $n+1$ relations in $n$ variables, described below, by Migliore and Mirò-Roig in \cite{Mi-Mi}, and more partial results in the general case by the same authors in \cite{Mi-Mi1}. In this paper we consider the same case as in \cite{Mi-Mi}, $n+1$ forms in $n$ variables. Our results can be described as follows. We can determine all graded Betti numbers of $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_{n+1})$, $g_i$ generic, at least if $\sum_{i=1}^{n+1}°(g_i)-n$ is even, often in more cases. Thus, given {\em any} set $\{ e_1,\ldots,e_n\}$, $e_i\ge2$ for all $i$, such that $°(g_i)=e_i$, $i=1,\ldots,n$, we get many numbers $D_j$, so that we can determine all graded Betti numbers of $\mathbb C[x_1,\ldots,x_n]/(g_1,\ldots,g_{n+1})$, $°(g_i)=e_i$, $1\le i\le n$, $°(g_{n+1})=D_j$. The main ingredients of the proof is a theorem by Pardue and Richert, \cite{Pa-Ri,Pa-Ri1}, and later by Diem,\cite{Di}, and a new short proof of a theorem on Hilbert series of artinian complete intersections by Reid, Roberts, and Roitman, \cite{R-R-R}. We also give examples of algebras with many so called "ghost terms" in the minimal resolution. |
| title | Some new Betti numbers of ideals generated by n+1 generic forms in n variables |
| topic | Commutative Algebra 13D02, 13D40, 13C13 |
| url | https://arxiv.org/abs/2503.16155 |