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Minimal free resolutions for homogeneous ideals with Betti numbers \(1\), \(n\), \(n\), \(1\). (English) Zbl 1441.13058

In this nicely written paper the authors study the standard generalized Gorenstein algebras of homological dimension three. Let \(R = \Bbbk[x_{1},\dots,x_{r}]\) with \(r\geq 3\) and \(\Bbbk\) is any field. A graded standard \(R\)-algebra \(R/I\) of homological dimension \(3\) is called generalized Gorensten algebra if the rank of the last syzygy module is \(1\) in its minimal graded free resolution. In particular, the authors investigate graded minimal resolutions of the type \[0 \rightarrow R(-s) \rightarrow \bigoplus_{j=1}^{n}R(-b_{j}) \rightarrow \sum_{i=1}^{n}R(-a_{i}) \rightarrow R.\] Now we say that a matrix \(M \in R^{n,m}\) is a presentation matrix if it is associated to a map \(\phi\) in a presentation of the form \[R^{m} \xrightarrow{\phi} R^{n} \rightarrow R.\] Now if \(M \in R^{n,m}\), \(n\leq m\), with rank \(\mathrm{rk }\, M = n-1\), and \(N\) is a submatrix of \(M\) of size \(n\times n-1\) of rank \(n-1\), then we define \(g_{i}(M) =g_{i}(N)/d_{N}\), where \(g_{i}(N)\) is the minor of \(N\) obtained by deleting the \(i\)-th row and \(d_{N} = \mathrm{GCD}(g_{1}(N),\dots, g_{n}(N))\). We set additionally \(\gamma(M) = (g_{1}(M), \dots, g_{n}(M))\). Observe that \(\gamma(M)\) generates an ideal \(I_{M}\) with \(\mathrm{depth} \, I_{M} \geq 2\). Moreover, \(\gamma(M)\) defines a map \(\gamma(M) : R^{n} \rightarrow R\). One can show that if \(M \in R^{n,n}\) is a square presentation matrix, that \(R/I_{M}\) is a generalized Gorenstein algebra whose free resolution is of the form \[0 \rightarrow R^{*} \xrightarrow{\gamma(\phi)^{*}} R^{n} \xrightarrow {\phi} R^{n} \xrightarrow{\gamma(\phi)} R \rightarrow R/I_{M} \rightarrow 0,\] where \(\phi\) is the map associated with the matrix \(M\). If the above resolution is minimal, than we say that \(M\) is a minimal presentation matrix.
The main structural result of the paper can be formulated as follows.
Theorem A. Let \(M \in R^{n,n}\) be a matrix of rank \(n-1\), \(\gamma(M) = (g_{1}, \dots, g_{n})\), \(\gamma(M^{T}) = (h_{1}, \dots, h_{n})\), and let \(J\) be the ideal generated by \(\gamma(M^{T})\). Denote by \(M^{C}\) the cofactor of matrix \(M\). The matrix \(M\) is a presentation matrix if and only if \(\mathrm{depth} \, J \geq 3\) and \(M^{C} =[u(g_{i}h_{j})]\), where \(u\) is a unit.
Theorem B. Let \(X \subset \mathbb{P}^{r}\) with \(r\geq 3\) be a closed projective scheme whose defining ideal \(I_{X}\) has a graded minimal free resolution of the form \[0 \rightarrow R \xrightarrow{\rho = (\tau,0)} R^{3} \oplus R^{n-3} \xrightarrow{(\alpha \kappa) \oplus \delta} R^{n} \rightarrow R,\] where \(\tau(1) = (h_{1},h_{2},h_{3})\) and \((h_{1},h_{2},h_{3})\) is a regular sequence, \(\kappa: R^{3} \rightarrow R^{3}\) is the Koszul map on \(h_{1},h_{2},h_{3}\), \(\alpha : R^{3} \rightarrow R^{n}\), \(\delta:R^{n-3} \rightarrow R^{n}\) are suitable maps. Let \(Z\) be the complete intersection defined by \(I(\rho) = I(\tau)\), and let \(S = V(\mathrm{det}(\alpha \oplus \delta))\). Denote by \(Y\) the scheme defined by \(I(\alpha \oplus \delta, \rho^{*})\). If \(\mathrm{codim} \, (S\cap Z) =4\), then \(X = Y \cup Z\).
Next, the authors focus on the case \(n=3\) providing an explicit characterization of the graded Betti numbers for generalized Gorenstein ideals having a graded minimal free resolution of the type \[0 \rightarrow F_{3} \xrightarrow{\rho}F_{2}\xrightarrow{\phi}F_{1} \xrightarrow{\psi} R \rightarrow R/I \rightarrow 0\] with \(\mathrm{rank}\, F_{1} = \mathrm{rank}\, F_{2} = 3\), and \(\mathrm{rank}\, F_{1} = 1\). Finally, in the last section, the authors provide a complete description of the graded Betti sequence for those schemes which have \(n\) generators and syzygies with concentraded degrees with \(n\) is odd.

MSC:

13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
14N20 Configurations and arrangements of linear subspaces
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
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References:

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