Syzygy ideals for determinantal ideals and the syzygetic Castelnuovo lemma.

*(English)*Zbl 0955.13005
Eisenbud, David (ed.), Commutative algebra, algebraic geometry, and computational methods. Proceedings of the conference on algebraic geometry, commutative algebra, and computation, Hanoi, Vietnam, 1996. Singapore: Springer. 247-258 (1999).

Let \(S=k[X_0,\dots,X_n]\) be a polynomial ring over a field \(k\), \(W=S_1\) the space of linear forms in \(S\) and \(I\) a homogeneous ideal of \(S\). By Koszul cohomology, \(\text{Tor}_p^S(I,k)_{p+q}\) is isomorphic to the middle cohomology of the complex: \(\bigwedge^{p+1} W\otimes I_{q-1} \to\bigwedge^p W\otimes I_q \to\bigwedge^{p-1} W\otimes I_{q+1}\). Let \(\tau\) be a non-zero element of \(\text{Tor}^S_p (I,k)_{p+q}\). It is represented by an element \(\sum X_{i_1}\wedge \cdots \wedge X_{i_p}\otimes f_{i_1, \dots,i_p} \in\text{Ker} (\bigwedge^p W\otimes I_q\to \bigwedge^{p-1}W \otimes I_{q+1})\). The syzygy ideal of \(\tau\) is, by definition, the ideal of \(S\) generated by the homogeneous element of \(I\) of degree \(<q\) and by the forms \(f_{i_1, \dots,i_p}\).

Assume, now, that \(I\) is the ideal generated by the \(g\times g\) minors of an \(f\times g\) matrix \(\varphi\) of linear forms, \(f\geq g\), and that \(I\) has the “expected” height \(f-g+1\). In this case, the Eagon-Northcott complex of \(\varphi\) is a minimal free resolution of \(I\). In particular, \(\text{Tor}_p(I,k)=0\) for \(p>f-g\) and \(\text{Tor}_{f-g} (I,k)\) is concentrated in degree \(f\). The main result of the paper asserts that if the syzygy ideal of each non-zero elements of \(\text{Tor}_{f-g} (I,k)\) is \(I\) then \(\varphi\) is 1-generic in the sense of D. Eisenbud [Am. J. Math. 110, No. 3, 541-575 (1988; Zbl 0681.14028)]. The authors also show that the converse is true if \(g=2\) and conjecture that it is true in general.

As an application, they give a new direct proof of the generalization of the syzygetic Castelnuovo lemma to non-reduced 0-dimensional schemes proved by M. L. Green [J. Differ. Geom. 19, 125-171 (1984; Zbl 0559.14008)] in the reduced case. This generalization was proved, independently, by S. Ehbauer [in: Zero-dimensional schemes, Proc. int. Conf., Ravello 1992, 145-170 (1994; Zbl 0840.14032)] and K. Yanagawa [J. Algebra 170, No. 2, 429-439 (1994; Zbl 0838.14040)].

For the entire collection see [Zbl 0921.00043].

Assume, now, that \(I\) is the ideal generated by the \(g\times g\) minors of an \(f\times g\) matrix \(\varphi\) of linear forms, \(f\geq g\), and that \(I\) has the “expected” height \(f-g+1\). In this case, the Eagon-Northcott complex of \(\varphi\) is a minimal free resolution of \(I\). In particular, \(\text{Tor}_p(I,k)=0\) for \(p>f-g\) and \(\text{Tor}_{f-g} (I,k)\) is concentrated in degree \(f\). The main result of the paper asserts that if the syzygy ideal of each non-zero elements of \(\text{Tor}_{f-g} (I,k)\) is \(I\) then \(\varphi\) is 1-generic in the sense of D. Eisenbud [Am. J. Math. 110, No. 3, 541-575 (1988; Zbl 0681.14028)]. The authors also show that the converse is true if \(g=2\) and conjecture that it is true in general.

As an application, they give a new direct proof of the generalization of the syzygetic Castelnuovo lemma to non-reduced 0-dimensional schemes proved by M. L. Green [J. Differ. Geom. 19, 125-171 (1984; Zbl 0559.14008)] in the reduced case. This generalization was proved, independently, by S. Ehbauer [in: Zero-dimensional schemes, Proc. int. Conf., Ravello 1992, 145-170 (1994; Zbl 0840.14032)] and K. Yanagawa [J. Algebra 170, No. 2, 429-439 (1994; Zbl 0838.14040)].

For the entire collection see [Zbl 0921.00043].

Reviewer: I.Coandă (Bucureşti)

##### MSC:

13C40 | Linkage, complete intersections and determinantal ideals |

13D02 | Syzygies, resolutions, complexes and commutative rings |

13D25 | Complexes (MSC2000) |

##### Keywords:

determinantal ideals; matrices of linear forms; Koszul cohomology; syzygy ideal; Eagon-Northcott complex; syzygetic Castelnuovo lemma
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\textit{D. Eisenbud} and \textit{S. Popescu}, in: Commutative algebra, algebraic geometry, and computational methods. Proceedings of the conference on algebraic geometry, commutative algebra, and computation, Hanoi, Vietnam, 1996. Singapore: Springer. 247--258 (1999; Zbl 0955.13005)