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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].

##### MSC:
 13C40 Linkage, complete intersections and determinantal ideals 13D02 Syzygies, resolutions, complexes and commutative rings 13D25 Complexes (MSC2000)