The Resolution of the Ideal of 2 × 2 Minors of a 2 × nMatrix of Linear Forms
The Resolution of the Ideal of 2 × 2 Minors of a 2 × nMatrix of Linear Forms
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2 × n 线性形式矩阵的 2 × 2 次式理想值的解析
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发表时间：
1997
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通讯作者：
Michael L. Catalano
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作者：
Michael L. Catalano
设k是代数闭域，S = k [x1，…，xm].设M是S的线性形式的2 × n矩阵，I2（M）表示由M的2 × 2子式的行列式生成的理想.本文研究了S /I2（M）作为S -模的最小有限自由分解。M对应于某个m × n矩阵的二维向量空间L，即对应于一个矩阵束。这种矩阵束的Kronecker-Weierstrass理论为L提供了一个标准形，我们用这个标准形刻画了S /I2（M）的分解.特别地，如果L的广义元是内射的，我们通过反复应用马蹄引理，明确地构造了S /I2（M）的最小分解.对于任意M，我们将S /I2（M）的正则性表示为L的不变量的函数。
Abstract Let k be an algebraically closed field and let S = k [ x 1 ,…, x m ]. Let M be a 2 × n matrix of linear forms of S and let I 2 ( M ) denote the ideal generated by the determinants of the 2 × 2 minors of M . We study in this paper the minimal finite free resolution of S / I 2 ( M ) as an S -module. M corresponds to a certain 2-dimensional vector space L of m × n matrices, that is, to a matrix pencil. The Kronecker–Weierstrass theory of such matrix pencils provides a normal form for L , and we characterize the resolution of S / I 2 ( M ) in terms of this normal form. In particular, if the general element of L is injective, we explicitly construct the minimal resolution of S / I 2 ( M ) by repeated application of the horseshoe lemma. For any M , we express the regularity of S / I 2 ( M ) as a function of the invariants of L .