Generic lattice ideals
Generic lattice ideals
复制标题
通用格理想
DOI：
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发表时间：
1998
期刊：
影响因子：
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通讯作者：
B. Sturmfels
中科院分区：
文献类型：
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作者：
I. Peeva;B. Sturmfels
设S=k[x1，.。。，xn]是域k上的多项式环，i是S中的齐次理想。交换代数中的一个基本问题是构造S上S/i的最小自由分解f_i。当i是完全交时，分解是结构良好且简单的：在这种情况下，f_i是Koszul复形。完全交是生成元具有足够一般系数的理想，因此它们可以被认为是所有理想中的一般理想。然而，还有另一个完全不同的泛性概念：理想的生成元相对于它们的指数--而不是它们的系数--是通用的。这一观点是在[BPS]中针对单项理想而提出的。在本文中，我们引入了格理想的通用性的概念，它包括定义环簇的理想。如果L是Z的任一子格，则其在S中的结合格理想是IL：=<xa−x：a，b∈N and a−b∈L>，其中单项式记为x=x11··xan n，其中a=(A1，.。。、An)。如果格理想IL是由具有完全支撑的二项式生成的，即，
Let S = k[x1, . . . , xn] be a polynomial ring over a field k and I a homogeneous ideal in S. A basic problem in commutative algebra is to construct the minimal free resolution FI of S/I over S. The resolution is nicely structured and simple when I is a complete intersection: in this case FI is the Koszul complex. Complete intersections are ideals whose generators have sufficiently general coefficients, so they might be regarded as generic among all ideals. Yet there is another, entirely different, notion of genericity: ideals whose generators are generic with respect to their exponents – not their coefficients. This point of view was developed for monomial ideals in [BPS]. In the present work we introduce a notion of genericity for lattice ideals, which include ideals defining toric varieties. If L is any sublattice of Z, then its associated lattice ideal in S is IL := 〈xa − x : a,b ∈ N and a− b ∈ L 〉, where monomials are denoted x = x1 1 · · ·xan n for a = (a1, . . . , an). We call a lattice ideal IL generic if it is generated by binomials with full support, i.e.,