Maximal minimal resolutions
Maximal minimal resolutions
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最大最小分辨率
DOI：
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发表时间：
1999
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通讯作者：
K. Pardue
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作者：
S. Iyengar;K. Pardue
除了在F0处外，它在任何地方都是精确的，在F0处，上核是M。这样的决议总是存在的。如果R是一个Noether N-分次环，R 0是一个域，M是一个N-生成的Z-分次R-模，则可以用极小的方式构造一个分解，其中每个Fi是分次的，每个微分是0次齐次的。M的极小自由分解中的第i个自由模的秩是M的不变量，称为M的第i个贝蒂数，记为β i（M）。同样地，β Rij（M），Fi的齐次生成元的最小集合中的j次元素的个数，也是M的不变量，称为M的第（i，j）分次Betti数。本文研究了具有极大分次Betti数的模.更精确地说，考虑一个由对（R，M）组成的集合R，其中R是一个诺特N分次环，R 0是一个域，M是一个N-生成分次R-模。是否存在某个（R，M）∈ M使得对每个i和j以及每个（R，M）∈ M，β ′ ij（M）≥ β ij（M）？有一些微不足道的例子，其中的答案是肯定的（例如，只有一个元素）或没有（例如，所有这些对都包含在内）。下面的定理1和2给出了由希尔伯特级数的条件和R和M的深度定义的某些集合的肯定答案。定理3允许计算定理1和2中考虑的集合中的对的最大分次Betti数。在本文中，k表示域，Q表示多项式环k[x 1，. . .，xn]，其中通常的N-分级由degxi = 1给出。M的Hilbert级数为HM（s）= ∑ dimk Mis .我们在第二节中定义了d-字典序理想和子模。R-模M的Poincare级数为PM（s; t）= ∑ β ij（M）st.我们把系数在Z中的Laurent级数的系数不等式写成4。
that is exact everywhere except at F0, where the cokernel is M . Such a resolution always exists. If R is a noetherian N-graded ring, with R0 a field, and M is a finitely generated Z-graded R-module, then a resolution may be constructed in a minimal way in which each Fi is graded and each differential is homogeneous of degree 0. The rank of the ith free module in a minimal free resolution of M is an invariant of M , called the ith Betti number of M and denoted β i (M). Likewise, β R ij(M), the number of elements of degree j in a minimal set of homogeneous generators of Fi, is also an invariant of M , called the (i, j)th graded Betti number of M . In this paper we study modules with maximal graded Betti numbers. More precisely, consider a set Π consisting of pairs (R,M) where R is a noetherian N-graded ring, with R0 a field, and M is a finitely generated graded R-module. Is there some (R ,M ) ∈ Π such that β ′ ij (M ) ≥ β ij(M) for every i and j and every (R,M) ∈ Π? There are trivial examples in which the answer is yes (e.g., Π has only one element) or no (e.g., Π consists of all such pairs). Theorems 1 and 2 below give affirmative answers for certain sets Π defined by conditions on the Hilbert series and the depths of R and M . Theorem 3 allows one to compute the maximal graded Betti numbers for the pairs in the sets considered in Theorems 1 and 2. Throughout this paper, k denotes a field and Q the polynomial ring k[x1, . . . , xn] with the usual N-grading given by degxi = 1. The Hilbert series of a finitely generated graded Q-module M is HM (s) = ∑ dimk Mis . We define d-lexicographic ideals and submodules in Section 2. The Poincare series of an R-module M is P M (s; t) = ∑ β ij(M)s t. We write 4 for coefficient-wise inequality of Laurent series with coefficients in Z.