Betti splittings for powers of sums of ideals
Betti splittings for powers of sums of ideals
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理想之和的幂的贝蒂分裂
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
H. D. Nguyen
中科院分区：
文献类型：
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作者：
H. D. Nguyen
设$A$和$B$是域$k$上的标准分次多项式环，$I$和$J$分别是包含在$A$和$B$中的非零真齐次理想.用P表示R=A\otimes_k B中I和J的和。在k，I和J的合理条件下，给出了P的深度的精确公式，并利用I和J的数据描述了P的深度和幂的正则性.从而加强了H.T. H\`a、N.V. Trung和T.N.忠我们的主要技术结果表明，在上述条件下，对于所有s\ge 0$和所有n\ge 1$，简单分解I^sP^n=I^{s+1}P^{n-1}+I^sJ^n$产生I^sP^n$的贝蒂分裂。一个分解的理想$L$作为一个总和的两个子理想被称为贝蒂分裂，如果最小自由决议$L$是完全确定的那些被加数和他们的交集。
Let $A$ and $B$ be standard graded polynomial rings over a field $k$ and $I$ and $J$ be non-zero, proper homogeneous ideals contained in $A$ and $B$, respectively. Denote by $P$ the sum of $I$ and $J$ in $R=A\otimes_k B$. Under reasonable conditions on $k, I$ and $J$, we provide exact formulas and describe the asymptotic behavior of the depth and the regularity of the powers of $P$ in terms of the data of $I$ and $J$. Thereby, we strengthen previous work of H.T. H\`a, N.V. Trung and T.N. Trung. Our main technical result says that, under the aforementioned conditions, for all $s\ge 0$ and all $n\ge 1$, the simple decomposition $I^sP^n=I^{s+1}P^{n-1}+I^sJ^n$ yields a Betti splitting for $I^sP^n$. A decomposition of an ideal $L$ as a sum of two subideals is called a Betti splitting if the minimal free resolution of $L$ is completely determined by those of the summands and their intersection.
DOI：
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发表时间：
2006
期刊：
Integrable systems, geometry, and topology, AMS/IP Studies of Advanced Mathematics, American Mathematical Society 36
影响因子：
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作者：
FURUYA;Jun;Martin Guest
通讯作者：
Martin Guest