Duality and de Rham cohomology for graded D-modules
Duality and de Rham cohomology for graded D-modules
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DOI：
10.1016/j.aim.2018.07.005
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发表时间：
2017-05
影响因子：
1.7
通讯作者：
N. Switala;Wenliang Zhang
中科院分区：
文献类型：
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作者：
N. Switala;Wenliang Zhang
考虑多项式环R= k [x 1，…，x n]（k是特征为零的域）上的一个分级D模M的（分级）矩阵对偶D (M)，并证明了它可以给定一个D模的结构，即当dim k (H) D R i (M)是有限时，H D R i (M)是H D R n - i （D (M)）的k对偶。因此，我们证明了如果M是一个梯度d模，使得H d R n (M)是有限维k空间，那么dim k （H d R n (M)）是存在满射d -线性同态M→E s的最大整数s，其中E是上局部上同态模H (x 1，…，x n) n (R)。这将Hartshorne和Polini关于形式幂级数环的最新结果推广到多项式环的情况；我们还应用相同的思想循环来提供其结果的替代证明。当M是一个有限生成的梯度d模，使得dim k (H) R (i)是有限的，我们进一步推广了上述结果，表明H (R) n - i (M)是Ext (d) i （M， E）的k对偶。
We consider the (graded) Matlis dual D (M) of a graded D-module M over the polynomial ring R= k [x 1,…, x n](k is a field of characteristic zero), and show that it can be given a structure of D-module in such a way that, whenever dim k H d R i (M) is finite, then H d R i (M) is k-dual to H d R n− i (D (M)). As a consequence, we show that if M is a graded D-module such that H d R n (M) is a finite-dimensional k-space, then dim k(H d R n (M)) is the maximal integer s for which there exists a surjective D-linear homomorphism M→ E s, where E is the top local cohomology module H (x 1,…, x n) n (R). This extends a recent result of Hartshorne and Polini on formal power series rings to the case of polynomial rings; we also apply the same circle of ideas to provide an alternate proof of their result. When M is a finitely generated graded D-module such that dim k H d R i (M) is finite, we generalize the above result further, showing that H d R n− i (M) is k-dual to Ext D i (M, E).