Let $R$ be a polynomial or power series ring over a field $k$. We study thelength of local cohomology modules $H^j_I(R)$ in the category of $D$-modulesand $F$-modules. We show that the $D$-module length of $H^j_I(R)$ is bounded bya polynomial in the degree of the generators of $I$. In characteristic $p>0$ weobtain upper and lower bounds on the $F$-module length in terms of thedimensions of Frobenius stable parts and the number of special primes of localcohomology modules of $R/I$. The obtained upper bound is sharp if $R/I$ is anisolated singularity, and the lower bound is sharp when $R/I$ is Gorenstein and$F$-pure. We also give an example of a local cohomology module that hasdifferent $D$-module and $F$-module lengths.
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机译：令$ R $为在域$ k $上的多项式或幂级数环。我们在$ D $ -modules和$ F $ -modules类别中研究局部同调模块$ H ^ j_I（R）$的长度。我们证明，$ H ^ j_I（R）$的$ D $模块长度在$ I $的生成度中受多项式限制。在特征$ p> 0 $中，根据Frobenius稳定部分的维数和$ R / I $局部同调性模块的特殊素数的数量，获得$ F $模块长度的上限和下限。如果$ R / I $是孤立的奇点，则得到的上限是尖锐的，而当$ R / I $是Gorenstein和$ F $ -pure时，下界则是尖锐的。我们还给出了一个局部同调模块的示例，该模块具有不同的$ D $-模块和$ F $-模块长度。
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