On the modules of m-integrable derivations in non-zero characteristic

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• 作者：Luis Narvá ; ez Macarro¹ ; ^{narvaez@algebra.us.es}
• 关键词：14F10 ; 13N15 ; 14B05
• 刊名：Advances in Mathematics
• 出版年：2012
• 期刊代码：62_00018708
• 类别：mt
• 出版时间：20 March, 2012
• 卷：229
• 期：5
• 页码：2712-2740
• 文件大小：275 K

摘要

Let k be a commutative ring and A a commutative k-algebra. Given a positive integer m, or , we say that a k-linear derivation 未 of A is m-integrable if it extends up to a Hasse-Schmidt derivation of A over k of length m. This condition is automatically satisfied for any m under one of the following orthogonal hypotheses: (1) k contains the rational numbers and A is arbitrary, since we can take ; (2) k is arbitrary and A is a smooth k-algebra. The set of m-integrable derivations of A over k is an A-module which will be denoted by . In this paper we prove that, if A is a finitely presented k-algebra and m is a positive integer, then a k-linear derivation 未 of A is m-integrable if and only if the induced derivation is m-integrable for each prime ideal . In particular, for any locally finitely presented morphism of schemes and any positive integer m, the S-derivations of X which are locally m-integrable form a quasi-coherent submodule such that, for any affine open sets and , with , we have and for each . We also give, for each positive integer m, an algorithm to decide whether all derivations are m-integrable or not.