论完全相交的局部除数类群

Pub Date : 2024-09-10 DOI:10.1016/j.jpaa.2024.107804
Daniel Windisch
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引用次数: 0

摘要

萨缪尔在 1961 年猜想,一个(诺特)局部完全交环在至多三维上是一个 UFD,它本身也是一个 UFD。据说格罗thendieck 发明了局部同调学来证明这一事实。根据 UFD 无非是一个具有微分子类群的 Krull 域(在诺特情况下是一个正域)这一理念,我们仔细研究了塞缪尔-格罗thendieck 定理,并证明了以下概括:(1)当且仅当 A 是标度至多为 1 的正域时,A 是正域。(2)假设 A 是正域和完全交集。我们利用这一事实来描述一个积分诺特局部完全交方案 X 的魏尔组和卡蒂埃分维组之间的差距,并在这种情况下推广经典结果,即如果 X 是局部 UFD,这两个概念是重合的。
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On local divisor class groups of complete intersections

Samuel conjectured in 1961 that a (Noetherian) local complete intersection ring that is a UFD in codimension at most three is itself a UFD. It is said that Grothendieck invented local cohomology to prove this fact. Following the philosophy that a UFD is nothing else than a Krull domain (that is, a normal domain, in the Noetherian case) with trivial divisor class group, we take a closer look at the Samuel–Grothendieck Theorem and prove the following generalization: Let A be a local Cohen–Macaulay ring.

  • (1)

    A is a normal domain if and only if A is a normal domain in codimension at most 1.

  • (2)

    Suppose that A is a normal domain and a complete intersection. Then the divisor class group of A is a subgroup of the projective limit of the divisor class groups of the localizations Ap, where p runs through all prime ideals of height at most 3 in A.

We use this fact to describe for an integral Noetherian locally complete intersection scheme X the gap between the groups of Weil and Cartier divisors, generalizing in this case the classical result that these two concepts coincide if X is locally a UFD.
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