On local divisor class groups of complete intersections
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866910563929423872 |
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| author | Windisch, Daniel |
| author_facet | Windisch, Daniel |
| contents | 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.
(i) $A$ is a normal domain if and only if $A$ is a normal domain in codimension at most $1$.
(ii) 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 $A_p$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_05270 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On local divisor class groups of complete intersections Windisch, Daniel Commutative Algebra Algebraic Geometry 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. (i) $A$ is a normal domain if and only if $A$ is a normal domain in codimension at most $1$. (ii) 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 $A_p$, 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. |
| title | On local divisor class groups of complete intersections |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2303.05270 |