论文标题
紧密闭合，连贯性和单个元素的本地化
Tight closure, coherence, and localization at single elements
论文作者
论文摘要
在本说明中，提出了一个条件（\ emph {open持久性}），在该条件下，在该方案$ x $上对全球部分的consodules（分别为理想的）（分别为理想）的（预）闭合操作可以扩展到（封闭式封闭式操作）对一致的$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ shea的（of）的（封闭式封闭操作）。 $ \ MATHCAL {O} _X $）。给出了第二个条件（\ emph {glueability}），以使这种操作表现得很好。结果表明，对于满足两种条件的操作，在单个元素上使用定位的操作是否与新操作保持准固定的问题相当。结果表明，这两种条件都符合紧密闭合及其一些重要变体，从而产生了几何反映，即紧密闭合是否定位在单个元素上。出现了一种新的奇异性类型（\ emph {semi f-regularity}），它位于f-reculacority和弱f-regormuline之间。该论文以（1）半F型和弱F型重合的情况结尾，以及（2）在不暗示对主要猜想的解决方案而无法重合的情况下。
In this note, a condition (\emph{open persistence}) is presented under which a (pre)closure operation on submodules (resp. ideals) over rings of global sections over a scheme $X$ can be extended to a (pre)closure operation on sheaves of submodules of a coherent $\mathcal{O}_X$-module (resp. sheaves of ideals in $\mathcal{O}_X$). A second condition (\emph{glueability}) is given for such an operation to behave nicely. It is shown that for an operation that satisfies both conditions, the question of whether the operation commutes with localization at single elements is equivalent to the question of whether the new operation preserves quasi-coherence. It is shown that both conditions hold for tight closure and some of its important variants, thus yielding a geometric reframing of the open question of whether tight closure localizes at single elements. A new singularity type (\emph{semi F-regularity}) arises, which sits between F-regularity and weak F-regularity. The paper ends with (1) a case where semi F-regularity and weak F-regularity coincide, and (2) a case where they cannot coincide without implying a solution to a major conjecture.
