Local cohomology and base change
Local cohomology and base change
复制标题
局部上同调和基数变化
DOI：
10.1016/j.jalgebra.2017.09.036
复制
发表时间：
2018
影响因子：
0.9
通讯作者：
Smith, Karen E.
中科院分区：
文献类型：
--
作者：
Smith, Karen E.
设X ∈ f S是Noether格式的态射，S是约化的。对于S上有限X的任何闭子概型Z，令j表示开浸入X <$Z <$X。则对X <$Z上的任何凝聚层F和任何指标r≥ 1，层F <$（R r j <$F）在S上是一般自由的，并与基变交换。我们通过证明一个关于局部上同调的相关陈述来证明这一点：设R是Noether整环A上的Noether代数，设I <$R是一个理想，使得R/I是一个A-模的n-生成。设M是R-生成的R-模。则存在一个非零g∈ A使得局部上同调模H I r（M）<$A g在A g上是自由的，并且对于任何通过A g的环映射A→ L，我们有H I r（M）<$A L <$H I <$A L r（M <$A L）对所有r。
Abstract Let X⟶ f S be a morphism of Noetherian schemes, with S reduced. For any closed subscheme Z of X finite over S, let j denote the open immersion X∖ Z↪ X. Then for any coherent sheaf F on X∖ Z and any index r≥ 1, the sheaf f⁎(R r j⁎ F) is generically free on S and commutes with base change. We prove this by proving a related statement about local cohomology: Let R be Noetherian algebra over a Noetherian domain A, and let I⊂ R be an ideal such that R/I is finitely generated as an A-module. Let M be a finitely generated R-module. Then there exists a non-zero g∈ A such that the local cohomology modules H I r (M)⊗ A A g are free over A g and for any ring map A→ L factoring through A g, we have H I r (M)⊗ A L≅ H I⊗ A L r (M⊗ A L) for all r.