Determinant functors on exact categories and their extensions to categories of bounded complexes
Determinant functors on exact categories and their extensions to categories of bounded complexes
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精确范畴上的行列式函子及其对有界复形范畴的扩展
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发表时间：
2002
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通讯作者：
F. Knudsen
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作者：
F. Knudsen
在这篇论文中，我重新审视了[KM]中处理得并不令人满意的一个主题。这里使用的方法更自然、更通用。我们证明的定理是格洛腾迪克在 1973 年 5 月 19 日的一封信中向我提出的（见附录 B），它指出精确范畴的派生范畴上的行列式范畴通过限制与精确范畴本身的行列式范畴等效。 [KM] 问题是这样产生的。考虑以下类别。这些对象是固定方案（位点）X 上 OX 模的局部自由有限准相干滑轮的有界复合体。两个此类复合体的态射 Mor (A,B) 是从 A 到 B 的同伦类胚芽束的全局部分的群。如果我们将可逆层分配给每个复合体 f(A) = (⊗ i∈Z max ∧ A2i ) ⊗ (⊗ i∈Z max ∧ A2i+1 )−1 ,
In this paper I revisit a theme unsatisfactorily treated in [KM]. The methods used here are more natural and more general. The theorem we prove was suggested to me by Grothendieck in a letter dated May 19, 1973 (see Appendix B), and it states that the category of determinants on the derived category of an exact category is equivalent via restriction to the category of determinants on the exact category itself. Here is how the problem comes about [KM]. Consider the following category. The objects are bounded complexes of locally free finite quasi-coherent sheaves of OX-modules on a fixed scheme (site) X. The morphism Mor (A,B) of two such complexes is the group of global sections of the sheaf of germs of homotopy classes of homomorphisms from A to B. If we assign to every complex the invertible sheaf f(A) = (⊗ i∈Z max ∧ A2i ) ⊗ (⊗ i∈Z max ∧ A2i+1 )−1 ,