We use the anti-equivalence between Cohen-Macaulay complexes and coherent sheaves on formal schemes to shed light on some older results and prove new results. We bring out the relations between a coherent sheaf M satisfying an S_2 condition and the lowest cohomology N of its "dual" complex. We show that if a scheme has a Gorenstein complex satisfying certain coherence conditions, then in a finite étale neighborhood of each point, it has a dualizing complex. If the scheme already has a dualizing complex, then we show that the Gorenstein complex must be a tensor product of a dualizing complex and a vector bundle of finite rank. We relate the various results in [S] on Cousin complexes to dual results on coherent sheaves on formal schemes.
我们在连通诺特泛悬链线有限维分离概型的有限型映射范畴上构造一个正则伪函子，取值于Cousin复形范畴。这个伪函子是对Grothendieck对偶伪函子^！到具有Cohen-Macaulay复形作为对象的导出范畴的全子范畴（一旦余维函数被固定，则子范畴等价于Cousin复形的范畴）。具体地说，对于Cousin复形M和上面的任何模式映射f：X -> Y，存在一个函子导范畴映射γ：f^# M -> f^！M在Cousin复形范畴f^# M \iso E（f^！（其中E是Cousin函子）。γ本身是同构，如果复形f^！M是Cohen-Macaulay--当映射f是平坦的或者复形M是单射的时候都是如此。此外，f^#取残差（分别为单射）复形在Y上到残差（分别）。单射）复形;因此，伪函子^#推广并规范了哈特索恩《剩余与对偶》第六章中提出的剩余复形的“方差理论”。“此外，我们推广了loc.cit.的留数定理，p.369通过定义分次模Tr_f（M）：f_*f^# M -> M（局部剩余之和）的函子迹映射，使得只要f是真的，Tr_f（M）是复形的映射，并且（f^# M，Tr_f（M））对表示Cousin复形C的函子Hom（f_*C，M）。
If K is a number field, arithmetic duality theorems for tori and complexes of tori over K are crucial to understand local-global principles for linear algebraic groups over K. When K is a global field of positive characteristic, we prove similar arithmetic duality theorems, including a Poitou-Tate exact sequence for Galois hypercohomology of complexes of tori. One of the main ingredients is Artin-Mazur-Milne duality theorem for fppf cohomology of finite flat commutative group schemes.
本文给出了交换环上Grothendieck对偶的一种新的研究方法。我们的方法是基于刚性对偶复形的思想，这是由Van den Bergh在非交换代数几何的背景下引入的。本文对文献[YZ 5]中的刚性方法进行了改进，使之适用于一般交换基环。在本文中，我们获得了许多重要的本地功能的Grothendieck对偶，但设法避免冗长和困难的兼容性验证。我们的结果适用于有限维正则Noether基环上的本质有限型代数，因此也适用于算术环。在[Ye4]中，这些结果将被用来构造和研究概型上的刚性对偶复形。
In this article we explain the theory of rigid residue complexes in commutative algebra and algebraic geometry, summarizing the background, recent results and anticipated future results. Unlike all previous approaches to Grothendiec Duality, the rigid approach concentrates on the construction of rigid residue complexes over rings, and their intricate yet robust properties. The geometrization, i.e. the passage to rigid residue complexes on schemes and Deligne-Mumford (DM) stacks, by gluing, is fairly easy. In the geometric part of the theory, the main results are the Rigid Residue Theorem and t…
Let $\mathbb{X}$ be a semiseparated Noetherian scheme with a dualizing complex $D$. We lift some well-known triangulated equivalences associated with Grothendieck duality to Quillen equivalences of model categories. In the process we are able to show that the Gorenstein flat model structure, on the category of quasi-coherent sheaves on $\mathbb{X}$, is Quillen equivalent to the Gorenstein injective model structure. Also noteworthy is that we extend the recollement of Krause to hold without the Noetherian condition. Using a set of flat generators, it holds for any quasi-compact semiseparated sc…
In this article, we prove that there is a canonical Verdier self-dual intersection space sheaf complex for the middle perversity on Witt spaces that admit compatible trivializations for their link bundles, for example toric varieties. If the space is an algebraic variety our construction takes place in the category of mixed Hodge modules. We obtain an intersection space cohomology theory, satisfying Poincaré duality, valid for a class of pseudomanifolds with arbitrary depth stratifications. The main new ingredient is the category of Künneth complexes; these are cohomologically constructible co…
We define cohomological complexes of locally compact abelian groups associated with varieties over $p$-adic fields and prove a duality theorem under some assumption. Our duality takes the form of Pontryagin duality between locally compact motivic cohomology groups.
In this paper we present a new approach to Grothendieck duality on schemes. Our approach is based on the idea of rigid dualizing complexes, which was introduced by Van den Bergh in the context of noncommutative algebraic geometry. We obtain most of the important features of Grothendieck duality, yet manage to avoid lengthy and difficult compatibility verifications. Our results apply to finite type schemes over a regular noetherian finite dimensional base ring, and hence are suitable for arithmetic geometry.
Poincaré duality for pro-étale $\mathbf{Q}_p$-local systems
Morningside Center of Mathematics and State Key Laboratory of Mathematics Sciences, Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院，中国科学院数学与系统科学研究所）；CNRS, IMJ-PRG, Sorbonne Université（法国国家科学研究中心，IMJ-PRG，索邦大学）；Institut for Matematiske Fag, Københavns Universitet（哥本哈根大学数学系）；University of Maryland, Department of Mathematics（马里兰大学，数学系）
We generalize the adjunction between the functors $Rf_*$ and $f^!$ of derived categories of quasi-coherent sheaves for proper morphisms $f\colon X \to Y$ of Noetherian schemes to the following situation: Let $f$ be a finite type morphism and let $Z' \subseteq X$ and $Z \subseteq Y$ be closed subsets such that $f$ restricts to a proper morphism $f'\colon Z'\to Z$ of $f$. Then the functor $Rf_*$ is left adjoint to $RΓ_{Z'}f^!$ when considered as functors between complexes supported on $Z'$ or $Z$.
Let $X$ be a separated scheme of dimension $d$ of finite type over a perfect field $k$ of positive characteristic $p$. In this work, we show that Bloch's cycle complex $\mathbb{Z}^c_X$ of zero cycles mod $p^n$ is quasi-isomorphic to the Cartier operator fixed part of a certain dualizing complex from coherent duality theory. From this we obtain new vanishing results for the higher Chow groups of zero cycles with mod $p^n$ coefficients for singular varieties.
We present an obstruction theoretic inductive construction of intersection space pairs, which generalizes Banagl's construction of intersection spaces for arbitrary depth stratifications. We construct intersection space pairs for pseudomanifolds with compatible trivial structures at the link fibrations; this includes the case of toric varieties. We define intersection space complexes in an axiomatic way, similar to Goresky-McPherson axioms for intersection cohomology. We prove that if the intersection space exists, then the pseudomanifold has an intersection space complex whose hypercohomology…
We study consequences and applications of the folklore statement that every double complex over a field decomposes into so-called squares and zigzags. This result makes questions about the associated cohomology groups and spectral sequences easy to understand. We describe a notion of `universal' quasi-isomorphism, investigate the behaviour of the decomposition under tensor product and compute the Grothendieck ring of the category of bounded double complexes over a field with finite cohomologies up to such quasi-isomorphism (and some variants). Applying the theory to the double complexes of smo…
