scholarly journals Relation between two twisted inverse image pseudofunctors in duality theory

2014 ◽  
Vol 151 (4) ◽  
pp. 735-764 ◽  
Author(s):  
Srikanth B. Iyengar ◽  
Joseph Lipman ◽  
Amnon Neeman
Keyword(s):  
Finite Type ◽  
Duality Theory ◽  
Inverse Image ◽  
Base Change ◽  
Hochschild Homology ◽  
Fundamental Class ◽  
Compactly Supported ◽  
Canonical Map ◽  
Flat Base ◽  
Derived Functors

Grothendieck duality theory assigns to essentially finite-type maps $f$ of noetherian schemes a pseudofunctor $f^{\times }$ right-adjoint to $\mathsf{R}f_{\ast }$, and a pseudofunctor $f^{!}$ agreeing with $f^{\times }$ when $f$ is proper, but equal to the usual inverse image $f^{\ast }$ when $f$ is étale. We define and study a canonical map from the first pseudofunctor to the second. This map behaves well with respect to flat base change, and is taken to an isomorphism by ‘compactly supported’ versions of standard derived functors. Concrete realizations are described, for instance for maps of affine schemes. Applications include proofs of reduction theorems for Hochschild homology and cohomology, and of a remarkable formula for the fundamental class of a flat map of affine schemes.

scholarly journals Uniformizing the Moduli Stacks of Global G-Shtukas

2019 ◽  
Author(s):  
E Arasteh Rad ◽  
Urs Hartl
Keyword(s):  
Finite Type ◽  
Moduli Spaces ◽  
Function Field ◽  
Group Scheme ◽  
Base Change ◽  
Generic Fiber ◽  
Smooth Projective Curve ◽  
Langlands Program ◽  
Moduli Stacks ◽  
Divisible Groups

Abstract This is the 2nd in a sequence of articles, in which we explore moduli stacks of global $\mathfrak{G}$-shtukas, the function field analogs for Shimura varieties. Here $\mathfrak{G}$ is a flat affine group scheme of finite type over a smooth projective curve $C$ over a finite field. Global $\mathfrak{G}$-shtukas are generalizations of Drinfeld shtukas and analogs of abelian varieties with additional structure. We prove that the moduli stacks of global $\mathfrak{G}$-shtukas are algebraic Deligne–Mumford stacks separated and locally of finite type. They generalize various moduli spaces used by different authors to prove instances of the Langlands program over function fields. In the 1st article we explained the relation between global $\mathfrak{G}$-shtukas and local ${{\mathbb{P}}}$-shtukas, which are the function field analogs of $p$-divisible groups. Here ${{\mathbb{P}}}$ is the base change of $\mathfrak{G}$ to the complete local ring at a point of $C$. When ${{\mathbb{P}}}$ is smooth with connected reductive generic fiber we proved the existence of Rapoport–Zink spaces for local ${{\mathbb{P}}}$-shtukas. In the present article we use these spaces to (partly) uniformize the moduli stacks of global $\mathfrak{G}$-shtukas for smooth $\mathfrak{G}$ with connected fibers and reductive generic fiber. This is our main result. It has applications to the analog of the Langlands–Rapoport conjecture for our moduli stacks.

scholarly journals Bivariance, Grothendieck duality and Hochschild homology, II: The fundamental class of a flat scheme-map

2014 ◽  
Vol 257 ◽  
pp. 365-461 ◽  
Author(s):  
Leovigildo Alonso Tarrío ◽  
Ana Jeremías López ◽  
Joseph Lipman
Keyword(s):  
Hochschild Homology ◽  
Fundamental Class

scholarly journals Higher orbital integrals, Shalika germs, and the Hochschild homology of Hecke algebras

2001 ◽  
Vol 26 (3) ◽  
pp. 129-160 ◽  
Author(s):  
Victor Nistor
Keyword(s):  
Hecke Algebras ◽  
Hochschild Homology ◽  
Cyclic Homology ◽  
Detailed Calculation ◽  
Orbital Integrals ◽  
Compactly Supported ◽  
Shalika Germs ◽  
Definition Of ◽  
Compactly Supported Functions

We give a detailed calculation of the Hochschild and cyclic homology of the algebra𝒞c∞(G)of locally constant, compactly supported functions on a reductivep-adic groupG. We use these calculations to extend to arbitrary elements the definition of the higher orbital integrals introduced by Blanc and Brylinski (1992) for regular semi-simple elements. Then we extend to higher orbital integrals some results of Shalika (1972). We also investigate the effect of the “induction morphism” on Hochschild homology.

scholarly journals INVARIANT HYPERSURFACES

2020 ◽  
pp. 1-27
Author(s):  
Jason Bell ◽  
Rahim Moosa ◽  
Adam Topaz
Keyword(s):  
Rational Function ◽  
Finite Type ◽  
Characteristic Zero ◽  
Positive Characteristic ◽  
Inverse Image ◽  
The Other ◽  
Partial Result ◽  
Rational Maps ◽  
Special Cases ◽  
The One

The following theorem, which includes as very special cases results of Jouanolou and Hrushovski on algebraic $D$ -varieties on the one hand, and of Cantat on rational dynamics on the other, is established: Working over a field of characteristic zero, suppose $\unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}:Z\rightarrow X$ are dominant rational maps from an (possibly nonreduced) irreducible scheme $Z$ of finite type to an algebraic variety $X$ , with the property that there are infinitely many hypersurfaces on  $X$ whose scheme-theoretic inverse images under $\unicode[STIX]{x1D719}_{1}$ and $\unicode[STIX]{x1D719}_{2}$ agree. Then there is a nonconstant rational function $g$ on $X$ such that $g\unicode[STIX]{x1D719}_{1}=g\unicode[STIX]{x1D719}_{2}$ . In the case where $Z$ is also reduced, the scheme-theoretic inverse image can be replaced by the proper transform. A partial result is obtained in positive characteristic. Applications include an extension of the Jouanolou–Hrushovski theorem to generalised algebraic ${\mathcal{D}}$ -varieties and of Cantat’s theorem to self-correspondences.

scholarly journals Matrix factorizations via Koszul duality

2014 ◽  
Vol 150 (9) ◽  
pp. 1549-1578 ◽  
Author(s):  
Junwu Tu
Keyword(s):  
Duality Theory ◽  
Hochschild Homology ◽  
Matrix Factorizations ◽  
Koszul Duality ◽  
Technical Tool ◽  
Graded Algebras ◽  
Differential Graded Algebras ◽  
Homological Perturbation ◽  
The Relationship

AbstractIn this paper we prove a version of curved Koszul duality for $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{Z}/2\mathbb{Z}$-graded curved coalgebras and their cobar differential graded algebras. A curved version of the homological perturbation lemma is also obtained as a useful technical tool for studying curved (co)algebras and precomplexes. The results of Koszul duality can be applied to study the category of matrix factorizations $\mathsf{MF}(R,W)$. We show how Dyckerhoff’s generating results fit into the framework of curved Koszul duality theory. This enables us to clarify the relationship between the Borel–Moore Hochschild homology of curved (co)algebras and the ordinary Hochschild homology of the category $\mathsf{MF}(R,W)$. Similar results are also obtained in the orbifold case and in the graded case.

scholarly journals Phantom depth and flat base change

2005 ◽  
Vol 134 (02) ◽  
pp. 313-321
Author(s):  
Neil M. Epstein
Keyword(s):  
Base Change ◽  
Flat Base
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