scholarly journals Derived invariants of irregular varieties and Hochschild homology

2014 ◽  
Vol 8 (3) ◽  
pp. 513-542
Author(s):  
Luigi Lombardi
Keyword(s):  
Hochschild Homology ◽  
Irregular Varieties

scholarly journals LINEAR SYSTEMS ON IRREGULAR VARIETIES

2019 ◽  
Vol 19 (6) ◽  
pp. 2087-2125 ◽  
Author(s):  
Miguel Ángel Barja ◽  
Rita Pardini ◽  
Lidia Stoppino
Keyword(s):  
Continuous Function ◽  
Line Bundle ◽  
Linear Systems ◽  
Projective Variety ◽  
Geometrical Properties ◽  
Complex Projective Variety ◽  
Wide Range ◽  
Image Position ◽  
Irregular Varieties ◽  
Complex Projective

Let $X$ be a normal complex projective variety, $T\subseteq X$ a subvariety of dimension $m$ (possibly $T=X$) and $a:X\rightarrow A$ a morphism to an abelian variety such that $\text{Pic}^{0}(A)$ injects into $\text{Pic}^{0}(T)$; let $L$ be a line bundle on $X$ and $\unicode[STIX]{x1D6FC}\in \text{Pic}^{0}(A)$ a general element.We introduce two new ingredients for the study of linear systems on $X$. First of all, we show the existence of a factorization of the map $a$, called the eventual map of $L$ on $T$, which controls the behavior of the linear systems $|L\otimes \unicode[STIX]{x1D6FC}|_{|T}$, asymptotically with respect to the pullbacks to the connected étale covers $X^{(d)}\rightarrow X$ induced by the $d$-th multiplication map of $A$.Second, we define the so-called continuous rank function$x\mapsto h_{a}^{0}(X_{|T},L+xM)$, where $M$ is the pullback of an ample divisor of $A$. This function extends to a continuous function of $x\in \mathbb{R}$, which is differentiable except possibly at countably many points; when $X=T$ we compute the left derivative explicitly.As an application, we give quick short proofs of a wide range of new Clifford–Severi inequalities, i.e., geographical bounds of the form $$\begin{eqnarray}\displaystyle \text{vol}_{X|T}(L)\geqslant C(m)h_{a}^{0}(X_{|T},L), & & \displaystyle \nonumber\end{eqnarray}$$ where $C(m)={\mathcal{O}}(m!)$ depends on several geometrical properties of $X$, $L$ or $a$.

scholarly journals DETECTING TORSION IN SKEIN MODULES USING HOCHSCHILD HOMOLOGY

2006 ◽  
Vol 15 (02) ◽  
pp. 259-277 ◽  
Author(s):  
MICHAEL McLENDON
Keyword(s):  
Tensor Product ◽  
Spectral Sequence ◽  
Boundary Surface ◽  
Hochschild Homology ◽  
Heegaard Splitting ◽  
Common Boundary ◽  
Skein Modules ◽  
Skein Module ◽  
Hochschild Complex

Given a Heegaard splitting of a closed 3-manifold, the skein modules of the two handlebodies are modules over the skein algebra of their common boundary surface. The zeroth Hochschild homology of the skein algebra of a surface with coefficients in the tensor product of the skein modules of two handlebodies is interpreted as the skein module of the 3-manifold obtained by gluing the two handlebodies together along this surface. A spectral sequence associated to the Hochschild complex is constructed and conditions are given for the existence of algebraic torsion in the completion of the skein module of this 3-manifold.

scholarly journals Hochschild homology and global dimension

2009 ◽  
Vol 41 (3) ◽  
pp. 473-482 ◽  
Author(s):  
Petter Andreas Bergh ◽  
Dag Madsen
Keyword(s):  
Global Dimension ◽  
Hochschild Homology

The excision theorems in Hochschild and cyclic homologies

2014 ◽  
Vol 144 (2) ◽  
pp. 305-317
Author(s):  
Guram Donadze ◽  
Manuel Ladra
Keyword(s):  
Hochschild Homology ◽  
Excision Property ◽  
Crossed Modules ◽  
Simplicial Algebras

We study the excision property for Hochschild and cyclic homologies in the category of simplicial algebras. We extend Wodzicki's notion of H-unital algebras to simplicial algebras and then show that a simplicial algebra I* satisfies excision in Hochschild and cyclic homologies if and only if it is H-unital. We use this result in the category of crossed modules of algebras and provide an answer to the question posed in the recent paper by Donadze et al. We also give (based on work by Guccione and Guccione) the excision theorem in Hochschild homology with coefficients.

scholarly journals Properties of the Secondary Hochschild Homology

2018 ◽  
Vol 25 (02) ◽  
pp. 225-242
Author(s):  
Jacob Laubacher
Keyword(s):  
Exact Sequence ◽  
Morita Equivalence ◽  
Hochschild Homology ◽  
Low Dimension

In this paper we study properties of the secondary Hochschild homology of the triple (A, B, ε) with coefficients in M. We establish a type of Morita equivalence between two triples and show that H•((A, B, ε); M) is invariant under this equivalence. We also prove the existence of an exact sequence which connects the usual and the secondary Hochschild homologies in low dimension, allowing one to perform easy computations. The functoriality of H•((A, B, ε); M) is also discussed.

Finite generation of Hochschild homology algebras

2000 ◽  
Vol 140 (1) ◽  
pp. 143-170 ◽  
Author(s):  
Luchezar L. Avramov ◽  
Srikanth Iyengar
Keyword(s):  
Hochschild Homology ◽  
Finite Generation

scholarly journals Residues and homology for pseudodifferential operators on foliations

2004 ◽  
Vol 94 (1) ◽  
pp. 75 ◽  
Author(s):  
M.-T. Benameur ◽  
V. Nistor
Keyword(s):  
Diophantine Approximation ◽  
Pseudodifferential Operators ◽  
Hochschild Homology ◽  
Foliated Manifold ◽  
Homology Groups ◽  
General Properties ◽  
Poisson Homology ◽  
Foliated Manifolds

We study the Hochschild homology groups of the algebra of complete symbols on a foliated manifold $(M,F)$. The first step is to relate these groups to the Poisson homology of $(M,F)$ and of other related foliated manifolds. We then establish several general properties of the Poisson homology groups of foliated manifolds. As an example, we completely determine these Hochschild homology groups for the algebra of complete symbols on the irrational slope foliation of a torus (under some diophantine approximation assumptions). We also use our calculations to determine all residue traces on algebras of pseudodifferential operators along the leaves of a foliation.

scholarly journals Hochschild homology relative to a family of groups

2008 ◽  
Vol 8 (2) ◽  
pp. 693-728
Author(s):  
Andrew Nicas ◽  
David Rosenthal
Keyword(s):  
Hochschild Homology
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