Adams operations in Hochschild and cyclic homology of de Rham algebra and free loop spaces

K-Theory ◽  
1991 ◽  
Vol 4 (3) ◽  
pp. 269-287 ◽  
Author(s):  
D. Burghelea ◽  
Z. Fiedorowicz ◽  
W. Gajda
Keyword(s):  
Cyclic Homology ◽  
Loop Spaces ◽  
De Rham ◽  
Adams Operations

Loop spaces, cyclic homology and the Chern character

1989 ◽  
pp. 95-108
Author(s):  
E. Getzler ◽  
J.D.S. Jones ◽  
S.B. Petrack
Keyword(s):  
Chern Character ◽  
Cyclic Homology ◽  
Loop Spaces

scholarly journals Periodic cyclic homology and derived de Rham cohomology

2019 ◽  
Vol 4 (3) ◽  
pp. 505-519
Author(s):  
Benjamin Antieau
Keyword(s):  
Cyclic Homology ◽  
De Rham Cohomology ◽  
De Rham

Cyclic homology and de Rham cohomology

1991 ◽  
Vol 208 (1) ◽  
pp. 489-502 ◽  
Author(s):  
Gerald Lodder
Keyword(s):  
Cyclic Homology ◽  
De Rham Cohomology ◽  
De Rham

The chiral de rham complex and positivity of the equivariant signatures of some loop spaces

2004 ◽  
Vol 113 (3) ◽  
pp. 359-370 ◽  
Author(s):  
V. Gorbounov ◽  
F. Malikov
Keyword(s):  
Loop Spaces ◽  
De Rham Complex ◽  
Rham Complex ◽  
De Rham

scholarly journals Derived de Rham Complex and Cyclic Homology.

1996 ◽  
Vol 79 ◽  
pp. 176
Author(s):  
Javier Majadas
Keyword(s):  
Cyclic Homology ◽  
De Rham Complex ◽  
Rham Complex ◽  
De Rham

scholarly journals Algèbres simplicialesS1-équivariantes, théorie de de Rham et théorèmes HKR multiplicatifs

2011 ◽  
Vol 147 (6) ◽  
pp. 1979-2000 ◽  
Author(s):  
Bertrand Toën ◽  
Gabriele Vezzosi
Keyword(s):  
Algebraic Topology ◽  
Homotopy Theory ◽  
Chern Character ◽  
Loop Spaces ◽  
Derived Algebraic Geometry ◽  
De Rham ◽  
Dg Algebra ◽  
Algebraic De Rham Cohomology ◽  
Decomposition Theorems ◽  
Equivariant Functions

AbstractThis work establishes a comparison between functions on derived loop spaces (Toën and Vezzosi,Chern character, loop spaces and derived algebraic geometry, inAlgebraic topology: the Abel symposium 2007, Abel Symposia, vol. 4, eds N. Baas, E. M. Friedlander, B. Jahren and P. A. Østvær (Springer, 2009), ISBN:978-3-642-01199-3) and de Rham theory. IfAis a smooth commutativek-algebra andkhas characteristic 0, we show that two objects,S1⊗Aand ϵ(A), determine one another, functorially inA. The objectS1⊗Ais theS1-equivariant simplicialk-algebra obtained by tensoringAby the simplicial groupS1:=Bℤ, while the object ϵ(A) is the de Rham algebra ofA, endowed with the de Rham differential, and viewed as aϵ-dg-algebra(see the main text). We define an equivalence φ between the homotopy theory of simplicial commutativeS1-equivariantk-algebras and the homotopy theory of ϵ-dg-algebras, and we show the existence of a functorial equivalence ϕ(S1⊗A)∼ϵ(A) . We deduce from this the comparison mentioned above, identifying theS1-equivariant functions on the derived loop spaceLXof a smoothk-schemeXwith the algebraic de Rham cohomology of X/k. As corollaries, we obtainfunctorialandmultiplicativeversions of decomposition theorems for Hochschild homology (in the spirit of Hochschild–Kostant–Rosenberg) for arbitrary semi-separatedk-schemes. By construction, these decompositions aremoreovercompatible with theS1-action on the Hochschild complex, on one hand, and with the de Rham differential, on the other hand.

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