Abstract - This work establishes a comparison between functions on derived loop spaces (To en and Vezzosi, Chern character, loop spaces and derived algebraic geometry, in Algebraic topology: the Abel symposium 2007, Abel Symposia, vol. 4, eds N. Baas, E. M. Friedlander, B. Jahren and P. A. Ostvaer (Springer, 2009), ISBN:978-3-642-01199-3) and de Rham theory. If A is a smooth commutative k-algebra and k has characteristic 0, we show that two objects, S1 \otimes A and \epsilon(A), determine one another, functorially in A. The object S1 \otimes A is the S1-equivariant simplicial k-algebra obtained by tensoring A by the simplicial group S1 := BZ, while the object \epsilon(A) is the de Rham algebra of A, endowed with the de Rham differential, and viewed as a mixed-dg-algebra (see the main text). We define an equivalence \phi between the homotopy theory of simplicial commutative S1-equivariant k-algebras and the homotopy theory of mixed-dg-algebras, and we show the existence of a functorial equivalence \phi(S1 \otimes A) \simeq \epsilon(A). We deduce from this the comparison mentioned above, identifying the S1-equivariant functions on the derived loop space LX of a smooth k-scheme X with the algebraic de Rham cohomology of X/k. As corollaries, we obtain functorial and multiplicative versions of decomposition theorems for Hochschild homology (in the spirit of Hochschild-Kostant-Rosenberg theorem) for arbitrary semi-separated k-schemes. By construction, these decompositions are moreover compatible with the S1-action on the Hochschild complex, on one hand, and with the de Rham dierential, on the other hand.

Algebres simpliciales S^1-equivariantes, theorie de de Rham et theoremes HKR multiplicatifs / G. Vezzosi; B. Toen. - In: COMPOSITIO MATHEMATICA. - ISSN 0010-437X. - STAMPA. - Volume 147 / Issue 06:(2011), pp. 1979-2000. [10.1112/S0010437X11005501]

Algebres simpliciales S^1-equivariantes, theorie de de Rham et theoremes HKR multiplicatifs

VEZZOSI, GABRIELE;
2011

Abstract

Abstract - This work establishes a comparison between functions on derived loop spaces (To en and Vezzosi, Chern character, loop spaces and derived algebraic geometry, in Algebraic topology: the Abel symposium 2007, Abel Symposia, vol. 4, eds N. Baas, E. M. Friedlander, B. Jahren and P. A. Ostvaer (Springer, 2009), ISBN:978-3-642-01199-3) and de Rham theory. If A is a smooth commutative k-algebra and k has characteristic 0, we show that two objects, S1 \otimes A and \epsilon(A), determine one another, functorially in A. The object S1 \otimes A is the S1-equivariant simplicial k-algebra obtained by tensoring A by the simplicial group S1 := BZ, while the object \epsilon(A) is the de Rham algebra of A, endowed with the de Rham differential, and viewed as a mixed-dg-algebra (see the main text). We define an equivalence \phi between the homotopy theory of simplicial commutative S1-equivariant k-algebras and the homotopy theory of mixed-dg-algebras, and we show the existence of a functorial equivalence \phi(S1 \otimes A) \simeq \epsilon(A). We deduce from this the comparison mentioned above, identifying the S1-equivariant functions on the derived loop space LX of a smooth k-scheme X with the algebraic de Rham cohomology of X/k. As corollaries, we obtain functorial and multiplicative versions of decomposition theorems for Hochschild homology (in the spirit of Hochschild-Kostant-Rosenberg theorem) for arbitrary semi-separated k-schemes. By construction, these decompositions are moreover compatible with the S1-action on the Hochschild complex, on one hand, and with the de Rham dierential, on the other hand.
2011
Volume 147 / Issue 06
1979
2000
G. Vezzosi; B. Toen
File in questo prodotto:
File Dimensione Formato  
AbstractS^1algebras.pdf

accesso aperto

Tipologia: Altro
Licenza: Open Access
Dimensione 127.11 kB
Formato Adobe PDF
127.11 kB Adobe PDF
S^1algebras-in-print.pdf

accesso aperto

Tipologia: Versione finale referata (Postprint, Accepted manuscript)
Licenza: Open Access
Dimensione 803.83 kB
Formato Adobe PDF
803.83 kB Adobe PDF

I documenti in FLORE sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificatore per citare o creare un link a questa risorsa: https://hdl.handle.net/2158/394522
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 31
  • ???jsp.display-item.citation.isi??? 29
social impact