The tangent complex and Hochschild cohomology of $\mathcal {E}_n$-rings

  title={The tangent complex and Hochschild cohomology of \$\mathcal \{E\}\_n\$-rings},
  author={John Francis},
  journal={Compositio Mathematica},
  pages={430 - 480}
  • J. Francis
  • Published 1 April 2011
  • Mathematics
  • Compositio Mathematica
Abstract In this work, we study the deformation theory of ${\mathcal {E}}_n$-rings and the ${\mathcal {E}}_n$ analogue of the tangent complex, or topological André–Quillen cohomology. We prove a generalization of a conjecture of Kontsevich, that there is a fiber sequence $A[n-1] \rightarrow T_A\rightarrow {\mathrm {HH}}^*_{{\mathcal {E}}_{n}}\!(A)[n]$, relating the ${\mathcal {E}}_n$-tangent complex and ${\mathcal {E}}_n$-Hochschild cohomology of an ${\mathcal {E}}_n$-ring $A$. We give two… 

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