Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf sheaves simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 25th 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI am trying to write up an elementary exposition for how the Hochschild chain complex for a commutative associate algebra is the normalized chains/Moore complex of the simplicial algebra that one gets by [[tensoring]] the algebra $A$ with the simplicial set $\Delta[1]/\partial \Delta[1]$: $$ C_\bullet(A,A) = N_\bullet( (\Delta[1]/\partial \Delta[1]) \cdot A ) \,. $$ I would like to get feedback on whether or not my exposition is in fact understandable in an elementary way. The section that contains this material is the section <a href="http://ncatlab.org/nlab/show/Hochschild+cohomology#SimplicialCircleAlgebra">The simplicial circle algebra</a> at the entry [[Hochschild cohomology]]. Just this one section. It's not long. It describes first the simplicial set $\Delta[1]/\partial \Delta[1]$, then discusses how the coproduct in $CAlg_k$ is given by the tensor product over $k$, and deduces from that what the simplicial algebra $(\Delta[1]/\partial \Delta[1])$ is like. After taking the normalized chains of that, the result is Pirashvili's construction of a chain complex from a simplicial set and a commutative algebra. I just think it is important to amplify that this construction of Pirashvili's is a categorical tensoring=copower operation. Because that connects the construction to general abstract constructions. That's what the beginning of the above entry is about. But for the moment I would just like to make the elementary exposition of the tensoring operation itself pretty and understandable.

   I am trying to write up an elementary exposition for how the Hochschild chain complex for a commutative associate algebra is the normalized chains/Moore complex of the simplicial algebra that one gets by tensoring the algebra $A$ with the simplicial set $\Delta[1]/\partial \Delta[1]$:

   $$C_\bullet(A,A) = N_\bullet( (\Delta[1]/\partial \Delta[1]) \cdot A ) \,.$$

   I would like to get feedback on whether or not my exposition is in fact understandable in an elementary way.

   The section that contains this material is the section

   The simplicial circle algebra

   at the entry Hochschild cohomology. Just this one section. It’s not long.

   It describes first the simplicial set $\Delta[1]/\partial \Delta[1]$, then discusses how the coproduct in $CAlg_k$ is given by the tensor product over $k$, and deduces from that what the simplicial algebra $(\Delta[1]/\partial \Delta[1])$ is like.

   After taking the normalized chains of that, the result is Pirashvili’s construction of a chain complex from a simplicial set and a commutative algebra. I just think it is important to amplify that this construction of Pirashvili’s is a categorical tensoring=copower operation. Because that connects the construction to general abstract constructions. That’s what the beginning of the above entry is about. But for the moment I would just like to make the elementary exposition of the tensoring operation itself pretty and understandable.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeDec 2nd 2010
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have further expanded the section <a href="http://nlab.mathforge.org/nlab/show/Hochschild+cohomology#IdWithK%C3%A4hlerForOrdAlgebra">Identification with Kähler differential forms</a> and have now some discussion in <a href="http://nlab.mathforge.org/nlab/show/Hochschild+cohomology#SimplicialCircleActionForOrdAlgebra">The simplicial circle action</a>. This gives so far at least a hint of how to prove that under the identification $(Spec A)^{S^1} \simeq Spec(C_\bullet(A,A))$ and the HKR-theorem, the canonical circle action induced the de Rham differential.

   I have further expanded the section

   Identification with Kähler differential forms and have now some discussion in The simplicial circle action. This gives so far at least a hint of how to prove that under the identification $(Spec A)^{S^1} \simeq Spec(C_\bullet(A,A))$ and the HKR-theorem, the canonical circle action induced the de Rham differential.