Traces of differential forms and Hochschild homology.

*(English)*Zbl 0675.13019
Lecture Notes in Mathematics, 1368. Berlin etc.: Springer-Verlag. 111 p. DM 25.00 (1989).

The paper starts with studying the Hochschild homology \(H_{\bullet}:\quad A\to ADG\) from the category A of commutative algebras to the category ADG of anti-commutative DG-algebras. This functor is extended to the category Topal of topological algebras. Some relations between differential algebras and Hochschild homology modules are established. Let \(\Omega^{\bullet}:\quad A\to ADG\) be the functor “universal differential algebra”. The functorial homomorphism \(\theta^{\bullet}:\quad \Omega^{\bullet}\to H_{\bullet}\) can be extended to the full subcategory Topal\(\sim\) of all objects of Topal for which the differential algebra \^R[d\^R] is complete. If R is a \({\mathbb{Q}}\)-algebra or if \(\Omega^ 1_{R/k}\) is finite then a left inverse \(\delta^{\bullet}_{(R/k,\tau)}\) of \(\theta^{\bullet}_{(R/k,\tau)}\) is constructed explicitly. - A trace for the Hochschild homology of topological algebras is constructed and the following theorem is proved:

Let X be the class of all pairs \(((S,\tau ')/(R,\tau),\kappa)\) with the following properties:

i. \(\kappa\) is a noetherian ring and (R,\(\tau)\)//\(\kappa\) is a topological algebra.

ii. S/R is an algebra and \(\tau '\) is the linear topology on S induced by \(\tau\).

iii. The \(\tau '\)-adic completion (Ŝ,\({\hat \tau}{}')\) of \((S,\tau ')\) is a finite and free module over the \(\tau\)-adic completion (\^R,\({\hat \tau}\)) of (R,\(\tau)\).

iv. The topology \({\hat \tau}{}'\) on S is the linear topology induced by \({\hat \tau}\).

Then there exists a system of canonical morphisms \(tr_{(S/R,r)}:\quad H_{\bullet}(S/\kappa,\tau ')\to H_{\bullet}(S/\kappa,\tau),((S,\tau ')/(R,\tau),\kappa)\) in X, satisfying some axioms formulated in terms of Hochschild homology.

These results are used to construct the pretrace of Angeniol. Then the trace maps for differential forms are constructed for some classes of forms and it is shown that they satisfy the trace axioms. It is not known whether the trace maps for finite locally complete intersections can be defined in a similar way via Hochschild homology. Nevertheless these traces are closely related to the traces of Hochschild homology. It is shown that the various definitions of trace coincide on the intersection of the classes where they are defined. Lipman’s definition of residues is extended to topological Hochschild homology and universal finite differential forms, and this theory is used to derive under suitable reducedness assumptions a slightly weaker version of the trace formula. This formula is used to deduce the residue axiom (R4) “transitivity” for Lipman’s residue symbol.

Let X be the class of all pairs \(((S,\tau ')/(R,\tau),\kappa)\) with the following properties:

i. \(\kappa\) is a noetherian ring and (R,\(\tau)\)//\(\kappa\) is a topological algebra.

ii. S/R is an algebra and \(\tau '\) is the linear topology on S induced by \(\tau\).

iii. The \(\tau '\)-adic completion (Ŝ,\({\hat \tau}{}')\) of \((S,\tau ')\) is a finite and free module over the \(\tau\)-adic completion (\^R,\({\hat \tau}\)) of (R,\(\tau)\).

iv. The topology \({\hat \tau}{}'\) on S is the linear topology induced by \({\hat \tau}\).

Then there exists a system of canonical morphisms \(tr_{(S/R,r)}:\quad H_{\bullet}(S/\kappa,\tau ')\to H_{\bullet}(S/\kappa,\tau),((S,\tau ')/(R,\tau),\kappa)\) in X, satisfying some axioms formulated in terms of Hochschild homology.

These results are used to construct the pretrace of Angeniol. Then the trace maps for differential forms are constructed for some classes of forms and it is shown that they satisfy the trace axioms. It is not known whether the trace maps for finite locally complete intersections can be defined in a similar way via Hochschild homology. Nevertheless these traces are closely related to the traces of Hochschild homology. It is shown that the various definitions of trace coincide on the intersection of the classes where they are defined. Lipman’s definition of residues is extended to topological Hochschild homology and universal finite differential forms, and this theory is used to derive under suitable reducedness assumptions a slightly weaker version of the trace formula. This formula is used to deduce the residue axiom (R4) “transitivity” for Lipman’s residue symbol.

Reviewer: E.V.Pankrat’ev

##### MSC:

13N05 | Modules of differentials |

12H05 | Differential algebra |

12G05 | Galois cohomology |

13D03 | (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) |