Dennis trace map

Builds on Hochschild homology Group homology Plus construction Ring Kähler differentials

Definition

Let $R$ be a ring, let $GL(R) = \bigcup_k GL_k(R)$ , and let $n \ge 1$ . Write $K_n(R) = \pi_n(BGL(R)^+)$ , using the plus construction. The Dennis trace map $K_n(R) \to HH_n(R)$ is the composite of four maps:

the Hurewicz map $\pi_n(BGL(R)^+) \to H_n(BGL(R)^+; \Z) = H_n(GL(R); \Z)$ ;
the map $H_n(G; \Z) \to HH_n(\Z[G])$ for $G = GL(R)$ , induced on chains by $(g_1, \ldots, g_n) \mapsto (g_1 \cdots g_n)^{-1} \otimes g_1 \otimes \cdots \otimes g_n$ ;
the map $HH_n(\Z[GL(R)]) \to HH_n(M(R))$ induced by the ring map $\Z[GL(R)] \to M(R) = \bigcup_k M_k(R)$ ;
the trace isomorphism $HH_n(M(R)) \cong HH_n(R)$ of Morita invariance.

Here $H_n(G; \Z)$ is group homology and $HH_n$ is Hochschild homology over $\Z$ .

Remarks

Example: in degree $1$ , a unit $u$ of $R$ gives $u \in GL_1(R)$ , and its image in $HH_1(R)$ is the class of $u^{-1} \otimes u$ . For $R$ commutative, $HH_1(R) \cong \Omega^1_{R/\Z}$ by $a \otimes b \mapsto a \, db$ , so the Dennis trace sends $u$ to $u^{-1} du$ , a logarithmic Kähler differential. Non-example of injectivity: for $R = \Q$ , one has $\Omega^1_{\Q/\Z} = 0$ , so the Dennis trace on $K_1(\Q) = \Q^\times$ is zero although $K_1(\Q)$ is not. The Dennis trace factors through negative cyclic homology $HC^-_n(R)$ , and its topological refinement is the map $K(R) \to THH(R)$ .

Reading path

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Group homology concept
Hochschild homology concept
Kähler differentials concept
Plus construction concept
Ring concept
Dennis trace map definition

Tag	`EW1ZC30V49`
URI	`m2://EW1ZC30V49`
Kind	definition
Writing	W0 stub
MSC	19D55, 16E40
Defines	Dennis trace map
Uses	Hochschild homology, Group homology, Plus construction, Ring, Kähler differentials
Changed	3 November 2026

Dennis trace map

Definition

Remarks

Reading path

Further reading

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