Example

In particular the above may be applied when E is one of the graded algebras T(M), S(M) or $\bigwedge(M)$ for an A-module M and P an A-module (with the trivial graduation). To find explicitly the bimodule structures thus obtained, note that the elements of degree $-n$ of Homgr_A(T(M), P) (resp. Homgr_A(S(M), P), resp. Homgr_A($\bigwedge(M)$, P)) are identified with the n-linear mappings (resp. symmetric n-linear mappings, resp. alternating n-linear mappings) of $M^n$ into P. It suffices to express the products

(resp. $f \sqcup (x_1 x_2 \ldots x_p)$), resp. $f \sqcup (x_1 \wedge x_2 \wedge \cdots \wedge x_p)$) for every finite sequence $(x_i)_{1 \leq i \leq p}$ of elements of M and the analogues for the left inner product. It follows immediately from the definitions that

if $p > n$ and that, for $p \leq n$, $f \sqcup (x_1 \otimes x_2 \otimes \cdots \otimes x_p)$ (resp.

is the (n - p)-linear mapping defined by

For $p > n$, there are also in Homgr_A(S(M), P) (resp. Homgr_A($\bigwedge(M)$, P)) formulae (44) with $x_1 \otimes x_2 \otimes \cdots \otimes x_p$ replaced by $x_1 x_2 \ldots x_p$ (resp. $x_1 \wedge x_2 \wedge \cdots \wedge x_p$). For $p \leq n$, the same substitutions in (45) define the symmetric (n - p)-linear mappings $f \sqcup (x_1 x_2 \ldots x_p)$ and $(x_1 x_2 \ldots x_p) \sqcup f$ (resp. the alternating (n - p)-linear mappings

When $n = p$, the above products are equal to the constant function on $M$ equal to $f(x_1, \ldots, x_p)$.

If $u : M \to N$ is an $A$-module homomorphism, $T(u) : T(M) \to T(N)$ is a graded $A$-algebra homomorphism, then it follows from what we have seen above that $(T(u))^\sim$ is a $T(M)$-homomorphism of the $(T(N), T(N))$-bimodule $\mathrm{Homgr}_A(T(N), P)$ into the $(T(M), T(M))$-bimodule $\mathrm{Homgr}_A(T(M), P)$, relative to the ring homomorphism $T(u)$. There are analogous results for $(S(u))^\sim$ and $(\wedge(u))^\sim$.