Corollary

a) Suppose that the field K has characteristic 0. The homomorphisms $\theta$ and $\theta_K$ are injective. Two semisimple A-modules of finite dimension over K are isomorphic if and only if their characters are equal.

b) Suppose that the field K is perfect of nonzero characteristic $p$. Then the homomorphism $\theta_K$ is injective, and the kernel of $\theta$ is $pR_K(A)$. Let E be a semisimple A-module of finite dimension over K. The linear form Tr$_E$ is zero if and only if there exists an A-module F such that E is isomorphic to $F^p$.

Let $\mathscr{S}_K$ be the set of classes of simple A-modules of finite dimension over K. The elements [S], where S runs through $\mathscr{S}_K$, form a basis of the $\mathbf{Z}$-module $R_K$(A), so the elements $1\otimes [S]$ form a basis of the K-vector space $K\otimes_{\mathbf{Z}}R_K(A)$ (VIII, p. 195).

Suppose that the field K is perfect. By Proposition 6 and the corollary of VIII, p. 383, the elements $\theta ([S]) =\theta_K(1\otimes [S]) =$ Tr$_S$ of $A^*$ are linearly independent over K. It follows that the homomorphism $\theta_K$ is injective and that the kernel of the homomorphism $\theta$ consists of the elements $\sum_{S\in\mathscr{S}_K}n_S[S]$ of $R_K(A)$ such that $n_S\cdot 1_K= 0$ for every $S\in \mathscr{S}_K$. The kernel of $\theta$ is therefore equal to $pR_K$(A), where $p$ is the characteristic of K. In particular, if K has characteristic 0, then the homomorphism $\theta$ is injective.

The last assertion of a) follows from the corollary of VIII, p. 190.

Suppose that the assumptions of b) are satisfied, and let E be a semisimple A-module of finite dimension over K. If there exists an A-module F such that E is the direct sum of $p$ submodules isomorphic to F, then the module F is semisimple and finite-dimensional over K and we have Tr$_E=p$ Tr$_F= 0$. Conversely, suppose Tr$_E= 0$. We have $[E]\in pR_K$(A), so the multiplicity of every simple module $S\in \mathscr{S}_K$ in E is a multiple of $p$ (VIII, p. 190, Proposition 7); we can write it as $p n_S$ with $n_S\in \mathbf{N}$. The family $(n_S)$ has finite support. Set $F =\oplus_{S\in\mathscr{S}_K}S^{n_S}$. We then have $[E] = [F^p]$, so that E and $F^p$ are isomorphic (VIII, p. 190, Corollary).

Let E be an A-module of finite dimension over K. Let $a\in A$. Denote by $\chi_E(a; T)$ the determinant of the endomorphism $1_E+ Ta_E$ of the K[T]-module $E[T] = K[T]\otimes_KE$ (III, §8, No. 11, p. 541). We have the relation

(10) $\chi_E(a; T)\equiv 1 +$ Tr$_E(a)T$ (mod $T^2K[T]$)

(loc. cit., formula (49)). Since this polynomial has constant term equal to 1, it is an invertible formal power series (IV, §4, No. 4, p. 30). Moreover, if

is an exact sequence of A-modules, all finite-dimensional over K, then we have $\chi_E(a; T) =\chi_{E'}(a; T)\chi_{E''}(a,T)$ (III, §8, No. 7, p. 534, formula (31)). By the definition of the Grothendieck group $R_K(A)$ (VIII, p. 194) and its universal property (VIII, p. 186, Proposition 4), there exists a unique homomorphism $\chi_a$ from the group $R_K(A)$ to the multiplicative group 1 + TK[[T]] such that $\chi_a([E]) =\chi_E(a; T)$ for every A-module E of finite dimension over K. We deduce from formula (10) the relation

(11) $\chi_a(x)\equiv 1 +\theta (x)(a)T$ (mod $T^2K[[T]]$)

for $x\in R_K(A)$ and $a\in A$.

If E and $E'$ are two A-modules of finite dimension over K with isomorphic semisimplifications, then we have $\chi_E(a; T) =\chi_{E'}(a; T)$.