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If $kG$ is not semisimple, it is a non-semisimple Frobenius algebra and has infinite global dimension always in that case, see for example the books of Lam on rings and modules where the global dimension of a Frobenius algebra is determined. So yes, finite global dimension is equivalent to being semisimple (global dimension zero). For a proof that nonsemisimple Frobenius algebras have infinite global dimension one can use that the syzygy functor $\Omega^1$ is a stable equivalence and thus $\Omega^i(M)$ is always non-zero for all $i>0$ if $\Omega^1(M)$ is not projective.

If $kG$ is not semisimple, it is a non-semisimple Frobenius algebra and has infinite global dimension always in that case, see for example the books of Lam on rings and modules where the global dimension of a Frobenius algebra is determined. So yes, finite global dimension is equivalent to being semisimple (global dimension zero). For a proof that nonsemisimple Frobenius algebras have infinite global dimension one can use that the syzygy functor $\Omega^1$ is a stable equivalence and thus $\Omega^i(M)$ is always non-zero for all $i>0$ if $\Omega^1(M)$ is not projective (this shows in fact the stronger statement that the finitistic dimension is zero, which implies that the global dimension is infinite when the algebra is not semisimple).

If $kG$ is not semisimple, it is a non-semisimple Frobenius algebra and has infinite global dimension always in that case, see for example the books of Lam on rings and modules where the global dimension of $kG$ is determined. So yes, finite global dimension is equivalent to being semisimple (global dimension zero). For a proof that nonsemisimple Frobenius algebras have infinite global dimension one can use that the syzygy functor $\Omega^1$ is a stable equivalence and thus $\Omega^i(M)$ is always non-zero for all $i>0$ if $\Omega^1(M)$ is not projective.

If $kG$ is not semisimple, it is a non-semisimple Frobenius algebra and has infinite global dimension always in that case, see for example the books of Lam on rings and modules where the global dimension of a Frobenius algebra is determined. So yes, finite global dimension is equivalent to being semisimple (global dimension zero). For a proof that nonsemisimple Frobenius algebras have infinite global dimension one can use that the syzygy functor $\Omega^1$ is a stable equivalence and thus $\Omega^i(M)$ is always non-zero for all $i>0$ if $\Omega^1(M)$ is not projective.

If $kG$ is not semisimple, it is a non-semisimple Frobenius algebra and has infinite global dimension always in that case, see for example the books of Lam on rings and modules where the global dimension of $kG$ is determined. So yes, finite global dimension is equivalent to being semisimple (global dimension zero).

If $kG$ is not semisimple, it is a non-semisimple Frobenius algebra and has infinite global dimension always in that case, see for example the books of Lam on rings and modules where the global dimension of $kG$ is determined. So yes, finite global dimension is equivalent to being semisimple (global dimension zero). For a proof that nonsemisimple Frobenius algebras have infinite global dimension one can use that the syzygy functor $\Omega^1$ is a stable equivalence and thus $\Omega^i(M)$ is always non-zero for all $i>0$ if $\Omega^1(M)$ is not projective.