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Your sequence can't be exact as both det(E) and det(G) have rank 1. Det transforms exact sequences into tensor products see this questionthis question.

Your sequence can't be exact as both det(E) and det(G) have rank 1. Det transforms exact sequences into tensor products see this question.

Your sequence can't be exact as both det(E) and det(G) have rank 1. Det transforms exact sequences into tensor products see this question.

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Your sequence can't be exact as both det(E) and det(G) have rank 1. Det transforms exact sequences into tensor products see this question.