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Let $A = \bigoplus_{k \in \mathbb{Z}} A_k$ be a not necessarily commutative $\mathbb{Z}$-graded unital algebra over a field $\mathbb{K}$, and assume that it is strongly gradedstrongly graded: $$ A_kA_l = A_{k+l}. $$ In general can it happen that the multiplication does not give an isomorphism $$ A_k \otimes_{A_0} A_l \simeq A_{k+l}? $$ The map will be surjective since we are assuming the gradind to be strong, but will it be injective?

Let $A = \bigoplus_{k \in \mathbb{Z}} A_k$ be a not necessarily commutative $\mathbb{Z}$-graded unital algebra over a field $\mathbb{K}$, and assume that it is strongly graded: $$ A_kA_l = A_{k+l}. $$ In general can it happen that the multiplication does not give an isomorphism $$ A_k \otimes_{A_0} A_l \simeq A_{k+l}? $$ The map will be surjective since we are assuming the gradind to be strong, but will it be injective?

Let $A = \bigoplus_{k \in \mathbb{Z}} A_k$ be a not necessarily commutative $\mathbb{Z}$-graded unital algebra over a field $\mathbb{K}$, and assume that it is strongly graded: $$ A_kA_l = A_{k+l}. $$ In general can it happen that the multiplication does not give an isomorphism $$ A_k \otimes_{A_0} A_l \simeq A_{k+l}? $$ The map will be surjective since we are assuming the gradind to be strong, but will it be injective?

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$\mathbb{Z}$-graded algebras and tensor products

Let $A = \bigoplus_{k \in \mathbb{Z}} A_k$ be a not necessarily commutative $\mathbb{Z}$-graded unital algebra over a field $\mathbb{K}$, and assume that it is strongly graded: $$ A_kA_l = A_{k+l}. $$ In general can it happen that the multiplication does not give an isomorphism $$ A_k \otimes_{A_0} A_l \simeq A_{k+l}? $$ The map will be surjective since we are assuming the gradind to be strong, but will it be injective?