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Let $A$ be a noetherian, graded ring and $M$ be a projective, graded $A$-module. Denote by $M_{\ge k}:=\oplus_{d \ge k} M_k$$M_{\ge d}:=\oplus_{\ell \ge d} M_\ell$ the sub-module of $M$. Is $M_{\ge k}$$M_{\ge d}$ again a projective $A$-module?
Let $A$ be a noetherian, graded ring and $M$ be a projective, graded $A$-module. Denote by $M_{\ge k}:=\oplus_{d \ge k} M_k$ the sub-module of $M$. Is $M_{\ge k}$ again a projective $A$-module?
Let $A$ be a noetherian, graded ring and $M$ be a projective, graded $A$-module. Denote by $M_{\ge d}:=\oplus_{\ell \ge d} M_\ell$ the sub-module of $M$. Is $M_{\ge d}$ again a projective $A$-module?
Let $A$ be a noetherian, graded ring and $M$ be a projective, graded $A$-module. Denote by $M_{\ge k}:=\oplus_{d \ge k} M_k$ the sub-module of $M$. Is $M_{\ge k}$ again a projective module$A$-module?
Let $A$ be a noetherian, graded ring and $M$ be a projective, graded $A$-module. Denote by $M_{\ge k}:=\oplus_{d \ge k} M_k$ the sub-module of $M$. Is $M_{\ge k}$ again a projective module?
Let $A$ be a noetherian, graded ring and $M$ be a projective, graded $A$-module. Denote by $M_{\ge k}:=\oplus_{d \ge k} M_k$ the sub-module of $M$. Is $M_{\ge k}$ again a projective $A$-module?
Let $A$ be a noetherian, graded ring and $M$ be a projective, graded $A$-module. Denote by $M_{\ge k}:=\oplus_{d \ge k} M_k$ the sub-module of $M$. Is $M_{\ge k}$ again a projective module?
