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In atiyah's book on commutative algebra page 106 it says that elements in graded modules can be written uniquely as a sum of homogeneous elements. More precisely:
If [tex]A = \oplus^{\infty}_{n=0} A_n[/tex] is a graded ring, and [tex]M = \oplus^{\infty}_{n=0} M_n[/tex] is a graded A-module, then an element [tex]y \in M[/tex] can be written uniquely as a finite sum [tex]\Sigma y_n[/tex], where [tex]y_n \in M_n[/tex].
I can't see how that can be right. Consider any ring A. Then [tex]A = \oplus^{\infty}_{n=0} A[/tex] is a graded ring, since [tex]AA \subseteq A[/tex]. Let M be any non-zero A-module. Then [tex]M = \oplus^{\infty}_{n=0} M[/tex] is a graded A-module, since [tex]AM \subseteq M[/tex]. However, it is clear that if we pick a non-zero [tex]x \in M[/tex], then [tex]x =2x+(-x)[/tex], where [tex]2x \in M_1 =M[/tex], and [tex]-x \in M_2 = M[/tex]. Maybe x should be on some "normal form" in the graded module, but how do we decide what the normal form is?
Have I got it wrong or is this an error in the book? I think the necessary and sufficient condition is that the M_n's are disjoint.
If [tex]A = \oplus^{\infty}_{n=0} A_n[/tex] is a graded ring, and [tex]M = \oplus^{\infty}_{n=0} M_n[/tex] is a graded A-module, then an element [tex]y \in M[/tex] can be written uniquely as a finite sum [tex]\Sigma y_n[/tex], where [tex]y_n \in M_n[/tex].
I can't see how that can be right. Consider any ring A. Then [tex]A = \oplus^{\infty}_{n=0} A[/tex] is a graded ring, since [tex]AA \subseteq A[/tex]. Let M be any non-zero A-module. Then [tex]M = \oplus^{\infty}_{n=0} M[/tex] is a graded A-module, since [tex]AM \subseteq M[/tex]. However, it is clear that if we pick a non-zero [tex]x \in M[/tex], then [tex]x =2x+(-x)[/tex], where [tex]2x \in M_1 =M[/tex], and [tex]-x \in M_2 = M[/tex]. Maybe x should be on some "normal form" in the graded module, but how do we decide what the normal form is?
Have I got it wrong or is this an error in the book? I think the necessary and sufficient condition is that the M_n's are disjoint.
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