ScalingStacks

Proposition 38 For any graded $A$-module $M$ we have direct sum decompositions of $A\otimes M,$ considered as a graded $A$-module

	$\displaystyle A\otimes M$	$\displaystyle=$	$\displaystyle\Delta_{M}M\oplus\iota_{M}M$		(193)
	$\displaystyle A\otimes M$	$\displaystyle=$	$\displaystyle\iota_{M}M\oplus(\Delta_{M}-\iota_{M}m_{M}\Delta_{M})M$		(194)
	$\displaystyle A\otimes M$	$\displaystyle=$	$\displaystyle\Delta_{M}M\oplus(\iota_{M}-c\iota_{M}m_{M}\Delta_{M})M$		(195)