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This is a little exercise problem from Peeva's book on graded syzygies.

Let $\phi:N\to T$ be a homomorphism of graded $R$-modules. If $f=f_1+\cdots+f_n\in N$ and $f_i$ are its homogeneous components, then $\phi(f_i)$ are homogeneous components of $\phi(f)$ (in the context $R$ is a quotient of a polynomial ring by a graded ideal.)

It shouldn't be hard, but I'm having trouble figuring out how to use the grading of $T$.

D. Huang
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1 Answers1

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By definition, a morphism of graded modules maps $N_k$ into $T_k$ for each $k$, which is to say that it maps homogeneous elements to homogeneous elements of the same degree. In particular, it preserves the homogeneous components.

Bruno Joyal
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    Hmm, I wonder why it's "by definition". We know what module homomorphisms are and how graded modules are defined. Shouldn't this "definition" be something we can proof from those knowledge? – D. Huang Sep 10 '13 at 02:42
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    @D.Huang: Often mathematicians say "homomorphism of graded modules" when they mean for the map to be a "graded homomorphism" and not just a "homomorphism" whose domain and codomain happen to be graded. – Jim Sep 10 '13 at 02:46
  • @Jim I see. I should've read the definition more carefully. This problem is obvious then. Thanks! – D. Huang Sep 10 '13 at 02:52