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In Robert Ash's notes a chain map is defined by the next relation: $f_{n-1}\circ d_n = d_n\circ f_n $; while in Charles Weibel's book on page 2, it's defined as follows: $u_{n-1}\circ d_n = d_{n-1} \circ u_n$, where $u$ and $f$ are the chain maps.

So which terminology is the right one? It seems to me in both cycles go to cycles and boundaries go to boundaries but in different index; in Ash's notation we have $z\in Z_n(C) \Rightarrow f_n(z)\in Z_n(D)$ and $b\in B_n(C) \Rightarrow f_n(b) \in B_n(D)$. But in Weibel's: $z \in Z_n(C) \Rightarrow u_n(z) \in Z_{n-1}(D)$ and $b \in B_n(C) \Rightarrow u_n(b)\in B_{n-1}(D)$.

Is there any difference between the two notations in the fact that $H_n(f)$ or $H_n(u)$ are functors from $Ch(mod-R)$ to $mod-R$. (It's the $n$-th homology functor).

http://www.math.uiuc.edu/~r-ash/Algebra/Supplement.pdf

http://www.math.unam.mx/javier/weibel.pdf

MathematicalPhysicist
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1 Answers1

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I think Weibel has a typo. Using his notation, he has supposed that $u_n: C_n \to D_n$ and $d_{n-1} : D_{n-1} \to D_{n-2}$. Then the composition $d_{n-1} \circ u_n$ is not defined.

posilon
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