Questions tagged [homological-algebra]

Homological algebra studies homology and cohomology groups in a general algebraic setting, that of chains of vector spaces or modules with composable maps which compose to zero. These groups furnish useful invariants of the original chains.

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Three questions about Properties of Ext functor.

I have three questions about the Ext functor's properties. (i) $Ext(H \oplus H',G) = Ext(H,G) \oplus Ext(H',G)$ (ii) $Ext(H,G) = 0$ if $H$ is free (iii) $Ext(\mathbb{Z_n}, G) = G/nG$ There is a ...
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Question on a proof about spectral sequences from exact couples

I am going through Proposition 2.9 in User's guide in spectral sequences (2nd edition) by McCleary. This is a proof on defining spectral sequences using the language of exact couples. Towards the end ...
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Given filtered complex $K=\oplus_n K^n$ with gradation $n$ and filtration $\{K_p\}$ of $K$, $K_p\cap K^n$ is filtration of $K^n$?

Let $K$ be a filtered graded complex s.t. $K=\oplus_{n\in Z}K^n$ and $\{K_p\}$ filtration of $K$. Define $K_p^n=K_p\cap K^n$. $\textbf{Q:}$ Why $\{K_p^n\}_{p\in Z}$ forms filtration of $K^n$? The ...
Let $0\rightarrow M'\rightarrow M \rightarrow M''\rightarrow 0$ be a short-exact sequence of modules (over a ring $R$). By a projective resolution of this sequence, we mean (according to Jacobson ...