Exercise 15, § 2

Algebra, X, p. 176

Let $E$ be a spectral sequence of A-modules, approaching a graded A-module G (Exercise 14). a) Suppose that $E_2^{p,q} = 0$ for $p < 0$ . Now prove that the mappings $\alpha_r$ induce for every $q$ injective A-homomorphisms:

E_{\infty}^{0,q} \to \cdots \to E_3^{0,q} \to E_2^{0,q}

whence, by composition with $(\beta^{0q})^{-1}$ and with the mapping of passing to the quotient, an A-homomorphism $\gamma^q : G^q \to E_2^{0,q}$ . b) Suppose that $E_2^{pq} = 0$ for $q < 0$ . Now prove that the mappings $\alpha_r^{-1}$ induce surjective homomorphisms:

E_2^{p,0} \to E_3^{p,0} \to \cdots \to E_{\infty}^{p,0}

whence by composition with $\beta^{p,0}$ an A-homomorphism $\delta^p : E_2^{p,0} \to G^p$ . c) Suppose that the hypotheses of a) and b) are satisfied, that is to say $E_2^{pq} = 0$ if $p < 0$ or $q < 0$ . Now prove that the sequence

0 \to E_2^{1,0} \xrightarrow{\delta^1} G^1 \xrightarrow{\gamma^1} E_1^{0,1} \xrightarrow{d_2} E_2^{2,0} \xrightarrow{\delta^2} G^2

is exact. d) Suppose that $E_2^{pq} = 0$ if $p > 0$ or $q > 0$ ; define an exact sequence analogous to that of c). e) Suppose that there exist integers $r, d$ , with $d \geqslant r \geqslant 2$ , such that $E_r^{pq} = 0$ if $p \neq 0, d$ . Define an exact sequence:

\cdots \to E_r^{d,n-d} \to G^n \to E_r^{0,n} \to E_r^{d,n+1-d} \to G^{n+1} \to \cdots .

If analogously $E_r^{pq} = 0$ for $q \neq 0, d$ , now prove that there exists an exact sequence:

\cdots \to E_r^{n,0} \to G^n \to E_r^{n-d,d} \to E_r^{n+1,0} \to \cdots .

f) Suppose that $E_2^{pq} = 0$ if $q \neq 0$ (resp. $p \neq 0$ ). Now prove that the homomorphism

\delta^n : E_2^{n,0} \to G^n \text{ (resp. } \gamma^n : G^n \to E_2^{0,n})

is bijective for every $n$ .

Solution

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