Lemma 23.12.3. Let $(A, \text{d}, \gamma )$, $d \geq 1$, $f \in A_{d - 1}$, and $A\langle T \rangle $ be as in Lemma 23.6.8.

If $d = 1$, then there is a long exact sequences

\[ \ldots \to H_0(A) \xrightarrow {f} H_0(A) \to H_0(A\langle T \rangle ) \to 0 \]For $d = 2$ there is a bounded spectral sequence $(E_1)_{i, j} = H_{j - i}(A) \cdot T^{[i]}$ converging to $H_{i + j}(A\langle T \rangle )$. The differential $(d_1)_{i, j} : H_{j - i}(A) \cdot T^{[i]} \to H_{j - i + 1}(A) \cdot T^{[i - 1]}$ sends $\xi \cdot T^{[i]}$ to the class of $f \xi \cdot T^{[i - 1]}$.

Add more here for other degrees as needed.

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