Lemma 23.12.3 (0GZ6)—The Stacks project

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.

Proof. For $d = 1$, we have a short exact sequence of complexes

\[ 0 \to A \to A\langle T \rangle \to A \cdot T \to 0 \]

and the result (1) follows easily from this. For $d = 2$ we view $A\langle T \rangle $ as a filtered chain complex with subcomplexes

\[ F^ pA\langle T \rangle = \bigoplus \nolimits _{i \leq p} A \cdot T^{[i]} \]

Applying the spectral sequence of Homology, Section 12.24 (translated into chain complexes) we obtain (2). $\square$

The Stacks project