Remark 63.8.4 (0F7E)—The Stacks project

Remark 63.8.4. Consider a commutative diagram

\[ \xymatrix{ X'' \ar[r]_{k'} \ar[d]_{f''} & X' \ar[r]_ k \ar[d]_{f'} & X \ar[d]^ f \\ Y'' \ar[r]^{l'} \ar[d]_{g''} & Y' \ar[r]^ l \ar[d]_{g'} & Y \ar[d]^ g \\ Z'' \ar[r]^{m'} & Z' \ar[r]^ m & Z } \]

of schemes whose vertical arrows are proper and whose horizontal arrows are separated and locally quasi-finite. Let us label the squares of the diagram $A$, $B$, $C$, $D$ as follows

\[ \begin{matrix} A & B \\ C & D \end{matrix} \]

Then the maps of Lemma 63.8.1 for the squares are (where we use $Rf_* = f_*$, etc)

\[ \begin{matrix} \gamma _ A : l'_! \circ f''_* \to f'_* \circ k'_! & \gamma _ B : l_! \circ f'_* \to f_* \circ k_! \\ \gamma _ C : m'_! \circ g''_* \to g'_* \circ l'_! & \gamma _ D : m_! \circ g'_* \to g_* \circ l_! \end{matrix} \]

For the $2 \times 1$ and $1 \times 2$ rectangles we have four further maps

\[ \begin{matrix} \gamma _{A + B} : (l \circ l')_! \circ f''_* \to f_* \circ (k \circ k')_* \\ \gamma _{C + D} : (m \circ m')_! \circ g''_* \to g_* \circ (l \circ l')_! \\ \gamma _{A + C} : m'_! \circ (g'' \circ f'')_* \to (g' \circ f')_* \circ k'_! \\ \gamma _{B + D} : m_! \circ (g' \circ f')_* \to (g \circ f)_* \circ k_! \end{matrix} \]

By Lemma 63.8.3 we have

\[ \gamma _{A + B} = \gamma _ B \circ \gamma _ A, \quad \gamma _{C + D} = \gamma _ D \circ \gamma _ C \]

and by Lemma 63.8.2 we have

\[ \gamma _{A + C} = \gamma _ A \circ \gamma _ C, \quad \gamma _{B + D} = \gamma _ B \circ \gamma _ D \]

Here it would be more correct to write $\gamma _{A + B} = (\gamma _ B \star \text{id}_{k'_!}) \circ (\text{id}_{l_!} \star \gamma _ A)$ with notation as in Categories, Section 4.28 and similarly for the others. Having said all of this we find (a priori) two transformations

\[ m_! \circ m'_! \circ g''_* \circ f''_* \longrightarrow g_* \circ f_* \circ k_! \circ k'_! \]

namely

\[ \gamma _ B \circ \gamma _ D \circ \gamma _ A \circ \gamma _ C = \gamma _{B + D} \circ \gamma _{A + C} \]

and

\[ \gamma _ B \circ \gamma _ A \circ \gamma _ D \circ \gamma _ C = \gamma _{A + B} \circ \gamma _{C + D} \]

The point of this remark is to point out that these transformations are equal. Namely, to see this it suffices to show that

\[ \xymatrix{ m_! \circ g'_* \circ l'_! \circ f''_* \ar[r]_{\gamma _ D} \ar[d]_{\gamma _ A} & g_* \circ l_! \circ l'_! \circ f''_* \ar[d]^{\gamma _ A} \\ m_! \circ g'_* \circ f'_* \circ k'_! \ar[r]^{\gamma _ D} & g_* \circ l_! \circ f'_* \circ k'_! } \]

commutes. This is true because the squares $A$ and $D$ meet in only one point, more precisely by Categories, Lemma 4.28.2 or more simply the discussion preceding Categories, Definition 4.28.1.

The Stacks project

Comments (0)

Post a comment