Lemma 53.15.6 (0C1N)—The Stacks project

Lemma 53.15.6. Let $k$ be an algebraically closed field. If $f : X' \to X$ is the squishing of a tangent vector $\vartheta $ as in Example 53.15.3, then there is an exact sequence

\[ (k, +) \to \mathop{\mathrm{Pic}}\nolimits (X) \to \mathop{\mathrm{Pic}}\nolimits (X') \to 0 \]

and the first map is injective if $X'$ is proper and reduced.

Proof. The map $\mathop{\mathrm{Pic}}\nolimits (X) \to \mathop{\mathrm{Pic}}\nolimits (X')$ is surjective by Varieties, Lemma 33.38.7. Using the short exact sequence

\[ 0 \to \mathcal{O}_ X^* \to f_*\mathcal{O}_{X'}^* \xrightarrow {\vartheta } x_*k \to 0 \]

of Example 53.15.3 we obtain

\[ H^0(X', \mathcal{O}_{X'}^*) \xrightarrow {\vartheta } k \to H^1(X, \mathcal{O}_ X^*) \to H^1(X, f_*\mathcal{O}_{X'}^*) \]

We have $H^1(X, f_*\mathcal{O}_{X'}^*) \subset H^1(X', \mathcal{O}_{X'}^*)$ (for example by the Leray spectral sequence, see Cohomology, Lemma 20.13.4). Hence the kernel of $\mathop{\mathrm{Pic}}\nolimits (X) \to \mathop{\mathrm{Pic}}\nolimits (X')$ is the cokernel of the map $\vartheta : H^0(X', \mathcal{O}_{X'}^*) \to k$. Because $k$ is algebraically closed any regular function on a reduced connected proper scheme over $k$ comes from an element of $k$, see Varieties, Lemma 33.9.3. Thus the final statement of the lemma. $\square$

Comments (2)

Comment #3005 by James Waldron on November 21, 2017 at 19:25

In the statement of Lemma 49.15.6, I think that $f$ should be a map $f:X'\to X$ and not $f:X\to X'$ . f

Comment #3127 by Johan on February 01, 2018 at 13:46

Thanks, fixed here.

Comments (2)

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