Let \((X, \mco_X)\) be a ringed space, let \(\mci\) be a sheaf of ideals with \(\mci^2=0\), and let \(X_0\) be the ringed space \((X, \mco_X/\mci)\). Show that there is an exact sequence of sheaves of abelian groups on \(X\), \[ 0 \to \mci \to \mco_X^* \to \mco_{X_0}^* \to 0, \] where \(\mco_X^*\) (respectively, \(\mco_{X_0}^*\)) denotes the sheaf of (multiplicative) groups of units in the sheaf of rings \(\mco_X\) (respectively, \(\mco_{X_0}\)), the map \(\mci \to \mco_X^*\) is defined by \(a \mapsto 1+a\), and \(\mci\) has its usual (additive) group structure. Conclude there is an exact sequence of abelian groups \[ \cdots \to H^1(X, \mci) \to \Pic X \to \Pic X_0 \to H^2(X, \mci) \to \cdots. \]
Solution
All structure rings are commutative, and \(X\) and \(X_0\) have the same underlying topological space. Write \(\rho:\OO_X\to\OO_{X_0}\) for the quotient morphism and \(\rho^\times\) for its restriction to the sheaves of units. The zero symbols in the displayed exact sequences denote trivial abelian groups; the neutral element in a unit group is \(1\).
1. The map \(e:\mci\to\OO_X^\times\), \(a\mapsto1+a\), is an injective homomorphism of abelian sheaves with image \(\ker\rho^\times\).
For sections \(a,b\) of \(\mci\) on the same open set, their product is zero because \(\mci^2=0\). Thus \[ (1+a)(1+b)=1+a+b,\qquad(1+a)(1-a)=1. \] The first equality proves the additive-to-multiplicative homomorphism property, and the second proves that \(1+a\) is a unit. The map commutes with restriction and is injective, since \(1+a=1\) implies \(a=0\).
Its image reduces to \(1\) in the quotient sheaf. Conversely, a section \(u\) of \(\OO_X^\times\) whose image is the identity satisfies \(u-1\in\ker\rho=\mci\). It is therefore \(e(u-1)\). This proves the kernel identification as sheaves.
2. The morphism \(\rho^\times\) is surjective as a morphism of sheaves.
The stalk of the sheaf of units at \(x\) is the unit group of the ring \(\OO_{X,x}\). Indeed, a unit germ and its inverse have representatives whose product is \(1\) after restricting to a sufficiently small common neighborhood; those representatives are inverse unit sections there. Hence it suffices to prove that units lift across the stalk quotient \(B\to B/J\), where \(B=\OO_{X,x}\) and \(J=\mci_x\) has \(J^2=0\).
Let \(\bar b\in(B/J)^\times\) and choose lifts \(b,c\in B\) of \(\bar b\) and its inverse. Then \(bc=1+a\) for \(a\in J\). Since \((1+a)^{-1}=1-a\), the element \(c(1-a)\) is an inverse for \(b\). Thus \(b\) is a unit lifting \(\bar b\). This proves surjectivity at every stalk and therefore surjectivity of sheaves. Together with step 1 it proves the required short exact sequence.
3. Its cohomology sequence gives the required Picard exact sequence, whose middle map is \([L]\mapsto[L\otimes_{\OO_X}\OO_{X_0}]\).
Apply the long exact sequence of derived global sections to the short exact sequence of abelian sheaves in steps <1>1–<1>2. Its beginning is \[ \begin{aligned} 0&\to H^0(X,\mci)\to\Gamma(X,\OO_X^\times) \to\Gamma(X,\OO_{X_0}^\times)\\ &\to H^1(X,\mci)\to H^1(X,\OO_X^\times) \to H^1(X,\OO_{X_0}^\times) \to H^2(X,\mci)\to\cdots. \end{aligned} \] The Picard and unit-cohomology identification applies to both ringed spaces and replaces the two degree-one unit groups by \(\Pic X\) and \(\Pic X_0\). To identify the map between them, trivialize \(L\) on an open cover and write its transition units as \(g_{ij}\). Tensoring with \(\OO_{X_0}\) replaces these by their images \(\rho^\times(g_{ij})\), which are exactly the images used by the cohomology map. Thus the middle map is the stated reduction of invertible sheaves, and the exact segment is \[ \cdots\to H^1(X,\mci)\to\Pic X\to\Pic X_0 \to H^2(X,\mci)\to\cdots. \] In particular, an invertible sheaf on \(X_0\) lifts precisely when its connecting class in \(H^2(X,\mci)\) is zero. No assertion that the map on global units is surjective was needed.
4. Q.E.D.
Steps <1>1–<1>2 prove exactness of the sheaf sequence, and step 3 gives the cohomological and Picard conclusions.