Let \(X\) be a topological space, let \(\mcf\) be a sheaf of abelian groups on \(X\), and let \(Y_1,Y_2\) be two closed subsets of \(X\). Then there is a long exact sequence of cohomology with supports \[ \begin{aligned} \cdots & \to H_{Y_1 \intersect Y_2}^i(X, \mcf) \to H_{Y_1}^i(X, \mcf) \oplus H_{Y_2}^i(X, \mcf) \to H_{Y_1 \union Y_2}^i(X, \mcf) \\ & \to H_{Y_1 \intersect Y_2}^{i+1}(X, \mcf) \to \cdots \end{aligned} \]
Solution
For a closed subset \(Y\), write \(\Gamma_Y(A)=\Gamma_Y(X,A)\) for the sections of an abelian sheaf \(A\) with support in \(Y\). These are precisely the sections whose restriction to \(X\setminus Y\) is zero, and their right derived functors are \(H_Y^i(X,-)\), as in P-AGH323SUPPORTS. Put \(U_i=X\setminus Y_i\) for \(i=1,2\).
1. For every abelian sheaf \(A\), the sequence\[ 0\longrightarrow\Gamma_{Y_1\cap Y_2}(A) \xrightarrow{a\mapsto(a,-a)}\Gamma_{Y_1}(A)\oplus\Gamma_{Y_2}(A) \xrightarrow{(b,c)\mapsto b+c}\Gamma_{Y_1\cup Y_2}(A) \]is exact at the first two nonzero terms.
A section supported in the intersection is supported in each \(Y_i\), so the first map is defined and injective. A sum of sections supported respectively in \(Y_1,Y_2\) vanishes outside their union, so the second map is defined. The composite is zero. Conversely, if \(b+c=0\), then \(b=-c\) is supported in both \(Y_1\) and \(Y_2\), and thus in their intersection. The pair \((b,c)\) is the image of this section under the first map. This proves exactness and specifies the signs.
2. If \(A\) is flasque, the last map in step 1 is surjective.
Let \(s\in\Gamma_{Y_1\cup Y_2}(A)\). Its restriction to \(U_1\cap U_2=X\setminus(Y_1\cup Y_2)\) is zero. Consequently the section \(0\) on \(U_1\) and the section \(s|_{U_2}\) agree on their overlap. They glue to a section \(t\) on \[ U_1\cup U_2=X\setminus(Y_1\cap Y_2). \] Since \(A\) is flasque, extend \(t\) to a global section \(b\) of \(A\). It is zero on \(U_1\), so \(b\) is supported in \(Y_1\). Moreover, \(c=s-b\) is zero on \(U_2\), so \(c\) is supported in \(Y_2\). Thus \(s=b+c\) is in the image of the sum map, proving surjectivity. The choice of this decomposition need not be unique; the kernel describing that ambiguity is exactly the image of the first map in step 1.
3. Applying steps <1>1–<1>2 to an injective resolution yields the stated natural long exact sequence.
Choose an injective resolution \(\mcf\to I^\bullet\) in the category of abelian sheaves on \(X\). Each \(I^j\) is flasque [@Har10a, Lemma III.2.4]. Steps <1>1–<1>2 therefore give a short exact sequence of complexes \[ 0\to\Gamma_{Y_1\cap Y_2}(I^\bullet) \to\Gamma_{Y_1}(I^\bullet)\oplus\Gamma_{Y_2}(I^\bullet) \to\Gamma_{Y_1\cup Y_2}(I^\bullet)\to0. \] The difference and sum maps commute with the differentials because they are natural in the sheaf. By the definition of supported cohomology, the cohomology of each complex is the corresponding \(H_Y^i(X,\mcf)\). The long exact sequence of cohomology of a short exact sequence of complexes consequently gives the sequence in the statement. It begins in degree zero with the injective map of step 1 and continues through all nonnegative degrees. Its first two maps in each degree are induced by the signed inclusions and sum map; the third is the connecting homomorphism. Functoriality of the short exact sequence and comparison of injective resolutions make the resulting long exact sequence natural in \(\mcf\) [@Har10a, Chapter III, §1].
4. Q.E.D.
Steps <1>1–<1>2 establish exactness for flasque resolution terms, and step 3 derives the required Mayer–Vietoris sequence with its natural maps.