Presheaves that are not sheaves

(Gluing fails)

For a fixed abelian group \(A\), the constant presheaf with \(P(U)=A\) for \(U\ne\emptyset\), \(P(\emptyset)=0\), and identity restrictions between nonempty open sets fails the gluing axiom. On a disconnected \(U = U_1 \sqcup U_2\) the sections \(a \in A\) on \(U_1\) and \(b \neq a\) on \(U_2\) agree on the empty overlap, and no element of \(P(U)=A\) restricts to both. Its sheafification is the sheaf of locally constant \(A\)-valued functions.

(Bounded functions)

Let \(C\) be the sheaf of continuous real functions on \(\RR\) and \(B\subseteq C\) the presheaf of bounded continuous functions. \(B\) satisfies the identity axiom, being a subpresheaf of a sheaf, and fails gluing: the restrictions of \(f(x)=x\) to the intervals \((n-1,n+1)\) are bounded and agree on overlaps, while \(f\) is unbounded. The presheaf quotient \(U\mapsto C(U)/B(U)\) fails the identity axiom: the class of \(f(x)=x\) on \(\RR\) is nonzero, and its restriction to each interval \((n-1,n+1)\) is zero.

(Tensor presheaf)

On \(X = \PP^1_k\) take \(\mcf = \OO(1)\) and \(\mcg = \OO(-1)\). The presheaf \(P(U) = \mcf(U) \tensor_{\OO_X(U)} \mcg(U)\) has \(P(X) = k^2 \tensor_k 0 = 0\). On each standard chart \(U_i = D_+(x_i)\) both sheaves are trivial, so \(P(U_i) \cong \OO(U_i)\), and the sections \(x_0 \tensor x_0^{-1}\) on \(U_0\) and \(x_1 \tensor x_1^{-1}\) on \(U_1\) agree on \(U_0 \cap U_1\), since both equal the unit section there. They do not glue to a section of \(P(X) = 0\), so \(P\) is not a sheaf; its sheafification is \(\OO(1) \tensor \OO(-1) \cong \OO_X\), with \(H^0 = k\).