Algebraic geometry notes by Akhil Mathew

By Akhil Mathew

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More generally, if X is a locally ringed space an injective object in Mod(OX ) is flabby. (2) If 0 → F → F → F → 0 is exact and F, F are flabby, so is F . Proof. (1) Let U ⊂ X. Consider the constant sheaf Z on U . We consider the extension by zero, j! (Z) on X; we denote this by ZU . There is an injection of this sheaf into the constant sheaf Z on X. Now it is easy to see that to give a morphism Z → G for any sheaf G is the same as giving a global section of G. Similarly, to give a morphism ZU → G is the same as giving a section of G over U .

If we show that H is flabby, then by induction we will get the result. 13. (1) An injective sheaf is flabby. More generally, if X is a locally ringed space an injective object in Mod(OX ) is flabby. (2) If 0 → F → F → F → 0 is exact and F, F are flabby, so is F . Proof. (1) Let U ⊂ X. Consider the constant sheaf Z on U . We consider the extension by zero, j! (Z) on X; we denote this by ZU . There is an injection of this sheaf into the constant sheaf Z on X. Now it is easy to see that to give a morphism Z → G for any sheaf G is the same as giving a global section of G.

Let F, G be coherent subsheaves of a coherent sheaf H. Then the sum4 F + G and the intersection F ∩ G are both coherent. , the presheaf associated to U → F(U ) + G(U ) ⊂ H(U ), or the internal sum in the category. 32 3. COHERENCE AND FINITENESS CONDITIONS Proof. Indeed, F + G is the image of the coherent sheaf F ⊕ G in H, so is coherent by the previous corollary. Similarly, F ∩ G is the kernel of the map H → H/F ⊕ H/G, and this is also coherent. 8. The tensor product of two coherent sheaves F, G is coherent.

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