Abstract
In this chapter, we introduce the prerequisites needed later on (in Chapter 6) for the sheaf-theoretic description of intersection homology groups.
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Notes
- 1.
A functor \(F:{\mathcal A} \to {\mathcal B}\) of abelian categories is left exact if a short exact sequence 0âââAâ²ââAâââAâ³âââ0 in \({\mathcal A}\) is sent by F to an exact sequence 0âââF(Aâ²)âââF(A)âââF(Aâ³). F is right exact if F(Aâ²)âââF(A)âââF(Aâ³)âââ0 is exact. F is exact if F preserves exact sequences, i.e., 0âââF(Aâ²)âââF(A)âââF(Aâ³)âââ0 is exact.
- 2.
A Stein manifold is a complex manifold X so that every coherent \({\mathcal O}_X\)-sheaf \({\mathcal F}\) is acyclic. (Recall that \({\mathcal F}\) is coherent if it has locally a finite presentation \({\mathcal O}_X^n \to {\mathcal O}_X^m \to {\mathcal F} \to 0\), e.g., \({\mathcal F}\) is the locally free sheaf of sections of a holomorphic vector bundle.) For example, a closed complex submanifold of some \({\mathbb C}^N\) (e.g., a nonsingular affine complex variety) is a Stein manifold.
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Maxim, L.G. (2019). Brief Introduction to Sheaf Theory. In: Intersection Homology & Perverse Sheaves. Graduate Texts in Mathematics, vol 281. Springer, Cham. https://doi.org/10.1007/978-3-030-27644-7_4
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