Algebraic Analysis notes Lecture 11 (4 Feb 2019)

Notes for lecture 10 Last time: for an abelian category A, C(A) is the category of complexes in A. Say $latex f, g \in \mathrm{Hom}_{C(A)} (X, Y)$ are homotopic, f~g, if there are maps $latex s^i : X^i \to Y^i$ such that $latex f^i -g^i = d_Y s + sd_X$. Definition The homotopy category K(A) … Continue reading Algebraic Analysis notes Lecture 11 (4 Feb 2019)

Algebraic Analysis notes Lecture 10 (1 Feb 2019)

Notes for lecture 9 Last time: $latex \Gamma : Sh(X; k) \to k-mod$ global sections functor is left exact. We'll leave sheaves for now to look at derived categories. What do sheaves have to do with cohomology? Poincare Lemma: Let M be a manifold. Consider the following complex of sheaves: where d is the de … Continue reading Algebraic Analysis notes Lecture 10 (1 Feb 2019)

Algebraic Analysis notes Lecture 8 (28 Jan 2019)

Notes for lecture 7 Last time: we defined additive categories, and kernels for morphisms in additive categories. Definition: The cokernel of a morphism $latex \phi$ (if it exists) is the universal object $latex cok \phi$ with the dual universal property: Definition: an additive category C is abelian if (A4) for any $latex \phi \colon X … Continue reading Algebraic Analysis notes Lecture 8 (28 Jan 2019)

Algebraic Analysis notes Lecture 5 (16 Jan 2019)

Notes for Lecture 4 Last time: Hard Lefschetz gives an orthogonal decomposition $latex H^k(X, \mathbb C) = \bigoplus_{2r \leq k}^\bot \eta^r H_{prim}^{k-2r} (X; \mathbb C)$ with respect to a Hermitian form on $latex H^k (X, \mathbb C)$ defined using the Poincare pairing. On the other hand, Hodge theory gives a decomposition $latex H^k (X; \mathbb … Continue reading Algebraic Analysis notes Lecture 5 (16 Jan 2019)