Category Archives: Algebraic Geometry

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)ContinueContinue reading “Algebraic Analysis notes Lecture 11 (4 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 deContinueContinue reading “Algebraic Analysis notes Lecture 10 (1 Feb 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 XContinueContinue reading “Algebraic Analysis notes Lecture 8 (28 Jan 2019)”

I wanted to wait until a while after the due date to post the questions from the first homework to avoid potential perception of foul play. It was due last Friday, and I already turned it in, so I think I should be good now. I have my own answers, which I’ll post in commentsContinueContinue reading “Algebraic Analysis Homework 1”

Notes for lecture 6 Sheafification Last time: for a space X and a ring k, a presheaf is a functor $latex Op(X)^{op} \to k-mod$, and a sheaf is a presheaf F such that for any open U in X and any open cover $latex (U_\alpha)_{\alpha \in I}$ of U, the sequence is exact. If weContinueContinue reading “Algebraic Analysis notes Lecture 7 (25 Jan 2019)”

Notes for Lecture 5 Last time, for a given topological space X and a ring k, we define a presheaf to be a functor $latex F \colon Op(X)^{op} \to k-mod$. Definition: If F is a presheaf on X, and x is a point in X, then the stalk of F at x, is the k-moduleContinueContinue reading “Algebraic Analysis notes Lecture 6 (18 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; \mathbbContinueContinue reading “Algebraic Analysis notes Lecture 5 (16 Jan 2019)”

Notes for Lecture 3 Last time, we proved the Lefschetz hyperplane theorem. Lefschetz Hyperplane Theorem: $latex Let X^n \subset \mathbb P^N$ be a projective variety and $latex H \subset \mathbb P^N$ a hyperplane such that $latex U= X \setminus Y$ (where $latex Y = X \cap H$) is smooth. Then $latex H^k (X, Y; \mathbbContinueContinue reading “Algebraic Analysis notes Lecture 4 (14 Jan 2019)”

Notes for Lecture 2 Theorem: If $latex X \subset \mathbb C^N$ is a smooth affine complex algebraic variety of complex dimension n, then X has the homotopy type of a CW complex of real dimension n. Note that this theorem is saying that certain spaces of real dimension 2n are homotopic to spaces with halfContinueContinue reading “Algebraic Analysis notes Lecture 3 (11 Jan 2019)”

Lecture 1 notes here. Recall that in algebraic topology, we construct homotopy invariants, e.g. homology. Homology measures “global” topology, but its not very sensitive to local structure. However, Poincare noticed that if X is a closed oriented manifold of real dimension n, then $latex dim H_k(X, \mathbb R) = dim H_{n-k} (X, \mathbb R)$. PoincareContinueContinue reading “Algebraic Analysis notes Lecture 2 (9 Jan 2019)”