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 comments … Continue reading Algebraic Analysis Homework 1

# Month: January 2019

# Algebraic Analysis notes Lecture 7 (25 Jan 2019)

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 we … Continue reading Algebraic Analysis notes Lecture 7 (25 Jan 2019)

# Algebraic Analysis notes Lecture 6 (18 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-module … Continue reading Algebraic Analysis notes Lecture 6 (18 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)

# Algebraic Analysis notes Lecture 4 (14 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; \mathbb … Continue reading Algebraic Analysis notes Lecture 4 (14 Jan 2019)

# Algebraic Analysis notes Lecture 3 (11 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 half … Continue reading Algebraic Analysis notes Lecture 3 (11 Jan 2019)

# Algebraic Analysis notes Lecture 2 (9 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 $latex dim H_k(X, \mathbb R) = dim H_{n-k} (X, \mathbb R)$. Poincare … Continue reading Algebraic Analysis notes Lecture 2 (9 Jan 2019)