Sheaves and Algebraic Geometry
Read this topicExplanations 24
- Benson Farb, Representation Stability
By Victor Wang
- Matthew Baker, Hodge theory in combinatorics
By Victor Wang
- Curve theory references Standard references: Hartshorne, Qing Liu, Vakil. Stichtenoth is a good reference for the function field perspective, including a simple argument for asy…
By Victor Wang, Victor Wang, Alex Federspiel
- Flatness to-do Equivalent definitions of flatness, see e.g. Qing Liu... when is locally free a good way to think about it? Use both algebra (e.g. number theory) and geometry int…
By Victor Wang
- Sheaves of differentials Nice to have different perspectives. See even Neukirch for "arithmetic Riemann-Roch theory" (including a notion of genus of number field, and so on). Wa…
By Victor Wang
- Vector bundles vs. locally free sheaves See Martin Brandenburg's answer to MSE: Equivalence of categories of vector bundles and locally free sheaves. On the topic of monomorphis…
By Victor Wang
- Divisors, degree, and codimension one techniques See e.g. Kedlaya's notes. "Codimension one techniques" is kind of vague—but it could be useful to compile examples. Blowups MO: …
By Victor Wang, Victor Wang
- Visualize intersections of complex curves in four-dimensional real space? Should be transverse intersection, but can't currently wrap head around it.
By Victor Wang
- Basics often left undefined or details unmentioned See MSE: Does Hartshorne really not define things like the composition or restriction of morphisms of schemes? In setting up c…
By Victor Wang
- Normality, regularity, smoothness, flatness, etale, unramified: big picture How are these related? What are different ways of thinking about these (e.g. regularity vs. locally f…
By Victor Wang, Victor Wang
- Affine, separated, (universally) closed, projective, proper Examples and non-examples of each. Different ways of thinking about these. Connection to topology notions (contractib…
By Victor Wang, Victor Wang
- Pushforward and pullback (of sheaves?) Examples (both algebraic and geometric/topological), contexts, applications, properties, how to think about these. See e.g. Wikipedia on i…
By Victor Wang
- Ringed spaces, morphisms, and immersions Qing Liu, Section 2.2.2 "Ringed topological spaces" seems quite nice and concise. Maybe for newcomers could use more elaboration and exa…
By Victor Wang
- Spectrum of a ring: basics Essentially, go back to big picture: why the Zariski topology? Why is this useful? Why study this and not something else? Connect this with classical …
By Victor Wang
- Sheaf mechanism Why sheaves? Why is this natural? Basic examples of working with sheaves, e.g. stalk-to-local yoga, local-to-global gluing, partitions of unity, connection to cl…
By Victor Wang
- Basic non-relative scheme properties Reduced, irreducible, integral Connected, quasi-compact, locally Noetherian Maybe dimension in nice cases? Affine open subsets ("affine piec…
By Victor Wang
- Fiber products and their uses Many key geometric notions can be captured through fiber products. base change, family, fiber, reduction restriction (e.g. f−1(V)→Vf^{-1}(V) \to Vf…
By Victor Wang
- Basic relative properties of schemes and morphisms Finite, affine Locally of finite type Quasi-compact (and how this is used, e.g. paired with (locally) Noetherian hypothesis) D…
By Victor Wang
- Generic points and specialization How to think about this. Basically, it's a reinterpretation of Zariski topology. How is this used? For example, an open property true for a sin…
By Victor Wang
- Common types of sheaves Quasicoherent, finitely generated, coherent, locally free, invertible Flasque resolution, e.g. 0→OX→KX→KX/OX→00 \to \mathcal{O}_X \to \mathcal{K}_X \to \…
By Victor Wang
- Morphisms to Ad\mathbb{A}^dAd and Pd\mathbb{P}^dPd, ampleness, and quasiprojective revisited Compare: for a morphism f :X→Adf\colon X \to \mathbb{A}^df:X→Ad, we have f∗OAd=OXf^…
By Victor Wang, Victor Wang
- Global sections, distinguished (e.g. affine) sections, and Cech cohomology "Local-to-global" arguments are key. Vaguely speaking, this includes gluing, patching, checking global…
By Victor Wang, Victor Wang
- Some tricky issues I don't understand, but I also don't think they are that important: Naturality of the boundary map for derived functors seems annoying to prove. See Wikipedia…
By Victor Wang
- Riemann-Roch over arbitrary fields See MO: Is there a Riemann-Roch for smooth projective curves over an arbitrary field? or a book on function fields, like Stichtenoth, Algebrai…
By Victor Wang, Victor Wang
Problems 26
- Let FFF be a subspace of a topological space XXX. How can you tell if FFF is irreducible? Select all that apply.
- Select all properties of irreducible topological spaces XXX. (Source: Cf. Qing Liu, Chapter 2, Proposition 4.5)
- Let ϕ :A→B\phi\colon A\to Bϕ:A→B be a ring homomorphism. If ℘\wp℘ is a prime ideal of AAA, describe Spec(B℘/℘B℘)\operatorname{Spec}(B_\wp/\wp B_\wp)Spec(B℘/℘B℘), the set of …
- Affine morphisms are always which of the following? Select all that apply.
- Where does the term OX,x\mathcal{O}_{X,x}OX,x come from?
- Let I,JI,JI,J be two ideals of RRR. What are V(I)∩V(J)V(I)\cap V(J)V(I)∩V(J) and V(I)∪V(J)V(I)\cup V(J)V(I)∪V(J)?
- Let XXX be a scheme, and let f∈OX(X)f\in \mathcal{O}_X(X)f∈OX(X) be a global section of the structure sheaf of XXX. Which of the following sets must be open in XXX? Mark all th…
- Must RRR have a nontrivial idempotent eee, i.e. a solution to e2=ee^2=ee2=e with e≠0,1e\neq0,1e=0,1?
- Let AAA be a ring and let q\mathfrak{q}q be a prime ideal. Let X=Spec(A)X=\operatorname{Spec}(A)X=Spec(A) be the spectrum of AAA with the Zariski topology. Which of the followi…
- Let XXX be a topological space and Y={ξ}Y = \{\xi\}Y={ξ} a singleton subspace of XXX. (Here ξ\xiξ is just a point of XXX.) When does a point p∈Xp\in Xp∈X lie in the closure Y‾\o…
- If UUU and CCC are open and closed subsets of XXX, respectively, what can you say about the relative containment of p,ξp,\xip,ξ in UUU and CCC? Mark all true implications.
- Let AAA be a ring and X=PA1X = \mathbb{P}^1_AX=PA1. What is the sheaf of relative differentials ΩX/A1\Omega^1_{X/A}ΩX/A1 isomorphic to as an OX\mathcal{O}_XOX-module?
- But first, let's try to understand why ddd induces a derivation (modulo I2I^2I2). Which identities below do d :B→Id\colon B\to Id:B→I satisfy in the AAA-algebra B⊗ABB\otimes_A …
- Which of the following guarantees that F\mathcal{F}F is trivial, meaning F≅OX\mathcal{F}\cong \mathcal{O}_XF≅OX? Give the strongest (most general) correct answer.
- Let XXX be a scheme. Then XXX is necessarily a scheme over which of the following base schemes SSS?
- What is the data of a morphism G→π∗F\mathcal{G} \to \pi_\ast \mathcal{F}G→π∗F of sheaves over YYY equivalent to? Select all that apply.
- What is another way to express the intersection U∩XfU\cap X_fU∩Xf?
- If YYY is a scheme such that αX,Y\alpha_{X,Y}αX,Y is bijective for all schemes XXX, must YYY be affine?
- Is it true that π−1Yg=Xπ♯g\pi^{-1} Y_g = X_{\pi^\sharp g}π−1Yg=Xπ♯g?
- Compute the Picard group of the spectrum of the ring R[x,y]/(x2+y2−1).\mathbb{R}[x,y]/(x^2+y^2-1).R[x,y]/(x2+y2−1).
- Let I,JI,JI,J be two ideals of RRR. What are D(I)∩D(J)D(I)\cap D(J)D(I)∩D(J) and D(I)∪D(J)D(I)\cup D(J)D(I)∪D(J)?
- What can you say about this map? Select all that apply.
- Suppose III is a homogeneous ideal of SSS contained in S+S_+S+. Let JJJ be another homogeneous ideal of SSS. Which of the following statements relating D(J),V(J),D(I),V(I),D+(I…
- Is the argument correct? If not, what is the first incorrect step?
- Let k=k‾k = \overline{k}k=k be an algebraically closed field. Let π :X→Y\pi\colon X\to Yπ:X→Y be a birational morphism of curves over kkk. How are the Picard groups of X,YX,YX,…
- Let I,JI,JI,J be ideals in a (commutative, unital) ring AAA. Compute the AAA-algebra tensor product A/I⊗AA/JA/I\otimes_A A/JA/I⊗AA/J.