Foundations of algebraic geometry, aka schemes 2020

Show that a map \(X \to \mathbb P^n\) is the same as a line bundle \(L\) on \(X\) and a surjective map \(O_X^{n+1} \to L.\)

Suppose for simplicity that we are working over an algebraically closed field \(k\). I claim that the above statement is a precise version of: \(\mathbb P^n_k\) is the space of one-dimensional quotients of \(k^{n+1}\). Can you justify this claim? Hint: Take \(X = {\rm Spec}\ k\) to get a description of closed points of \(\mathbb P^n_k\).

1. (14.1.A) Show that the space of global sections of \(O(n)\) on \(\mathbb P^1_k = {\rm Proj}\ k[X,Y]\) is naturally isomorphic to \(k[X,Y]_n\), the degree \(n\) graded component of \(k[X,Y]\).
2. Let \(s = X^2Y(X + Y)\), considered as a global section of \(O(4)\) on \(\mathbb P^1_{\mathbb C}\). Find \({\rm div} (s)\).
3. Let \(s = \frac{101}{100}\), considered as a rational section of \(O\) on \({\rm Spec}\ \mathbb Z\). Find \({\rm div} (s)\).
4. Choose one of \(X = {\rm Spec}\ \mathbb Z\) or \(X = {\rm Spec}\ \mathbb A^1_k\). Identify the group of \(\{(L,s)\}\)/isomorphism.
5. Understand the proof of 14.2.1: \({\rm div}\) is injective on normal (Noetherian) \(X\).
6. (14.1.D) Sketch a proof of the fact that \({\rm Pic} \mathbb P^1_k = \mathbb Z\). Hint: The only invertible sheaf on \(\mathbb A^1\) is the trivial one.

7. Failure of Hartog’s lemma. Let us construct an \(X\) that is regular in codimension 1, but not normal. \(X\) can’t be a curve (why?). So let us try a surface. We take a regular surface and modify it only in codimension 2 and hope to break normality.

   Start with \(Y = \mathbb A^2\). Obtain \(X\) by gluing two distinct points of \(X\), say \(p = (1,0)\) and \(q = (0,1)\)

   • Can you make this precise? Hint: this is an affine variety, so try to first construct the ring of functions, and then take the spec.
   • Take any function \(f\) on \(\mathbb A^2\) with distinct values at \(p\) and \(q\), for example \(x-y\). Then \(f\) is a rational function on \(X\) that has no poles (\({\rm div} f = 0\)) but is not regular.

1. (Example of 14.2.E) Let \(D = 3[0] - [\infty]\) on \(X = \mathbb P^1\). Show that \(O_X(D)\) is an invertible sheaf. Show that it is isomorphic to \(O(2)\).
2. (14.2.E) Spend some time on the general result: For \((L,s)\), we have \(O(div s) \cong L\).
3. (14.2.7) Digest the diagram in 14.2.7.
4. (14.2.5) Prove that if \(D\) is locally principal, then \(O_X(D)\) is invertible.
5. (14.2.I) For a factorial \(X\), every divisor is locally principal.
6. Prove (using the divisor class group) that \({\rm Pic} \mathbb P^n_k = \mathbb Z\).
7. Prove (using the divisor class group) that \({\rm Pic} \mathbb P^1_k \times \mathbb P^1_k = \mathbb Z \oplus \mathbb Z\).
8. (14.2.K) Pic of the complement of a hypersurface.

Thursday

Choose your adventure.

Let \(k\) be an algebraically closed field.

1. Let \(X = \mathbb A^1_k\) and let \(p \in X\) be any closed point. Let \(I_p \subset O_X\) be the ideal sheaf of \(p\).
   1. Describe explicitly the module \(I_p(U)\) for every open \(U \subset X\).
   2. Prove that \(I_p\) is locally free of rank 1.
   3. Now take \(X = \mathbb P^1_k\) and define \(I_p\) as before. What are the global sections of \(I_p\)? That is, find \(\Gamma(X, I_p)\).
   4. What is the cokernel of the map \(I_p \to O_X\)?
   5. Give examples of
      • a coherent sheaf on \(X\) that is not locally free
      • a quasi-coherent sheaf on \(X\) that is not coherent
   6. Let \(X = {\rm Spec}\ k[x,y]/(xy)\) and \(p = (0,0)\). Show that the ideal sheaf \(I_p\) is not locally free. Is it quasi-coherent? Coherent?
2. Let \(i \colon U \to X\) be the open inclusion \(\mathbb A^1 \to \mathbb P^1\).
   1. Prove that \(i_* O_U\) is a quasi-coherent \(O_X\)-module. (Appealing to a general theorem is OK).
   2. Is \(i_* O_U\) coherent?
   3. What is the pushfoward look like for \(f \colon {\rm Spec}\ A \to {\rm Spec}\ B\)?
   4. Prove the general theorem alluded to above (13.3.F)
3. Let \(X = {\rm Proj}\ k[u,v][X,Y]/(vX-uY) \subset \mathbb A^2 \times \mathbb P^1\). Prove that \(\pi \colon X \to \mathbb P^1\) is a line bundle. That is:
   1. Construct an open cover \(U_i\) of \(\mathbb P^1\) and isomorphisms \(\pi^{-1} U_i \cong U_i \times \mathbb A^1\). (The standard open cover will work.)
   2. Find the transition function(s) \(U_i \cap U_j \times \mathbb A^1 \to U_i \cap U_j \times \mathbb A^1\) and show that they are linear in the \(\mathbb A^1\) coordinate. (The linearity ensures that the fibers of \(\pi\) get a consistent vector space structure.)
   3. Let \(I\) be the sheaf of sections of \(\pi\). Then \(I\) is locally free of rank 1. Find the global sections of \(I\).
   4. What are the transitions functions of the dual \(I^\vee\)?
   5. What are the global sections of \(I^\vee\)?

Friday

Adventure continued. But also:

1. Stalks and fibers

   Remark: A good (but not perfect) way to think about a sheaf \(F\) is as a collection of stalks \(F_p\) (which are modules).

   If \(F\) is (quasi)-coherent, an even simpler (but more imperfect) approximation is as a collection of fibers \(F|_p\) (which are vector spaces).

   Let \(F\) be a coherent sheaf on a locally Noetherian \(X\). What can be gleaned from the fibers?

   1. Let \(M = k[t]/(t-1)(t+1)\) and \(F = \widetilde M\) on \(X = \mathbb A^1_k = {\rm Spec}\ k[t] \). What are the stalks of \(M\) at every point of \(X\)? What are the fibers of \(M\) at every point of \(X\)? (Including the generic point).
   2. If \(F|_p = 0\), then \(F_p = 0\), and there exists an open neighborhood of \(p\) on which \(F = 0\).
   3. \(F \to G\) is surjective on fibers if and only if it is surjective. If \(F|_p \to G|_p\) is surjective, then \(F \to G\) is surjective in a neighborhood of \(p\). What can you say about injectivity?
   4. If \(s_1, …, s_n\) are are sections of \(F\) that generate \(F|_p\), then they generate \(F|_q\) for every \(q\) in a neighborhood of \(p\).
   5. The function \(p \mapsto dim(F|_p)\) is upper semicontinuous.
   6. If \(X\) is reduced and connected, then \(F\) is locally free if and only if the rank function is constant.
   7. Give an example where the above fails for non-reduced \(X\).
   8. Let \(X = \mathbb A^2\) and \(I\) the ideal sheaf of \((0,0)\). Compute the rank function for \(I\).

1. 9.1.B (Fibered products of affines = tensor product of rings)

2. Consider \(f \colon Y = {\rm Spec}\ \mathbb C[y] \to {\rm Spec}\ \mathbb C[x] = X\) induced by \(x \mapsto y^2\). Let \(p \colon {\rm Spec}\ \mathbb C \to X\) be the closed embedding of a point. The (scheme-theoretic) preimage of \(p\) is the closed embedding \(p \times_X Y \to Y\). Find the preimages of the point \(x = 0\) and \(x = a\) for \(a \neq 0\).

   Draw a picture.

3. Consider \(f \colon Y = {\rm Spec}\ \mathbb Q[y] \to {\rm Spec}\ \mathbb Q[x] = X\) induced by \(x \mapsto y^2\). Find the scheme-theoretic preimages of
   1. \(p \colon {\rm Spec}\ \mathbb Q \to Y\) given by \(y \mapsto 0\).
   2. \(p \colon {\rm Spec}\ \mathbb Q \to Y\) given by \(y \mapsto 1\).

   3. \(p \colon {\rm Spec}\ \mathbb Q \to Y\) given by \(y \mapsto 2\).

      Try drawing a picture.

4. Consider \(f \colon Y = {\rm Spec}\ \mathbb Z[i] \to {\rm Spec}\ \mathbb Z = X\). Find the scheme-theoretic preimages of the points (2), (3), and (5). Try drawing a picture.
5. 9.2.C (a) (Intersection as fibered product)
6. A “geometric point” of a scheme \(X\) is a map \({\rm Spec}\ k \to X\), where \(k\) is algebraically closed. Show that the fibers of \(f \colon Y = {\rm Spec}\ \mathbb Q[y] \to {\rm Spec}\ \mathbb Q[x] = X\) over geometric points are either a pair of distinct points or a non-reduced (double) point.

Friday

. . . the end of all our exploring Will be to arrive where we started And know the place for the first time.

– T. S. Eliot, “Little Gidding” (Four Quartets)

1. 9.3.F (The blowup, again) Can you reconcile our previous description as \({\rm Proj}\ k[x,y][u,v]/(xv-yu)\) with Ravi’s definition?
2. Show that the following properties are preserved under base change:
   1. open embedding
   2. closed embedding
   3. affine
   4. finite (affine + finite as a module) (optional)
3. 9.6.1 (Check that the Segre embedding is a closed embedding)
4. 9.6.B (Equations of the Segre embedding)
5. 6.4.A, 6.4.B (Maps of graded rings and Proj) But NOT all maps of Proj are of this form!
6. 6.4.D (Veronese)
7. 6.4.F

1. 6.3.B For affines, morphism of locally ringed spaces = induced by a ring map
2. 6.3.A Morphisms glue
3. 6.3.C Morphism is locally induced by ring maps
4. 6.3.F Morphism to Spec A
5. 6.3.I Spec Z is final
6. 6.3.H Structure map of Proj
7. 6.3.J Map from a point
8. 6.3.M (Some) morphisms to projective space

Friday

\({\mathbf A}^{n} = Spec Z[x_{1},\dots, x_{n}]\). \({\mathbf P}^{n} = Proj Z[X_{0}, \dots, X_{n}]\)

1. A different take on 6.3.M
   • Construct a morphism \(\mathbf{A}^{{n+1}} - 0 \to \mathbf{P}^{n}\).
   • Show that a morphism \(X \to \mathbf{A}^{{n+1}} - 0\) is the same as \(n+1\) global sections \(f_{0}, \dots f_{n}\) of \(O_{X}\) that do not have a common zero.
   • Conclude that such \(f_{i}\) give a map \(X \to \mathbf{P}^{n}\).
   • You now see why all maps \(X \to \mathbf{P}^{n}\) may not arise in this way.
2. Make sense of the map \(\mathbf{P}^{1} \to \mathbf{P}^{2}\) “defined by \([X^{2}:XY:Y^{2}]\)”
3. 8.1.D (Use 7.3.4 - being affine is affine local on the target). Closed embeddings = maps which locally Spec(A/I) -> Spec A
4. Let p be a point of X. Show that the natural map Spec k(p) -> X is a closed embedding if and only if p is a closed point.
5. Show that the map in (2) “\([X^{2}:XY:Y^{2}]\)” is a closed embedding.
6. Generalise (2) for the “rational normal curve”: \(\mathbf{P}^{1} \to \mathbf{P}^{n}\) given by \([X^{n}:X^{n-1}Y:\dots:Y^{n}]\).
7. Describe the sheaf of ideals in (4) if p is a closed point.
8. Read 8.1.H and be convinced that this should work.
9. Let S be a graded ring and I ⊂ S a homogeneous ideal. Construct a map Proj(S/I) -> Proj(S) and show that it is a closed embedding.
10. Is the map Spec C[t] -> Spec C[x,y] “given by \((t^{2},t^{3})\)” a closed embedding? A homeomorphism onto its image?

1. The basics

   Let \(A = A_0 \oplus A_1 \oplus A_2 \oplus \cdots \) be a graded ring.

   1. What is the set \({\rm Proj}\ A\)?
   2. Describe a base for the topology on \({\rm Proj}\ A\).
   3. Describe the structure sheaf of \({\rm Proj}\ A\) by describing its value on the basic open sets.
   4. Why is this a scheme?
   5. Write a finite affine cover (under suitable conditions).

2. Examples

   Let \(k\) be an algebraically closed field.

   1. Let \(P = {\rm Proj}\ k[X,Y,Z]\), variables in degree 1.
      1. What is your affine cover in this case (from (5))?
      2. What are the closed points of \(P\)? Interpret them as lines in \(k^{\oplus 3}\).
   2. Let \(P = {\rm Proj}\ k[t][X,Y,Z]\), where \(t\) has degree 0 and \(X,Y,Z\) have degree 1. Repeat (1) and (2). You will have to modify the interpretation in (2). Hint: There is a map \(P \to {\rm Spec}\ k[t]\) and it maps closed points to closed points.
   3. Only if you have time Let \(S = {\rm Proj}\ k[u,v][X,Y]/(Xv-Yu)\), where \(u,v\) have degree 0 and \(X,Y\) have degree 1. Repeat (1) and (2). Describe the map \(S \to {\rm Spec}\ k[u,v]\).

1. Playing with Proj

   Let \(k\) be an algebraically closed field.

   • Trivial base ring and no equations: Let \(A = k[X_0, \dots, X_n]\). Show that closed points of \({\rm Proj}\ A\) = Lines in \(k^{n+1}\) = \((n+1)\)-tuples of homogeneous coordinates \([x_0:\dots:x_n]\) where \(x_i \in k\) are not all zero.
   • Equations: Let \(A = k[X_0, \dots, X_n]/(F_1, \dots, F_m)\). Show that closed points of \({\rm Proj}\ A\) = \((n+1)\)-tuples of homogeneous coordinates satisfying all the equations \(F_i = 0\).
   • Non-trivial base rings: Let \(A = A_0 \oplus A_1 \oplus \cdots \).

     • Construct a map \(\pi \colon {\rm Proj}\ A \to {\rm Spec}\ A_0\). For now, describe the map on points. It can be upgraded to a map of schemes.

       Via \({\pi}\), we can think of \({\rm Proj}\ A\) as fibered over \({\rm Spec}\ A_0\), and try to understand the fibers.

     • Suppose \(A\) is finitely generated over \(A_0\). Then \(\pi\) will turn out to be a proper map. Proper maps send closed sets to closed sets. In particular, they send closed points to closed points. So closed points of \({\rm Proj} A\) are fibered over closed points of \({\rm Spec}\ A_0\).

       Take \(A = k[t][X_0, \dots, X_n]\), where \(t\) has degree 0 and the \(X_i\) have degree 1.

       • Show that the fiber of \({\rm Proj}\ A\) over a closed point of \({\rm Spec}\ k[t]\) is a copy of \({\rm Proj}\ k[X_0, \dots, X_n]\). For now, show this just for the sets of closed points, but it can be upgraded to a statement about schemes.
   • Non-trivial base rings and equations: Let \(A = k[s,t][X,Y,Z]/(X^2+sY^2+tZ^2)\), where \(s,t\) have degree 0 and \(X,Y,Z\) have degree 1.
     • Recall the map \(\pi \colon {\rm Proj}\ A \to {\rm Spec}\ k[s,t]\). What is its fiber over the point \((s-a, t-b)\)?
     • We say that the map \(\pi\) describes a family of projective \(k\)-schemes (in this case conics in \(\mathbb P^2_k\)) parametrized by \({\rm Spec}\ k[s,t]\). Do you see what this means?
     • Slogan: \({\rm Proj\ } A\) is a family of projective schemes parametrized by \({\rm Spec}\ A_0\).
   • Another non-trivial base ring and equations: Let \(A = k[u,v][X,Y]/(Xv-Yu)\).
     • Describe the closed points of \({\rm Proj}\ A\) using the map \(\pi\) to \({\rm Spec}\ k[u,v]\) (be careful!)
     • Describe the affine charts of \({\rm Proj}\ A\).
     • Is \({\rm Proj}\ A\) reducible or irreducible? Is it smooth or singular?
     • Try to draw a picture of \(\pi \colon {\rm Proj}\ A \to \mathbb A^2\).
     • Can you write the map \({\rm Proj}\ A \to \mathbb A^2\) on the affine charts?

Poll

Let \(X = {\rm Spec}\ {\mathbb C} \sqcup {\rm Spec}\ {\mathbb C} \sqcup {\rm Spec}\ {\mathbb C} \sqcup \cdots\) (infinite disjoint union). Let \(R = \mathbb C \times \mathbb C \times \mathbb C \times \cdots\) (infinite product).

/poll

• X is an affine scheme; in fact X = Spec R.
• X is an affine scheme, but it is not Spec R.
• X is not an affine scheme.

1. Suppose \(g \in A\) does not vanish on the open set \({\rm spec}\ A_f\) contained in \({\rm spec}\ A\). What can you say about \(g\)? Must it be a power of \(f\)?
2. 4.1.A: Show that \(A_f \to O(D(f))\) is an isomorphism.
3. We know that elements of \(A\) give functions on \({\rm spec}\ A\), and functions form a sheaf. Then why is there work involved in showing that \(O\) satisfies the sheaf axioms?
4. Prove the base-gluability of the structure sheaf.
5. 4.3.F: Stalks of the structure sheaf.
6. 4.1.D: Work through the construction of \(\widehat M\).
7. 4.1.5: Pitfall of a slick definition of \(O(U)\). What is the correct description of \(O(U)\)?
8. Draw
   • \({\rm Spec}\ {\mathbb C}[t]/t^2\)
   • \({\rm Spec}\ {\mathbb C}[t]/t^3\)
   • \({\rm Spec}\ {\mathbb C}[x,y]/(x^2,xy,y^2)\)
   • \({\rm Spec}\ {\mathbb C}[x,y]/(xy,y^2)\)
   • \({\rm Spec}\ {\mathbb Z}/(12)\)

1. from yesterday 4.1.5: Pitfall of a slick definition of \(O(U)\). What is the correct description of \(O(U)\)?
2. enlightening but confusing 4.3.A: Isomorphisms of affine schemes = isomorphisms of rings
3. easy 4.3.B: Distinguished open subschemes 4.3.C: Open subschemes Think about 4.3.4 – closed subschemes
4. easy 4.3.G: Functions on locally ringed spaces
5. easy 4.4.A: Gluing schemes
6. 4.4.B/C: Lines and planes with doubled origins.
7. do once in life 4.4.D: Charts of the projective space
8. 4.4.F: Closed points of the projective space More generally, let \(A\) be a finitely generated \(k\)-algebra. What are the closed points of \(\mathbb P^n_A\)?
9. Example 4.4.12.
10. chinese remainder theorem Example 4.4.11.

1. 3.2.H (Points of \({\mathbb Q}[x,y]\))
2. 3.2.J (Points of a quotient)
3. 3.2.K (Points of a localisation)
4. 3.2.L (Thinking about localisation geometrically)
5. Draw pictures of the maps
   • \({\rm spec}\ \mathbb Q \to {\rm spec}\ \mathbb Z\)
   • \({\rm spec}\ k[t]_{(t)} \to {\rm spec}\ k[t]\)
   • \({\rm spec}\ \mathbb C[x] \to {\rm spec}\ \mathbb R[x]\)
   • \({\rm spec}\ k[t] \to {\rm spec}\ k[x,y]/xy\) induced by \(x \mapsto 0, y \mapsto t\).
   • \({\rm spec}\ k[x,y]/xy \to {\rm spec}\ k[t]\) induced by \(t \mapsto x+y\).
   • \({\rm spec}\ k \to {\rm spec}\ k[t]\) induced by \(t \mapsto a\) for some \(a \in k\).
   • \({\rm spec}\ \mathbb C \to {\rm spec}\ \mathbb Q[t]\) induced by \(t \mapsto \sqrt{2}\).
6. Let \(A\) and \(B\) be finitely generated \(k\) algebras, and \(f \colon A \to B\) a map of \(k\) algebras.
   • Show that \(f\) sends a closed point (maximal ideal) of \({\rm spec}\ B\) to a closed point of \({\rm spec}\ A\).
   • Show that \(f\) may not send closed points to closed points, in general.
7. Let \(X = \mathbb A^n_{k}\). What are the residue fields of the closed points (max ideals) of \(X\) if \(k = \mathbb C\)? If \(k = \mathbb Q\)?
8. 3.2.R (Nilpotents and spec)
9. 3.2.S (Nilradical = intersection of prime ideals).
   • When is \(R_f = 0\)? (Only when \(f\) is a nilpotent.)
   • Aside: For some rings (“Jacobson rings”), nilradical = intersection of all maximal ideals. This means that if \(f\) and \(g\) take the same value at every closed point, then \(f-g\) is nilpotent. Fields are Jacobson (trivially) \(\mathbb Z\) is Jacobson (easy), finitely generated algebras over Jacobson rings are Jacobson. Can you come up with an example of a non-Jacobson ring (hint: its \({\rm spec}\) must have “few” closed points).

1. easy 3.6.I (closed points = maximal ideals) Related: What is the closure of a point \(p\)?
2. easy 3.4.I (\({\rm spec}\ A/I\) is closed in \({\rm spec}\ A\), and \({\rm spec}\ A_f\) is open.)
3. important 3.5.A (distinguished opens form a base)
4. 3.5.B/C, 3.6.G (coverings by distinguished opens and (quasi)-compactness)
5. 3.2.S (Nilradical = intersection of prime ideals). – skip proof at first
   • When is \(R_f = 0\)? (only when \(f\) is a nilpotent.)
   • Aside: For some rings (“Jacobson rings”), nilradical = intersection of all maximal ideals. This means that if \(f\) and \(g\) take the same value at every closed point, then \(f-g\) is nilpotent. Fields are Jacobson (trivially) \(\mathbb Z\) is Jacobson (easy), finitely generated algebras over Jacobson rings are Jacobson. Can you come up with an example of a non-Jacobson ring (hint: its \({\rm spec}\) must have “few” closed points).
6. 3.4.F (radical of I = intersection of primes containing I)
   • How do you intepret this statement geometrically?
7. 3.4.5 (nilpotents and spec)
8. 3.4.J (when does \(f\) vanish on \(V(I)\)?)
9. 3.5.E (inclusions among the distinguished opens)
10. I(.) and V(.) are mutual inverses Prove Theorem 3.7.1

Polls

Let \(X = \mathbb C^2\). Denote by \(O_X^*\) the sheaf of maps to \(\mathbb C^* = \mathbb C - \{0\}\). Let \(m_n \colon O_X^* \to O_X^*\) be the map \(f \mapsto f^n\).

/poll The map \(m_n\) above

• is a surjection of sheaves and also a surjection on all open sets.
• is a surjection of sheaves but not a surjection on all open sets.
• not a surjection of sheaves.

1. Let \(f \colon A \to B\) be a map of sheaves. Show using universal properties that the sheafification of the pre-sheaf cokernel of \(f\) is the cokernel of \(f\) in the category of sheaves.
2. 2.4.C (for sheaves, compatible germs = section)
3. 2.4.G (sheafification is unique up to a unique iso)
4. 2.4.I,J,K (verifying the construction of sheafification)
5. 2.4.M (sheafification does not change stalks)
6. 2.4.P (exponential map)
7. 2.5.B, C (sheaves on a base)
8. 2.5.D (gluing sheaves)
9. 2.6.D (exactness = exactness of stalks)
10. 2.6.F (left exactness of global sections)
11. Main takeaways
    • Sheaves of Ab groups forms an abelian cat. kernels, cokernels, images exist and behave like you expect.
    • How to construct? Construct naively, and sheafify!
    • How to detect? On stalks!
    • Enough to work on a base.
    • Everything holds for \(O_X\) modules.

1. important not logically necessary Definition: A differentiable manifold is a ringed space \((X, O_X)\) “which is locally isomorphic to \((U, O_U)\) where \(U \subset \mathbb R^n\) is an open set and \(O_U\) is the sheaf of differentiable real valued functions on \(U\)”.
   1. Make this precise (eliminate the “ ” marks). To do this, you will have to define the notion of an isomorphism of ringed spaces. More generally, try to define a morphism of ringed spaces.
   2. Show that this definition is equivalent to the usual definition of a differentiable manifolds (using charts and gluing maps). You will have to prove the (easy) fact that for \(U, V \subset \mathbb R^n\), xif \(\phi \colon U \to V\) is continuous, then it is differentiable if and only if differentiable functions pull back to differentiable functions.
2. - Draw \(spec k[t]_{(t)}\).
   • Draw \(spec Z_{(p)}\).
   • Draw \(spec \mathbb Z_p\) (the $p$-adics)
   • Draw \(spec k[[[[t]]]]\) (the ring of power series in \(t\)).
3. Describe \(\mathbb A^1_{\mathbb R}\) and \(\mathbb A^1_{\mathbb Q}\)
4. Draw \(spec \mathbb C[x,y]\).
5. Draw \(spec \mathbb C[x,y]/xy\).
6. - Draw \(spec \mathbb C[x]/x^2\)
   • Draw \(spec \mathbb C[x,y]/(x^2, xy, y^2)\)
7. Go back to sheaves and convince yourselves that the category of sheaves is equivalent to the category of sheaves on a base.

1. Points for discussion
   • What was easy, hard, confusing?
   • … insightful?
   • Examples!
   • 1.2.B (automorphism group)
   • 1.3.A (unique init/final obj)
   • What is the universal property for:
     • Disjoint union of two sets
     • The kernel of A -> B
     • The cokernel of A -> B
     • The one-point compactification
     • The metric completion
   • What are the initial/final objects in Top, Vec, Ab, Set? (create a poll?)

1. Pre-class poll

   Consider the category whose objects are algebraic field extensions of \(\mathbb Q\) (and whose morphisms are maps of fields).

   /poll The category above has

   • a final object, namely \(\overline{\mathbb Q}\).
   • no final object.
2. Points for discussion
   1. warm-up Let \(R = \mathbb Z\) and \(f = 2\).
      • Describe the elements of \(R_f\) explicitly.
      • Let \(K\) be a field of characteristic \(59\). Show that there is a unique map \(R_f \to K\).
   2. notation-alert! \(R = \mathbb Z\) and \(p = 2\).
      • Describe the elements of \(R_{(p)}\) (horrible notation, I know!)
      • What’s the analogue of the second part of the previous exercise?
   3. instructive Show (preferably using the universal property) that the localisation \(R_f\) is isomorphic to \(R[y]/(yf-1)\).
   4. Baby 1.3.C: localisation is injective for integral domains.
   5. geometry! important! Suppose \(R\) is the ring of functions of an affine variety \(X\). Then \(R_f\) is the ring of functions of <span class=“underline”><span class=“underline”><span class=“underline”>_</span></span></span>.
   6. Having done (5), can you “see” (4)?
   7. Can you construct an example where \(R \neq 0\) but \(R_f = 0\)?
   8. 1.3.E (\(\star\))