# Schemes

This semester I am teaching the course on schemes.

The lectures will be on Tuesday and Thursday from 11:40AM -- 12:55PM in Room 507.

The TA is Carl Lian who will be in the help room on Monday 10:30-12 and Thursday 4:30-6.

For last years course on schemes webpage click here. However, this year we will do something completely different. Please visit the first class of the course to find out more, or you can read about it in this blog post. It is very important to attend the lectures. You will not be able to pick up the material by just reading this webpage

Review of lectures and suggested readings

1. Lecture I: we discussed prerequisites and we found most participants had some knowledge of commutative algebra and sheaves on topological spaces. It turned out that most of the participants are interested in learning about schemes. We then started discussing the questions "What is a scheme?" and "What is a morphism of schemes?". We reviewed and discussed the following topics:
1. topological spaces and continuous maps of topological spaces,
2. categories of sheaves of different types on topological spaces,
3. pushforward and pullback of sheaves under a continuous map of topological spaces,
4. ringed spaces and morphisms of ringed spaces,
5. local rings and local homomorphisms of local rings,
6. locally ringed spaces and morphisms of locally ringed spaces,
7. the spectrum of a ring as a set and as a topological space,
8. the spectrum of C[x] where C is the complex numbers, and
9. the value of the structure sheaf on the spectrum of a ring on the basic opens of the spectrum.
Please review this material and prepare to ask questions about arguments or constructions you do not understand when you do this. Some links to material in the Stacks project:
1. Spectrum of a ring 00DY
2. Sheaves 006A
3. Sheaves of modules 01AC (this will come later)
4. Sheaves on a spectrum handout (pdf) (also see the fun result here: 0F1A)
5. Locally ringed spaces 01HA 01HD (skip closed immersions)
6. Affine schemes 01HR, 01HX
These references should be enough to keep you busy till Thursday. But I encourage you to read in *any* reference you want. In fact, it could be really fun to compare different arguments and different definitions we get from different references.
2. Lecture II: we discussed the following
1. functoriality of the spectrum: given a ring map how does this give you a map of spectra (as a map of sets, continuity, and on the structure sheaves),
2. surjections of rings give closed maps on spectra,
3. the localization of a ring at an element gives an open immersion of spectra,
4. fully faithfulness of the spectrum as a functor from the (opposite of the) category of rings to the category of locally ringed spaces,
5. Hausdorff topological spaces as those topological spaces whose diagonal is closed,
6. separated schemes as those schemes whose diagonal morphism is a closed immersion,
7. affine morphisms of schemes (just the definition; there's lots missing here),
8. closed immersions as affine morphisms which affine locally correspond to surjections of rings,
9. Spec(Z) is the final object in the category of schemes,
10. Spec(C) is the final object in the category of schemes over C (which means you replace rings everywhere by C-algebras in everything we've said sofar),
11. any non-affine scheme gives rise to a nonaffine morphism by looking at the morphism to Spec(Z),
12. P^1_Z was constructed and we computed the global sections of its structure sheaf,
13. the affine line with zero doubled was constructed and we computed the global sections of its structure sheaf,
14. we proved the previous two examples are not affine.
Please try to think of questions of a different flavor for next time!
3. Lecture III: we discussed the following
1. Yoneda lemma
2. fibre products in any category
3. examples of fibre products in sets, topological spaces, vector spaces,
4. pushouts are the dual notion to fibre products
5. pushouts of rings are tensor products
6. fibre products of affine schemes are spectra of tensor products of rings
7. example of fibre product of affine lines
8. fibre products as glueing fibre products of affine schemes
9. P^1 times P^1 as a closed subscheme of P^3 (in terms of points)
10. P^1 is separated using the above (kind of a cheat)
11. the morphism from punctured affine space to projective space
Please do some examples of the things we discussed yourself.
4. Lecture IV: we discussed the following
1. We reviewed the sheaf condition for the structure sheaf on an affine scheme, in particular we discussed if A is a ring and f, g are elements of A which generate the unit ideal in A, then the sequence with terms 0, A, A_f, A_g, A_{fg} is exact.
2. Let C be a category which has all fibre products and let P be a property of morphisms in C. We discussed what it means when we say a morphism in C is "universally P".
3. We discussed proper maps of topological spaces and universally closed maps of topological spaces and we discussed how these are the same, see 005M (our definitions were not literally the same as in the Stacks project... because the definitions in the Stacks project follow in this instance the conventions of Bourbaki which are slightly unnatural for us in this case).
4. We proved that the real line mapping to a point is not a universally closed map (and hence not proper).
5. Motivated by our discussion of proper maps of topological spaces we defined a morphism of schemes to be proper if it is finite type, separated, and universally closed.
6. We proved that the affine line over a field is not proper over the field.
7. We discussed affine classical varieties V over an algebraically closed field k, in particular we discussed
1. the Zariski topology on k^n,
2. V is an irreducible closed subset of k^n in Zariski topology,
3. regular functions on V are maps of sets from V to k which locally in the Zariski topology on V can be written as quotients of polynomials,
4. V corresponds to a prime ideal p of k[x_1, ..., x_n], namely the set of polynomials vanishing on V,
5. the k-algebra k[V] of regular functions on V is the same as the quotient of k[x_1, ..., x_n] by p
6. morphisms of classical affine algebraic varieties from V to V' correspond 1-to-1 to k-algebra maps k[V'] to k[V].
8. We discussed how a classical affine variety V over k correspond to an affine, integral scheme X of finite type over Spec(k) and we discussed how points of X correspond to closed (classical) subvarieties of V
9. In order to state the previous point we discussed how you can talk about connected topological spaces, irreducible topological spaces, connected components of topological spaces, irreducible components of topological spaces, generic points of irreducible topological spaces, sober topological spaces (each irreducible closed subset has a unique generic point)
10. We discussed how schemes are sober via the example of spectrum of a ring and we applied this to see the fact about points of the scheme X associated to the classical variety V mentioned above.
5. Lecture V: we discussed the following
1. we discussed the standard proof that the ring of regular functions on a classical affine algebraic variety V is the quotient of the polynomial ring by the (prime) ideal of functions vanishing on V,
2. given an affine scheme S = Spec(A) and an A-module M we defined the O_S-module \widetilde{M} associated to M,
3. we characterized \widetilde{M} as the unique sheaf of O_S-modules such that for any O_S_module F we have Hom_{sheaves of modules}(\widetilde{M}, F) = \Hom_A(M, F(S)),
4. in other words the functor \widetilde{M} is the left adjoint to the functor of global sections which goes from the category of all O_S-modules to the category of A-modules,
5. an important observation is that the functor which assigns to M the sheaf \widetilde{M} is exact; this follows because the stalk of \widetilde{M} at p is the localizations M_p and because localization is exact
6. given a scheme X we defined an O_X-module F to be quasi-coherent if for every affine open U = Spec(A) of X the restriction F|_U is of the form \widetilde{M} for some A-module M,
7. we observed that with this definition it isn't clear that given S = Spec(A) and an A-module M, the sheaf \widetilde{M} on Spec(A) is a quasi-coherent module!
8. more importantly perhaps, it isn't clear why, given an open covering of a scheme X by affine opens U_i = Spec(A_i), to check quasi-coherence of an O_X-module F, it suffices to check that F|_{U_i} is isomorphic to \widetilde{M_i} for some A_i-modules M_i,
9. anyway, we discussed pushforward and pullbacks of the modules \widetilde{M} under morphisms of affine schemes,
10. more precisely, given a ring map A → B we found that pushforward along Spec(B) → Spec(A) corresponds to the restriction functor Mod_B → Mod_A and we found that pullback along Spec(B) → Spec(A) corresponds to the base change functor Mod_A → Mod_B which sends M to M ⊗ B.
11. we finally were able to conclude that \widetilde{M} is indeed a quasi-coherent module (please make sure you understand how this follows)
12. Motivation for defining quasi-coherent modules: (1) these are the algebraic geometers version of modules over rings, (2) vector bundles are special types of quasi-coherent modules, (3) we can look at cohomology of quasi-coherent modules and try to find invariants of algebraic varieties in that manner.
Here are some references to material on quasi-coherent modules in the Stacks project
1. The approach in the Stacks project is to first define a notion of quasi-coherent module on any ringed space as those sheaves of modules which locally are the cokernel of a map of free modules. This is done in 01BD
2. Then it is shown that on an affine scheme one obtains exactly the sheaves \widetilde{M} from the lecture, see 01I6
3. Because quasi-coherent modules where defined in general (without referencing affine schemes or whatnot) we then see that a quasi-coherent module on a scheme is affine locally of the form we want (this is an advantage of doing things this way).
6. Lecture VI: we discussed the following
1. Very briefly we discussed why given a curve C and a point p of C there exists a regular function on C - {p} which blows up at p. The corresponding algebra statement for affine curves would be: (*) given a 1 dimensional Noetherian domain A and a maximal ideal p of A there exists an element f of the fraction field of A which is not contained in the local ring A_p but is contained in the local ring A_q for all prime ideals q of A which are not equal to p.
2. We gave the definition of a finite locally free O_X-module on a ringed space (X, O_X).
3. We defined an invertible O_X-module to be a finite locally free O_X-module of rank 1.
4. We defined the Picard group of X, denoted Pic(X), to be the set of isomorphism classes of invertible O_X-modules with addition given by tensor product.
5. We implicitly accepted the fact that this is indeed a group, i.e., that every invertible O_X-module has an inverse''. It turns out that the inverse of an invertible O_X-module L is given by the sheaf-hom from L to O_X, so L^{⊗ -1} = SheafHom_{O_X}(L, O_X).
6. We proved that if X is a scheme then every finite locally free module is quasi-coherent. In particular every invertible module is quasi-coherent.
7. We concluded if X = Spec(A), then the Picard group is the set of isomorphism classes of certain A-modules. Let's call the modules you obtain in this manner invertible A-modules.
8. We showed that if M is an invertible A-module, then for every prime ideal p of A the localization M_p is isomorphic to A_p.
9. The module M = \sum (1/p)Z, seen as a Z-submodule of Q, has the property that M_{(p)} is isomorphic to Z_{(p)} for all prime numbers p but M is not an invertible module (as we'll see soon).
10. We concluded that we were missing a property of invertible modules. After looking at the example above, we decided that we needed to look for finitely generated A-modules.
11. On a ringed space we defined a module to be of finite type if it can locally be generated by finitely many sections.
12. Lemma: Let A be a ring and let M be an A-module. TFAE: (i) M is a finite type A-module, (ii) the O_{Spec(A)}-module \widetilde{M} associated to M is of finite type.
13. A module M over a domain A is said to be torsion free if every nonzero element of A is a nonzerodivisor on M.
14. Let A be a domain and M an A-module. TFAE: (i) M is torsion free, (ii) M_p is a torsion free A-module for all prime ideals p of A.
15. We conlcude that over a domain an invertible module is torsion free.
16. (**) Let A be a domain and let M be a finite type torsion free A-module. Then M is isomorphic to a submodule of a free A-module.
17. Lemma: Let A be a domain and let M be an invertible A-module. Then M is isomorphic to an ideal of A (as an A-module). Proof: by the previous lemma we see that there is a nonzero map M → A. Localizing at the prime (0) of A we see that it is injective because M_{(0)} is a 1-dimensional vector space.
18. CONCLUSION: Let A be a PID. Then every invertible A-module is isomorphic to A as an A-module and the Picard group of Spec(A) is trivial.
19. Addendum: in fact the Picard group of Spec(A) is trivial if A is a UFD. The proof uses the same ingredients and then uses that if I is an ideal which is also an invertible module, then any minimal prime over I has height 1 and hence is principal (in a UFD).
20. The Picard groups of Spec(Z) and the affine spaces A^n_k over fields k (or even if k is a UFD) are all trivial.
In the discussion above, I never said that for general rings A the invertible modules are exactly the finite type A-modules all of whose localizations at primes are free of rank 1. This is true, but it is actually a bit tricky in the non-Noetherian case (see first reference below)!
1. Algebra: finite locally free modules, finite flat modules, finite projective modules, and relations between these, see 00NV especially the first lemma.
2. Counter examples 052H (finite flat, not finite locally free) 0CBZ (ideal whose localizations are free but not invertible).
3. The definition of an invertible O_X-module in the Stacks project is slightly different (better!). The difference vanishes for locally ringed spaces. See discussion in 01CR.
4. Correspondingly, for modules over rings we have the same thing, see 0AFW.
5. The Picard group of a UFD is trivial, see 0BCH.
7. Lecture VII: we discussed the following
1. very briefly the example of an ideal whose localizations are free of rank 1 but which is not invertible as a module,
2. pullbacks of quasi-coherent sheaves along morphisms of schemes are quasi-coherent,
3. Lemma : pushforwards of quasi-coherent sheaves along quasi-compact and separated morphisms are quasi-coherent,
4. we defined quasi-compact morphisms of schemes as those morphisms such that the inverse image of a quasi-compact open is a quasi-compact open,
5. any base change of a quasi-compact morphism is quasi-compact,
6. we defined separated morphisms as those morphisms whose diagonals are closed immersions,
7. any base change of a separated morphism is separated,
8. we proved that given a separated morphism of schemes X → Y and affine opens U, U' of X mapping into a common affine open of Y, then U ∩ U' is an affine open of X,
9. we briefly discussed how you can characterize separated morphisms in terms of intersections of affine opens + something on the rings,
10. we stated and proved that kernels, cokernels, and images of maps between quasi-coherent modules on schemes are quasi-coherent,
11. arbitrary direct sums of quasi-coherent modules are quasi-coherent,
12. using all of the above we proved the lemma on pushforwards of quasi-coherent modules,
13. we discussed the rings A_n = k[x, 1/(x - t_1)(x - t_2)...(x - t_n)] where t_1, ..., t_n are pairwise distinct elements of a field k,
14. we proved that the group of units A_n^* is isomorphic to the unit group k^* of k times a free abelian group of rank n,
15. we proved that A_n is not isomorphic to A_m if n is not equal m,
16. we observed that A_n is a PID for all n,
17. we stated but did not prove that if k is algebraically closed and A is a finite type k-algebra which is a Dedekind domain (equivalently A is normal of dimension 1) and if the fraction field of A is a purely transcendental extension of k, then A is isomorphic to A_n for some n and some t_1, ..., t_n pairwise distinct elements of k.
18. we concluded that if you do exercise 078S below, then you will have found an finite type k-algebra A which is a Dedekind domain whose fraction field is not isomorphic to a purely transcendental extension of k.
19. said another way we find there exists some extension field K/k with the following properties
1. K is finitely generated as a field extension of k,
2. the transcendence degree of K over k is 1,
3. K is not isomorphic to k(x) as a field extension of k.
Perhaps it would be a good idea to ask what this means geometrically about 1 dimensional affine varieties over k in a future lecture.
8. Lecture VIII: The motivating question for this lecture was: "What is the relationship between the Picard group and the divisor class group of a curve?" To answer this question we discussed the following
1. Let K be a number field with ring of integers O_K. We said:
1. The class group Cl(O_K) is the group of fractional ideals modulo principal fractional ideals,
2. A fractional ideal is a nonzero finitely generad O_K-submodule of K,
3. A principal fractional ideal (f) is just all O_K-multiples of f where f is a nonzero element of K,
4. Any fractional ideal is an invertible O_K-module,
5. Any invertible O_K-module is isomorphic (as a module) to a fractional ideal (this is because we showed that any invertible A-module is isomorphic to an ideal of A for a domain A),
6. Two fractional ideals give the same class in CL(O_K) if and only if they are isomorphic as O_K-modules,
7. Combining all of the above we conclude Pic(O_K) = Cl(O_K),
8. Any fractional ideal I can be uniquely written in the form p_1^{n_1} ... p_r^{n_r} where the p_i are pairwise distinct maximal ideals of O_K.
9. For a nonzero element f of K we have (f) = p_1^{n_1} ... p_r^{n_r} where n_i = v_i(f) where v_i : K^* → Z is the valuation on K associated to the discrete valuation ring O_{K, p_i}. Moreover we have v(f) = 0 if v = v_p is the valuation associated to a prime ideal not equal to any of the p_i.
10. We conclude that Cl(O_K) = Pic(O_K) = (free abelian group on maximal ideals of O_K)/ (group of principal divisors) where a principal divisor associated to f in K^* is the formal sum div(f) = sum_p v_p(f)[p].
2. Let k be a field. A variety is a scheme separated and of finite type over Spec(k) which is reduced and irreducible.
3. A scheme X is irreducible if and only if the underlying topological space of X is an irreducible topological space
4. A scheme X is reduced if and only if there are no nozero local sections of the structure sheaf of X which are nilpotent. Clearly, it is equivalent to assume the local rings O_{X, x} to be reduced rings (no nonzero nilpotents) for all points x of X.
5. A scheme X is integral if it is nonempty and for every nonempty affine open U = Spec(A) the ring A is a domain.
6. Lemma: a scheme X is integral iff X is reduced and irreducible Tag 01ON
7. Let X/k be a variety. The function field k(X) of X is
1. the stalk of the structure sheaf at the generic point of X, or
2. the residue field of the generic point of X, or
3. the fraction field of A if U = Spec(A) is any nonempty affine open subscheme of X, or
4. the fraction field of O_{X, x} for any point x of X.
You should double check that all of these things are the same.
8. The dimension of a scheme X is the Krull dimension of the underlying topological space of X.
9. The Krull dimension (sometimes called the combinatorial dimension) of a topological space X is the supremum of the lengths of chains if irreducible closed subsets.
10. If X is a sober topological space (for example the topological space underlying a scheme) then the Krull dimension is the supremum of the lengths n of chains x_n ⇝ x_{n - 1} ⇝ ... ⇝ x_0 of points x_i of X.
11. Let X be a topological space. Let x, y in X. We say
1. x specializes to y or
2. y is a specialization of x or
3. x is a generalization of y or
4. y generalizes to x,
notation x ⇝ y, if and only if y is in the closure of the subset {x} of X.
12. If the topological space X has an open covering by U_i then we have dim(X) = sup dim(U_i).
13. Fact. If X/k is a variety, then dim(X) is equal to the transcendence degree of k(X) over k. This follows from the corresponding algebra fact and the fact that we can reduce to affine varieties by the previous point.
14. A variety X is called a curve if it has dimension 1.
15. A variety X is called a surface if it has dimension 2.
16. A variety X is called a threefold if it has dimension 3.
17. Let X be an affine variety of dimension d over an algebraically closed field k. Write X = Spec(A) and write A = k[x_1, ..., x_n]/(f_1, ..., f_m). Consider the matrix T = (df_i/dx_j) of partial derivatives. We define the singular locus of X, denoted Sing(X) to be the closed subset of X cut out by the (n - d) x (n - d) minors of T.
18. We say x in X is a nonsingular point of X or a smooth point of X if x is not contained in Sing(X).
19. We proved that if x is a smooth point of X then the local ring O_{X, x} is regular. (Our argument only worked for closed points and it moreover really used that k is algebraically closed.)
20. Still over an algebraically closed ground field the converse is also true: if O_{X, x} is a regular local ring, then x is a smooth point of X.
21. Let X be a curve over an algebraically closed field. The following are equivalent
1. x is a nonsingular point of X,
2. x is a smooth point of X,
3. O_{X, x} is regular,
4. O_{X, x} is regular of dimension 1,
5. the maximal ideal m_x ⊂ O_{X, x} can be generated by 1 element,
6. O_{X, x} is a discrete valuation ring,
7. O_{X, x} is a normal domain,
8. O_{X, x} has finite global dimension,
9. the residue field kappa(x) of the local ring O_{X, x} has finite projective dimension over O_{X, x},
10. kappa(x) has projective dimension 1 over O_{X, x},
11. X → Spec(k) is smooth at x (this characterization wasn't discussed in the lecture; we'll come back to it)
22. From the discussion above, if X is a nonsingular curve (in other words, the singular locus is empty) over an algebraically closed field, then for any x in X we obtain a valuation v_x = ord_x : k(X)^* → Z by using that O_{X, x} is a discrete valuation ring and that k(X) = Frac(O_{X, x}), see above.
23. Let X be a curve. A Weil divisor on X is a (finite) formal linear combination sum n_x [x] of closed points x of X with integer coefficients n_x. The abelian group of Weil divisors is denoted Div(X).
24. Let X be a nonsingular curve over an algebraically closed field. There is a group homomorphism div : k(X)^* → Div(X) mapping f to div(f) = sum ord_x(f) [x].
25. Let X be a nonsingular curve over an algebraically closed field. The Weil divisor class group of X, denoted Cl(X), is the cokernel of the map div above. Thus we have an exact sequence k(X)^* → Div(X) → Cl(X) → 0.
OK, so there is a lot of new material here; take your time digesting this. Just one point I wanted to make about working over non-algebraically closed fields. Let X/k be an affine variety over a field k which is not algebraically closed. We say a point x of X is nonsingular if O_{X, x} is regular. We say X → Spec(k) is smooth at x if the matrix of partial derivatives of the equations for X has the "correct" rank at x. It is always true that "smooth at x" implies "nonsingular at x" but if k is a non-perfect field, then the reverse implication isn't true in general.
9. Lecture IX:
1. The affine line over the field F_p with p elements has infinitely many points if you think of it as a scheme.
2. Notation: if S is a scheme and we say "let X be a scheme over S" then we mean that X is a scheme which comes endowed with a morphism X → S. In this situation the morphism X → S is called the structure morphism. If k is a ring (often a field) and we say "let X be a scheme over k" then we mean that X is a scheme which comes endowed with a morphism X → Spec(k).
3. If X is a scheme over a scheme S, then X(S) denotes the set of morphisms S → X of schemes such that the composition with the structure morphism is the identity on S. We often call X(S) the set of S-valued points. If X is a scheme over a ring k, then we similarly denote X(k) the set of morphisms Spec(k) → X whose composition with the given morphism X → Spec(k) is the identity. We call an element of X(k) an k-valued point. If k is a field, we will call an element of X(k) a k-valued point.
4. An important variation on the theme above is that if we have X/S and S'/S, then X(S') denotes the set of morphisms S' → X such that the composition S' → X → S is equal the given morphism S' → S. Again we say that X(S') is the set of S'-valued points of X. Similarly, if we are given a ring map k → k' (often a field extension) we can consider the set X(k') of k'-valued points.
5. Let A be a local ring of dimension > 0 with residue field k. Then the morphism Spec(k) → Spec(A) is surjective on closed points but not surjective.
6. Let X be a scheme of finite type over a field k. For a point x of X we set δ(x) equal to the transcendence degree of the residue field of x over k. Then we have
1. δ(x) = 0 if and only if x is a closed point,
2. δ is a dimension function, that is
1. if x ↝ x' then δ(x) ≥ δ(x'), and
2. if x ↝ x' is an immediate specialization (the points aren't equal and there is no point strictly between), then δ(x) = δ(x') + 1,
3. if f : X → X' is a morphism of f.t. schemes/k, then δ(f(x)) ≤ δ(x),
4. δ(x) is equal to the dimension of the closure of {x} in X (you can easily deduce this from properties a and b),
5. many things can be added here, but they often trivially follow from a, b, c.
7. Let X be a scheme of finite type over a field k. Using the dimension function δ above it is easy to show if W is a locally closed subset of X then (a) the closed points of W are the closed points of X which are contained in W, and (b) these points are dense in W.
8. Let f : X → Y be a morphism of schemes of finite type over a field. Then f(closed point) = closed point. We proved this in the affine case by invoking the Hilbert Nullstellensatz but it also easily follows by using 6.a and 6.c above.
9. A morphism X → Y of schemes of finite type over a field is surjective if and only if it is surjective on closed points. The statement makes sense by the previous point; we proved it in class using the correspondence between points of the scheme X and irreducible closed subsets of the set of closed points of X. My suggestion: try to prove this using the properties of δ stated above alone; if it is not possible, please tell me so I can add the additioanl property we need to get this to work.
10. Let k be an algebraically closed field of characteristic not 2 or 3. Let X be the spectrum of k[x, y]/(y^2 - x^3 + x). By explicitly finding relations in Cl(X) and using the result of one of the exercises (slightly generalized) we proved that the map {closed points of X} → Cl(X) is injective and moreover every element but the zero element of Cl(X) is in the image. This shows that there is a natural abelian group structure on {closed points of X} ∪ {0} which is given by the usual addition law for points on elliptic curves. In particular, there is no need to check the associative property for this group law using formulae!
References: there is a lot of very general stuff related to this material in the Stacks project. For example, you can look up
1. dimension functions in Tag 02I8,
2. dimension theory of finite type algebras over a field in Tag 00OO Tag 07NB
3. Jacobson topological spaces (spaces with lots of closed points) in Tag 005T,
4. Jacobson rings in Tag 00FZ
10. Lecture X: The goal of this lecture was to finish off the discussion of the Picard group and the Weil divisor class group of a nonsingular curve.
1. The (ridiculously) general theory, see 0BE0 and 02SE, says that
1. for any locally Noetherian integral scheme there is a notion of Weil divisors and one can define the group Div(X) of Weil divisors,
2. for a nonzero element f of the function field of X there is an associated principal Weil divisor div(f) defined in terms of orders of vanishing of f at the generic points of Weil divisors,
3. the collection of all principal Weil divisors is a subgroup of Div(X) and the quotient of Div(X) by this subgroup is denoted Cl(X) and is called the Weil divisor class group of X,
4. for any invertible O_X-module L there exist nonzero meromorphic or rational sections s and we can define the Weil divisor div_L(s) associated to s in terms of the orders of vanishing of s at the generic points of Weil divisors,
5. this construction defines a group homomorphism c_1 : Pic(X) → Cl(X) which in general is neither injective nor surjective.
2. for nonsingular curves over algebraically closed fields the theory works in exactly the same way AND the resulting map c_1 : Pic(X) → Cl(X) is an isomorphism,
3. a nonzero rational (or meromorphic) section s of an invertible module L on a curve X is just a nonzero element of the stalk of L at the generic point of X,
4. we define div_L(s) = sum div_{P, L}(s) [P] where for every closed point P of X we choose a generator e_P of L in a neighbourhood of P, we write s = f_p e_p for some rational function f_P on X, and we set ord_{P, L}(s) = ord_P(f_P),
5. a fun exercise is to show that if div_L(s) = 0 then s is a regular nowhere vanishing section of L,
6. then one shows that the divisor class of div_L(s) is independent of the choice of s because s is well defined up to multiplication by a nonzero rational function f and because div_L(fs) = div(f) + div_L(s)
7. the map c_1 sends L to the divisor class of div_L(s) where s is as above,
8. you can check that c_1 is additive by showing that div_{L ⊗ L'}(s ⊗ s') = div_L(s) + div_{L'}(s'),
9. if c_1(L) = 0 the we see that div_L(s) + div(f) = 0 for some nonzero rational function f, hence div_L(fs) = 0. By the above we find that fs is a trivializing section of L, i.e., L defines the 0 element of Pic(X),
10. finally we showed that if I ⊂ O_X is the ideal sheaf of a closed point P of X, then c_1(I) = - [P], and hence the map c_1 is surjective.
11. From now on, given a Weil divisor D on X we set O_X(D) equal to the unique (up to unique isomorphism) invertible module which has a nonzero rational section 1_D such that div_{O_X(D)}(1_D) = D. For example, if D = sum -[P_i] for some pairwise distinct points P_1, ..., P_n, then O_X(D) is the ideal sheaf of {P_1, ..., P_n}.
12. We computed that Pic(P^1_k) = Z if k is an algebraically closed field by explicitly finding enough relations among the points (viewed as Weil divisors). We also observed that div(f) is always a divisor of degree 0.
13. Here the degree deg(D) of a Weil divisor D = sum n_i [P_i] (on a curve over an algebraically closed field) is defined as the sum of the integers n_i.
14. We speculated (?!?) that if X is a nonsingular projective curve over an algebraically closed field, the the degree of a principal Weil divisor should be 0.
11. Lecture XI: The goal of this lecture was to start talking about cohomology of sheaves, but we found ourselves discussing derived functors in general instead.
1. If F is a sheaf of O_X-modules on a locally ringed space X let us define the fibre of F at a point x of X to be the k(x)-vector space F_x/m_x F_x. This is not a universally accepted terminology. The fibre is a vector space over the residue field k(x).
2. We first discussed the following question: let X be a reduced scheme and let F be a finite type quasi-coherent O_X-module such that all fibres have the same finite dimension r. Show that F is finite locally free of rank r. To answer this first prove that around every point there is an open neighbourhood generated by r sections (use Nakayama and 01B8). Then use that there cannot be any relations between these local sections as X is a reduced scheme and we know each fibre has dimension r.
3. We also very briefly discussed the relationship between invertible modules and line bundles, between finite locally free modules and vector bundles as usually defined, and between quasi-coherent modules and vector bundles as defined in EGA and the Stacks project. In each case one gets an anti-equivalence of categories when properly formulated. The last correspondence is discussed in 01M1; observe how this is a bit of a cheat because on the geometric side we use an additional structure on the structure sheaf (namely a grading). There is a way to say all of this completely geometrically.
4. Let F : A → B be an additive functor between abelian categories. The formal definition of an abelian category can be found here.
5. Question: How does one define/construct the left/right derived functors of F?
6. Answer: use resolutions by nice objects.
7. A left resolution of an object M of A is an exact complex ... → P_1 → P_0 → M → 0 in A.
8. A right resolution of an object M of A is an exact complex 0 → M → I^0 → I^1 → ... in A.
9. We try to define L_kF(M) = H_k(F(P_*)) as the kth left derived functor of F.
10. We try to define R^qF(M) = H^q(F(I^*)) as the kth right derived functor of F.
11. In order to make this well defined and a functor, we need some functoriality property of resolutions. The usual solution to this is to take left resolutions by projectives and right resolutions by injectives.
12. Please look up the definition of projective and injective objects of an abelian category A. Please look up what it means to have enough projectives or enough injectives in A.
13. Please look up uniqueness up to homotopy of projective resolutions and injective resolutions. You can look here and here for example.
14. The upshot of this is that if A has enough projectives, then we obtain well defined left derived functors L_kF and if A has enough injectives, then we obtain well defined right derived functors R^qF.
15. If F is right exact, then L_0F = F.
16. If F is left exact, then R^0F = F.
17. We defined Tor^R_i(M, N) to be the ith left derived functor of the functor F(-) = M ⊗ (-). Then for R = k[x, y] the polynomial ring over a field k, we computed Tor^R_i(k, k) to be a k-vector space of dimension ..., 0, 1, 2, 1 in degrees ..., 3, 2, 1, 0.
18. Ask yourself some questions, such as what happens when you take the derived functors of an exact functor?
12. Lecture XII: We continued discussing cohomology.
1. First we briefly discussed the fact that the category Ab(X) of sheaves of abelian groups on a topological space X in general doesn't have enough projectives. Neither does the category Mod(O_X) of O_X-modules on a ringed space (X, O_X) nor the category QCoh(O_X) of quasi-coherent modules on a scheme X.
2. On the other hand, these abelian categories all have enough injective objects and hence we can define the qth cohomology group H^q(X, F) as the qth right derived functor R^qGamma(X, -) applied to the abelian sheaf F.
3. An object L of an abelian category A with enough injectives is said to be F-acyclic if R^qF(L) = 0 for q > 0. Leray's acyclicity lemma says that one may compute the right derived functors R^qF using right resolutions by acyclic objects. (It is a good exercise to try to prove this by a method called dimension shifting.)
4. A sheaf of abelian groups F on a topological space X is called flasque or flabby if the restriction maps are all surjective. If this is true then F is acyclic for the functor Gamma(X, -), in other words, H^q(X, F) = 0 for all q > 0. This is proved here (please don't read about the general construction of derived categories, just read the argument on the linked page and translate it into the language we have used in class).
5. Upshot: we may compute cohomology of abelian sheaves using flasque resolutions.
6. The construction of enough injective abelian sheaves (or injective sheaves of modules) and also of enough flasque abelian sheaves uses the following simple construction: given for each point x of a topological space an abelian group A_x we can consider the presheaf G defined by the rule U ↦ ∏_{x in U} A_x. Then G is a flasque abelian sheaf! If A_x is an injective abelian group (i.e., an injective object of the category of abelian groups) for all x, then G is an injective abelian sheaf (injective object in the category of abelian sheaves). Using this it is straightforward to embed any abelian sheaf in a flasque or in an injective abelian sheaf and I explained this a bit more in the lecture.
7. We prove the following lemma: given a continuous map f : X → Y of topological spaces, the pushforward f_*I of an injective abelian sheaf is an injective abelian sheaf. We used a clever lemma on adjoint functors to prove this, see 015Z
8. If 0 → F_1 → F_2 → F_3 → 0 is a short exact sequence of Ab(X), then we get a long exact cohomology sequence of cohomology involving boundary maps H^q(X, F_3) → H^{q + 1}(X, F_1) defined using the next point
9. Given a short exact sequence 0 → M_1 → M_2 → M_3 → 0 of an abelian category A having enough injectives, there exists a short exact sequence 0 → I_1^* → I_2^* → I_3^* → 0 of injective resolutions M_i → I_i^*. Then each of the short exact sequences 0 → I_1^q → I_2^q → I_3^q → 0 is a split short exact sequence. It follows that on applying an additive functor F : A → B we obtain a short exact sequence 0 → F(I_1^*) → F(I_2^*) → F(I_3^*) → 0 of complexes of B. Then taking cohomology we obtain the long exact sequence of right derived functors 0 → R^0F(M_1) → R^0F(M_2) → R^0F(M_3) → R^1F(M_1) → R^1F(M_2) → R^1F(M_3) → ...
10. A trickier thing to prove is that the boundary maps R^qF(M_3) → R^{q + 1}F(M_1) are well defined.
11. Going back to our cohomology of topological spaces, we want to show that cohomology has something to do with the topology of X. The key step to doing this is the following point.
12. Given an injective abelian sheaf I on X and an open U of X, then I|_U is an injective abelian sheaf on U. To prove this use the same clever lemma about abelian categories as above and use that restriction to U has an exact left adjoint, namely, extension by zero. See 01E0.
13. Thus given an injective resolution 0 → F → I^* of an abelian sheaf F on X the restriction 0 → F|_U → I^*|_U is one too. Hence there are obvious maps H^q(X, F) → H^q(U, F). We denote these maps ξ ↦ ξ|_U. Please make very sure you understand what this means.
14. Locality of cohomology: Given ξ in H^q(X, F) where q > 0 there is an open covering X = ⋃ U_i such that ξ|_{U_i} is zero for all i. Same link as above.
15. Cech cohomology: given an open covering 𝒰 : X = ⋃ U_i and a total ordering on the indices, we can define the Cech complex C^*(𝒰, F) whose degree p term is the product of the sections of F over the p = 1 fold intersections of distinct opens. We define H^q(𝒰, F) = H^q(C^*(𝒰, F)). (There should be a check'' on the H but I don't know how to do this in html.) Some refs: 01ED, 09UD, 01FG (this is the section discussing the canonical isomorphism between ordered and unordered Cech complexes -- it is terrible)
16. It is easy to show that H^0(X, F) = H^0(𝒰, F) for any open covering 𝒰. This corresponds exactly to the sheaf condition for F.
17. It turns out that there always is an injective map H^1(𝒰, F) → H^1(X, F) and that the image consists exactly of the elements ξ which die on the members of the covering. Hence locality of cohomology tells us that all elements of H^1(X, F) come from the first Cech cohomology group for some open covering of X. This is our first evidence that cohomology has something to do with the combinatorics of interesections of opens of X.
13. Lecture XIII: The question asked this lecture was: Why is the cohomology of a coherent sheaf on a proper variety finite dimensional? It turns out that this is a very difficult question. In fact, you can say that a lot of the theory of (quasi-)coherent modules and their cohomology builds towards answering this question. The discussion that follows is a "top down", in other words, we discuss the ingredients needed to prove the theorem in reverse order.
1. An important theorem is Chow's lemma. This lemma says that given a proper variety X there is a projective variety Y and a surjective morphism f : Y → X of varieties.
2. For the proof of the theorem it is desirable to have variants of Chow's lemma. A good variant is the following: let X be a separated and finite type scheme over a Noetherian ring A. Then there exists a morphism f : Y → X such that
1. f is a proper morphism of schemes,
2. f is an isomorphism over a dense open U of X and
3. Y is quasi-projective over A which we will define to mean that there exists a locally closed immersion from Y into projective space over A.
In this situation, the morphism f is actually projective (in all possible meanings of the word projective) and moreover, if X is proper over A, then Y will be projective over A, in other words the locally closed immersion into projective space will be a closed immersion. See 02O2
3. Another important ingredient is to use devissage of coherent modules. Let X be a Noetherian scheme. Consider the abelian category Coh(O_X) of coherent modules on X. Let P be a property of coherent modules. Assume the property P satisfies the following properties:
1. given a short exact sequence of coherent modules 0 → F_1 → F_2 → F_3 → 0 and P(F_i) holds for 2-out-of-3 then it holds for the third,
2. for every closed irreducible and reduced subscheme Z of X we have P(O_Z)
then P holds for every object of Coh(O_X). This is just an example satement; the idea is to use induction on the dimension of the support of objects of Coh(O_X) and to try to filter any coherent module such that the graded pieces are "easier". This is discussed in 01YC and in particular see 01YI for a slight generalization of the statement above (which is useful in the discussion below).
4. Another very important ingredient is the Leray spectral sequence and its friends. One version is that given a morphism f : Y → X of topological spaces and a sheaf of abelian groups F on Y, then we have a spectral sequence with E_2 terms E_2^{p, q} = H^p(X, R^qf_*F) converging to H^{p + q}(Y, F). Some remarks
1. The functors R^qf_* are the right derived functors of the pushforward functor f_* constructed exactly as in the previous lecture.
2. Oten the only thing that matters is the following consequence: there is a finite (separated and exhaustive) filtration on H^n(Y, F) whose graded pieces are subquotients of the E_2^{p, q} with p + q = n.
3. So for example, if we know that E^{p, q} is finite (as a module or something), then we'll know the same thing for H^n(X, F).
4. Suppose that we know for some n that H^n(Y, F) is finite and E_2^{p, q} is finite for pairs (p, q) with p < n and q > 0. Then E_2^{n, 0} is finite. (This is related to the construction of the subquotients in the statement above.)
5. Similarly given composable morphisms f and g one has a spectral sequence E_2^{p, q} = R^pg_*R^qf_*F converging to R^{p + q}(g \circ f)_*F.
5. How do we combine the above three ingredients to prove the finiteness for coherent modules on a proper variety X given a similar finiteness statement for projective morphisms? The idea is to choose, for every irreducible and reduced closed subvariety Z of X a morphism f : Y → Z as in the refined version of Chow's lemma. Then f_*O_Y is a coherent module on Z (this is already a nontrivial fact, see below) which is isomorphic to O_Z on an open of Z. Using the Leray spectral sequence we reduce to showing that R^qf_*O_Y is coherent for all q and that H^n(Y, O_Y) is finite for all n.
6. The discussion above reduces us to proving the following two statements
1. given a (locally) projective morphism f : X → Y of Noetherian schemes, then R^qf_*F is coherent on Y for F coherent on X, and
2. given a closed subscheme X of P^n_A where A is Noetherian and given a coherent O_X-module F, show that H^n(X, F) is a finite A-module for all n.
7. In 6a the sheaf R^qf_*F is the sheaf associated to the presheaf which sends V to H^q(f^{-1}V, F). Thus if we have 6b and if we define locally projective to mean locally embeddable in a projective space, then it is sort of clear how to deduce 6a from 6b.
8. A very important and very often used fact about cohomology of quasi-coherent modules is that H^n(X, F) = 0 for n > 0 when X is an affine scheme and F is a quasi-coherent module. I want to stress here that this is a nontrivial fact and stated in this generality it is not in Hartshorne IIRC.
9. Suppose f : X → Y is an affine morphism and F quasi-coherent on X. Then since R^qf_*F is the sheaf associated to the presheaf V ↦ H^q(f^{-1}V, F) and since we have a basis of affine opens V for which f^{-1}V is affine, then we see that R^qf_*F = 0 for q > 0.
10. To prove 6b we consider the closed embedding i : X → P^n_A and we use the Leray spectral sequence and the vanishing of higher direct images along the affine morphism i (recall this is how we defined closed immersions in class: as affine morphisms with an additional property) to see that H^n(X, F) = H^n(P_A^n, i_*F).
11. The previous point now reduces us to the following problem: given a coherent module F on P^n_A with A a Noetherian ring, show that H^q(P^n_A, F) is a finite A-module for all q. We need three further results before we can complete the proof.
12. If O(d) denotes the d-th twist of the structure sheaf on P^n_A, then we can explicitly compute the cohomology of this and it is a finite A-module in each degree.
13. If a scheme X has an open covering by n + 1 affine opens such that all (multiple) intersections of these opens are also affine (for example this is true if X is separated), then H^q(X, F) = 0 for q > n. This can be deduced by using Mayer-Vietoris n times starting with the vanishing of cohomology on affines we stated above. It is a good exercise to work this out yourself.
14. Given any coherent module F on P^n_A there exist integers r, d_1, ..., d_r and a surjection Π : O(d_1) ⊕ ... ⊕ O(d_r) → F.
15. The end of the proof is to use descending induction on the cohomological degree. Namely, we know the cohomology of F is zero in degrees > n. Thus the finiteness holds in degree n + 1 and higher. Denote G is the kernel of the surjection Π. This is a coherent module as the category of coherent modules is an abelian subcategory of the category of all modules. Then the long exact sequence of cohomology shows H^q(F) is sandwiched between H^q(O(d_1) ⊕ ... ⊕ O(d_r)) which is finite by direct computation and H^{q + 1}(G) which is finite by induction. This finishes the proof.
Observe that this step doesn't work if A is not Noetherian, but lots of the other things I said above also don't work for nonNoetherian schemes and modules. Anyway, this is a lot of material to grok in just one lecture. It seems to me we should (perhaps?) turn to some more computional questions next time to see what is really going on.
14. Lecture XIV: During this meeting we tried to have more of a discussion by limiting the length of the answers. This also means we covered more different topics and I may have forgotten a few in the list below; if you remember one of these, please send me a quick email. Thanks!
1. Question: What is the relationship between Pic(X) and H^1(X, O_X^*)? Answer: they are the same. More precisely, given any open covering 𝒰 : X = ⋃ U_i of a locally ringed space X there is a 1-to-1 correspondence between isomorphism classes of rank 1 locally free O_X-modules L which are trivial on each U_i and elements of H^1(𝒰, O_X^*). Namely, you map L to the Cech cohomology class of the cocycle defined by the gleuing functions. Please make sure you understand this fairly well because it is used very often.
2. Question: What is the analogue of the Whitney embedding theorem in algebraic geometry? Answer: there is none. More precisely, here are some different answers
1. give a base field k there isn't a known list of (proper) varieties such that every (proper) variety has a closed embedding into one of the list
2. it turns out that there is a countable collection of proper varieties such that every proper variety embeds into one of them, but this is a mere existence statement and nobody has any clue what the varieties on the list look like
3. in some sense the point of looking at (quasi-)projective varieties is that you can embed them into an easy/simple variety, namely projective space
3. Question: what are some tools that you use to compute cohomology of sheaves in algebraic geometry? Answer: often you are completely stuck. Question: what about excision? Answer: you can do cohomology with supports.
1. If Z is a closed subset of a topological space X with open complement U, then we always have a long exact sequence of compactly supported cohology H^i_c(U) → H^i_c(X) → H^i_c(Z) → H^{i + 1}_c(U)
2. The above is similar to the sequence H_i(U) → H_i(X) → H_i(X, U) → H_{i - 1}(U) in homology where one sometimes has excision
3. To do exision with cohomology as we've been working with in this course, you use cohomology with supports. The functors H^i_Z(X, -) are defined as the right derived functors of H^0_Z(X, -). Here H^0_Z(X, F) ⊂ H^0(X, F) is the subgroup of sections whose support is contained in the closed subset Z. This is a left exact functor. There is a long exact sequence H^i_Z(X, F) → H^i(X, F) → H^i(U, F) → H^{i + 1}_Z(X, F). Finally, if X' ⊂ X is an open subspace containing Z then we have H^i_Z(X', F) = H^i_Z(X, F) which is a very simple way to do excision.
4. Question: what is the relationship between Cohomology and Cech Cohomology? Answer: there are several ways to look at this.
1. The most satisfying answer to me is that there is a Cech-to-Cohomology spectral sequence. More precisely, given an open covering 𝒰 : X = ⋃ U_i of a topological space X and an abelian sheaf F on X there is a spectral sequence E_2^{p, q} = H^p(𝒰, H^q(-, F)) converging to H^{p + q}(X, F).
2. If F is a flasque sheaf, then all higher Cech cohomology groups are zero (for any open covering of any open).
3. Suppose we have a basis B for the topology of X closed under finite intersections and consisting of quasi-compact opens. If F is an abelian sheaf such that its higher Cech cohomology groups are zero for finite open covers of elements of B by elements of B, then H^q(U, F) = 0 for all U in B and all q > 0.
See for example 01EO and 09SV. You can prove b for finite open coverings without using the spectral sequence using a fun induction argument; this is a good excercise or you can read a proof of (something close to) this in Vakil's notes. Then you can prove c using b and a dimension shifting argument; ask me if you are in doubt.
5. Let X be an affine scheme, let 𝒰 : X = ⋃ U_i be a finite affine open covering, and let F be a quasi-coherent module. Then we proved that the higher Cech cohomologies of F with respect to 𝒰 are trivial.
6. Combinging the previous results we concluded that the higher cohomologies of a quasi-coherent module on an affine scheme are zero.
7. Given a curve C of degree d in P^2_k over a field k with homogeneous equation F of degree d we considered the short exact sequence 0 → O(-d) → O → O_C → 0 of coherent modules on P^2. Then we decided this would allow you to compute H^0 and H^1 of O_C by the long exact cohomology sequence and our knowledge of the cohomology groups of the twists of the structure sheaf on P^n_k.
15. Lecture XV: During this meeting we tried to discuss an example, rather than more general theory.
1. The question: how does the number of nodes on a curve influence the genus?
3. Before interpreting the answer given above, please consider that our definitions are as follows.
4. Let X be a proper $1$-dimensional scheme over a field k such that H^0(X, O_X) = k. Then the genus of X is g = \dim_k H^1(X, O_X).
5. This invariant is sometimes called the arithmetic genus of X.
6. If X is as in 4 and X is smooth over k, then everybody agrees that g is the genus of X.
7. My advisor Frans Oort and many other people have a different definition of a variety over a field k. These people say: besides being integral and separated, of finite type over k, it has to be the case that the base change X \times_{Spec(k)} \Spec(\overline{k}) to the algebraic closure \overline{k} is a variety too. Let us call such a thing a FO-variety.
8. We discussed how X = P^1_{Q(i)} is a variety over Q in the sense defined above (earlier in the course), but not a FO-variety over Q. Of course, if we view X as a variety over Q(i) then it is a FO-variety and a fortiori a variety.
9. We proved that P^1_{Q(i)} is a projective scheme over Q (if you don't remember how, then this is a great exercise).
10. Returning to the genus, suppose that X is a FO-curve over k. Then it turns out that there is a unique smooth and projective FO-curve X' over k which is birational to X. It turns out that X' is just the normalization of X in case X is projective (or equivalenty proper).
11. If X and X' are as in 10 then we define the geometric genus of X to be the genus of X' (as defined in 4 above).
12. In the lecture we computed an example of a pair C = X and X' as above as follows:
1. let k be any field of characteristic not 2 or 3,
2. let X' = P^1_k.
3. consider the map ψ X' → P^2_k given on homogeneous coordinates by [s, t] ↦ [s^3 + t^3 : s^2t : st^2]
4. because X' is proper over k the image of ψ is closed
5. we found the equation for the image being X_0X_1X_2 = X_1^3 + X_2^2.
6. denote C the curve in P^2 defined by the equation X_0X_1X_2 = X_1^3 + X_2^2
7. we considered the restriction of C to the open where X_0 is not zero
8. using affine coordinates x_1 = X_1/X_0 and x_2 = X_2/X_0 we obtained x_1x_2 = x_1^3 + x_2^3 as the equation for this affine piece
9. the inverse image of this open of C in X' = P^1_k is the open given by s^3 + t^3 not zero
10. this affine open is the spectrum of the ring (k[s, t, 1/(s^3 + t^3)])_0
11. this ring has generators A = s^3/(s^3 + t^3), B = s^2t/(s^3 + t^3), C = st^2/(s^3 + t^3), D = t^3/(s^3 + t^3),
12. equations among these are A + D = 1, AD = BC, B^2 = AC, C^2 = BD,
13. we computed the singularities of the curve x_1x_2 = x_1^3 + x_2^3 and we found it has a unique singular point corresponding to x_1 = x_2 = 0
14. the map ψ on the affine pieces above is given by the map x_1 ↦ B and x_2 ↦ C
15. the inverse image of the singular point is therefore the locus B = C = 0 in the affine curve above
16. this produces the points (A, B, C, D) = (1, 0, 0, 0) and (A, B, C, D) = (0, 0, 0, 1); these are the points 0 and ∞ on X' = P^1_k. This suggests that one gets the curve C by glueing 0 and ∞ to each other on X' = P^1_k (this is true in a very precise sense which you can ask me about later)
17. the genus of X' is 0
18. the genus of C is 1
13. finally we computed the genus of a plane curve C in P^2_k
14. first we defined a plane curve to be a closed subscheme of P^2_k which is also a curve (in the sense above)
15. then we proved that given a plane curve there exists an integer d and a homogeneous polynomial F of degree d such that C is the zero set of F
16. we mentioned the fact that in this situation it is actually the case that C is Proj(k[X_0, X_1, X_2]/(F)) as a scheme
17. then we observed that there is a short exact sequence of graded modules 0 → k[X_0, X_1, X_2](-d) → k[X_0, X_1, X_2] → k[X_0, X_1, X_2]/(F) → 0
18. given a graded module M the shift M(e) is the graded module with graded parts M(e)_d = M_{e + d}
19. there is an exact functor from graded modules M over the graded ring k[X_0, X_1, X_2] to quasi-coherent modules on P^2_k. If \widetilde{M} is the sheaf associated to M then we have \widetilde{M}(D(F)) = (M[1/F])_0
20. Applying the functor above to the short exact sequence of graded modules we obtain the short exact sequence 0 → O_{P^2_k}(-d) → O_{P^2_k} → O_C → 0
21. Taking the long exact cohomology sequence of this sequence we obtain an isomorphism H^1(C, O_C) → H^2(P^2_k, O_{P^2_k}(-d))
22. Since the dimension of H^2(P^2_k, O_{P^2_k}(-d)) is (d - 1)(d - 2)/2 we conclude the genus of C is (d - 1)(d - 2)/2.
16. Lecture XVI: Unfortunately, I am blanking a bit on the topics we discussed in this lecture. Please let me know if I am missing some.
1. We discussed the K\"unneth formula:
1. If X and Y are affine varieties over a field k, then the product XxY of X and Y, i.e., the fibre product of X and Y over Spec(k) in the category of schemes, is corresponds to A ⊗_k B if X and Y correspond to the finite type k-algebras A and B.
2. If F and G are quasi-coherent modules over X and Y as above, then pr_1^*F ⊗ pr_2^*G on XxY corresponds to the A ⊗_k B-module M ⊗_k N if F and G correspond to M and N.
3. The point above already proves the Kunneth formula for quasi-coherent modules over affine varieties! Moreover, it is the starting point for the proof in general, using Cech cohomology.
4. Now let X and Y be general varieties and choose affine open coverings X = ⋃ U_i with n terms and Y = ⋃ V_j with m terms. Denote C(F) and C(G) the Cech complex for X and Y, F and G, and these coverings. Then we get an affine open covering XxY = ⋃ U_i x V_j with nm terms. Moreover, using the affine case we see that the modules Gamma(U_{i_0...i_p} x V_{j_0...j_p}, pr_1^*F ⊗ pr_2^*G) which occur in the Cech complex C(F, G) of this affine open covering, are tensor products of the terms in C(F) and C(G). This is good...
5. It is *not* true that C(F, G) is the total complex of the double complex.
6. A first way around this, is explained in the proof of Tag 0BED. Essentially, you carefully construct the map and then you use induction on the degree of the cohomology (using an intersting fact about cohomology of quasi-coherent modules on varieties).
7. What is true is that C(F) and C(G) are complexes associated to simplicial k-vector spaces and that C(F, G) is the complex associated to the tensor product of these simplicial vector spaces in the category of simplicial vector spaces!
8. This fact combined with some simplicial arguments can be used to give a second proof of the Kunneth formula, but I don't have a good reference for this now.
2. Fun fact, which is used in one of the proofs above: Tag 0BDY tells us that on a quasi-compact scheme with affine diagonal taking cohomology of quasi-coherent modules is a universal delta functor. More, precisely, every quasi-coherent module has an embedding into a quasi-coherent module whose higher cohomology groups are zero. (This is not a triviality because we've defined cohomology using injective resolutions in the category of all O_X-modules and "most" O_X-modules aren't quasi-coherent.)
3. Using Kunneth, we computed all cohomology groups of all invertible modules on P^1xP^1. Please try to internalize this, because it comes up a lot!
4. Using this information, we discussed the genus of a curve C lying on P^1xP^1. This means C is a closed subscheme which is also a curve (a variety of dimension 1 and hence reduced and irreducible):
1. There is an exact sequence 0 → I → O_{P^1xP^1} → O_C → 0
2. I = O_{P^1xP^1}(-a, -b) for some integers a, b because I is an invertible module (as C is locally defined by a single equation; sorry here we use a different notation from the lecture)
3. SheafHom(I, O_{P^1xP^1}) = O_{P^1xP^1}(a, b) has a nonzero element and hence we conclude a, b are both ≥ 0
4. if k is algebraically closed then a = 0 implies b = 1 and b = 0 implies a = 1 and in both cases the curve C is isomorphic to P^1 (please think this through -- of course this case is not so interesting)
5. if neither a nor b is 0 then we conclude that H^0(C, O_C) is equal to k and we conclude that the genus of C is (a - 1)(b - 1)
5. We discussed how finding a curve C of bidegree (a, b) is the same as finding a bihomogeneous polynomial F in k[X_0, X_1, Y_0, Y_1] of bidgree (a, b) which is moreover irreducible (as a polynomial).
6. To see whether C corresponding to F is smooth you can dehomogenize the equation (in both sets of variables, each in 2 possible ways for a total of 4 affine pieces covering P^1xP^1) and do the thing with derivatives we did before (the Jacobian criterion for smoothness).
7. In fact, it is true for every genus g ≥ 0 and for every field there exists at least one curve (whose genus is defined) of that genus over that field.
8. Then there was a question as to how this works when you are looking at projective curve in some projective space defined by more than one equation. The answer is that it is tricky in general to compute the genus. There are bounds on the genus in terms of the degrees and number of equations, but there isn't a general formula.
9. If however, the curve is scheme theoretically cut out by n - 1 equations F_1,...,F_{n - 1} in P^n where the degree of F_i is d_i, then there is a formula. To prove the formula you use exactness of the Koszul complex (which I briefly discussed in the lecture -- ask me again if you are interested). The formula says that 2g - 2 = d_1 d_2 ... d_{n - 1} (d_1 + ... + d_{n - 1} - n - 1) where g is the genus.
17. Lecture XVII: We started this lecture with 20 questions where you guys had to guess what scheme I had in mind. It turned out that my schemes was the degree 5 Fermat surface over the field with 7 elements. After this we discussed some other topics:
1. When we say X : F = 0 where F is a homogeneous polynomial of degree d in T_0, ..., T_n over a field or ring k then we mean either of the following two equivalent things
1. X is the unique closed subscheme of P^n_k which on each standard affine piece D_+(T_i) = Spec(k[T_0/T_i, ..., T_n/T_i]) is given by the zero scheme of the polynomial T_i^{-d}F, or
2. X is the Proj of the graded ring k[T_0, ..., T_n]/(F)
2. in the situation above, we say that X is a hypersurface of degree d
3. when we say X : F_1 = ... = F_m = 0 where F_1, ..., F_m are homogeneous polynomials in T_0, ..., T_n over a field or ring k then we mean the analogous thing to what we said above; we will sometimes refer to this as saying that F_1, ..., F_m cut out X scheme theoretically (as opposed to set theoretically)
4. Question: when can you parametrize a variety?
5. Answer: this would mean the variety is rational!
6. Fact: given a variety X of dimension n over a field k, the following are equivalent
1. there is a nonempty open U of affine n space A^n_k and an open immersion U → X,
2. X and P^n_k have isomorphic open subschemes,
3. the function field k(X) of X is a purely inseparable extension of k
If this happens we say that X is rational over k
7. Fact: given a variety X of dimension n over a field k, the following are equivalent
1. there is a nonempty open U of affine n space A^n_k and a dominant morphism U → X,
2. there is an embedding k(X) → k(t_1, ..., t_n) over k
If this happens we say that X is unirational over k
8. Nontrivial facts: there are non-unirational varieties and there are unirational varieties which are not rational
9. A field extension K/k is finitely generated as a field extension if there is a finite subset E = {a_1, ..., a_n} ⊂ K such that any subfield K' ⊂ K which contains both k and E is equal to K
10. Let K/k be a finitely generated field extension. Then there is a variety X with k(X) = K. Namely, choose E = {a_1, ..., a_n} ⊂ K as above and denote A ⊂ K the k-subalgebra generated by E. Then X = Spec(A) works.
11. Let K/k be a finitely generated field extension. Then there is a projective variety X with k(X) = K. Namely, above we have seen that there is an affine variety U with k(U) = K. Then we can embed U into A^n_k for some n. Then we denote X ⊂ P^n_k the (scheme theoretic) closure of U. Then you check: (a) X is a variety, (b) U is open dense in X. Hence k(U) = k(X) and the proof is done. Please work this out and ask questions if you get stuck.
12. Some references for the two points above: Tag 0BXM Tag 01RR
13. After this general discussion we turned to a discussion for curves. First we need to understand better how function fields correspond to actual curves. Restricting to an algebraically closed ground field k we have
1. Let K/k be a finitely generated field extension of transcendence degree 1. Then there is a unique (up to unique isomorphism) smooth projective curve X over k with k(X) = K.
2. In the situation above X is called the smooth projective model of K.
3. If K_1 → K_2 is a morphism of finitely generated field extensions of k of transcendence degree 1, and if X_1 and X_2 denote their corresponding smooth projective models, then there is a unique nonconstant morphism X_2 → X_1 which induces the given map on function field.
This means that we have an anti-equivalence of categories between finitely generated field extension of k of transcendence degree 1 and smooth projective curves over k with nonconstant morphisms. Reference: Tag 0BXX
14. Fact: if X → Y is a nonconstant morphism of smooth projective curves over k algebraically closed, then the genus of X is ≥ the genus of Y.
15. Fact: if X is a smooth projective curves over k algebraically closed of genus 0 then X is isomorphic to P^1_k.
16. Conclusion from all of the above: if k is algebraically closed, then a curve is unirational if and only if it is rational if and only if the function field of X is isomorphic to k(t). Moreover if X is smooth and projective then X is isomorphic to P^1_k.
17. Question: how can we get closer to the cohomology we know and love from our understanding of P^1 and curves as Riemann surfaces and similarly for higher dimensional varieties?
18. Answer: in order to discuss this in a first year algebraic geometry course, the best thing is probably to talk about the module of differentials and the de Rahm complex. Here are some topics you could ask about next time:
1. how do we define Ω^1_{X/S}?
2. what is the algebraic de Rham complex?
3. what is coherent duality?
4. why is the dualizing sheaf related to the modules of differentials?
5. how can we use all of this to get a cohomology theory'' H^* for smooth projective varieties over an algebraically closed field k which for a smooth projective curve X gives betti numbers 1, 2g, 1, 0, 0, ... where g = g(X) is the genus?
18. Lecture XIX: We started this lecture with 20 questions where you guys had to guess what scheme I had in mind. It turned out that my schemes was the spectrum of the dual numbers over the complex numbers. After this we discussed some other topics:
1. Let k be a field. The dual numbers over k is the k-algebra k[ε] where ε is a variable whose square is zero.
2. Let X be a scheme over a field k. Let x be a point of X whose residue field is equal to k as a k-algebra. Then x is a closed point of X, see exercises. A tangent vector at x is an element of the k-vector space which is the k-linear dual to the k-vector space m_x/(m_x)^2 where m_x ⊂ O_{X, x} is the maximal ideal.
3. Fact: the set of all morphisms Spec(k[ε]) → X over Spec(k) is in natural bijection with the set of pairs (x, θ) where x is a closed point with residue field k and θ is a tangent vector at x.
4. This can be generalized, see 0B28 and please do some exercises in 029C.
5. As you can see in the reference just given this geometric notion of tangent vectors is related to an algebraic construction of differentials.
6. If R → A is a ring map, then we can define the module of differentials Ω_{A/R} as the target of the universal R-derivation d : A → Ω_{A/R}.
7. I'm afraid now you'll actually have to read a bit of algebra in order to understand what is going to follow next. Any good book on algebra will carefully introduce d : A → Ω_{A/R} and explain its properties. In the Stacks project you can read a bit here: 00RM
8. We proved that Ω_{A/R} if A = R[x_1, ..., x_n] is a free A-module with basis d(x_1), ..., d(x_n).
9. We indicated that Ω_{A/R} if A = R[x_1, ..., x_n]/(f_1, ..., f_m) is the A-module with generators d(x_1), ..., d(x_n) and relations ∑ ∂_i(f_j)d(x_i) = 0.
10. Fact: if A is a finite type algebra over a field k which is a domain of dimension n then A is smooth over k if and only if Ω_{A/k} is a finite locally free A-module of rank n.
11. Given a ring map R → A the de Rham complex of A over R is the differential graded R-algebra Ω_{A/R}^* whose terms are the exterior powers of Ω_{A/R} and whose differential is given by d(a_0 d(a_1) ∧ ... ∧ d(a_p)) = d(a_0) ∧ d(a_1) ∧ ... ∧ d(a_p). Please look this up in a good algebra book. In the Stacks project you can look here 07HX for the absolute case (namely where R = Z) and you can easily generalize this to the general case (you can use the lemma following the remark in the Stacks project to do this if you like).
12. The de Rham cohomology H^*_{dR}(A/R) of A over R is the cohomology of the de Rham complex of A over R.
13. In the rest of the lecture we computed the de Rham cohomology of A over R in a few cases:
1. If A = R we get H^0_{dR}(A/R) = A and zero in all other degrees
2. If k is a field of characteristic zero, then H^0_{dR}(k[x]/k) = k and we get zero in all other degrees
3. If k is a field of characteristic p, then H^0_{dR}(k[x]/k) = k[x^p] and H^1_{dR}(k[x]/k) = k[x^p]x^{p - 1}d(x) and we get zero in all other degrees
4. If k is a field of characteristic zero and A = k[x, x^{-1}] then we get H^0_{dR}(A/k) = k and H^1_{dR}(A/k) = k x^{-1}d(x) = k dlog(x) where dlog(x) is just a shorthand for the element x^{-1}d(x) in Ω_{A/k}.
19. Lecture XIX: The topic of this lecture was the relationship between schemes of finite type over the complex numbers C and the usual topology on C.
1. Denote Sch/C the category of schemes of finite type over C
2. Denote Top the category of topological spaces
3. A map f : X → Y of Top will be called an open embedding if f is a homeomorphism onto an open subset of Y
4. A map f : X → Y of Top will be called an closed embedding if f is a homeomorphism onto a closed subset of Y
5. The goal of this lecture was to construct a functor F : Sch/C → Top with the following properties:
1. F(X) = X(C) functorially in X (identification as a set)
2. F(A^n_C) = C^n with the standard topology
3. F sends closed immersions to closed embeddings
4. F sends open immersions to open embeddings
6. It turns out that there is a unique functor F having these properties. This functor is often denoted X ↦ X^{an} or simply X ↦ X(C).
7. Using this topology, we can do some fun things. For example
1. we can show that X is separated if and only if X(C) is Hausdorff
2. we can show that X is proper over Spec(C) if and only if X(C) is compact and Hausdorff
3. we can consider the singular cohomology of the topological space X(C) as a (contravariant) functor on Sch/C. Sometimes this is called the Betti cohomology, so H^*_{Betti}(X) = H^*_{sing}(X(C), Z).
4. Grothendieck proved that you can recover complexified Betti cohomology H^*_{Betti}(X) ⊗_Z C as the de Rham cohomology H^*_{dR}(X/C) for smooth varieties over C and in fact this was my motivation for talking about the construction of the topology on X(C) in this lecture. We will return to de Rham cohomology in the future.
8. The construction of the functor F works more generally when you have a field C which has a Hausdorff topology such that addition, substraction, multiplication, and taking inverse are continuous (on their domains).
9. First we prove a meta result: if we can construct a functor F satisfying properties a, b, c, d on the category Aff/C of finite type affine schemes over C, then it automatically extends uniquely to the whole category Sch/C. I omit the proof; hint: first extend the construction to separated objects of Sch/C by using that you get these by glueing affine schemes along affine schemes and after that extend to all objects of Sch/C by using that you get any X by glueing affine opens along separated opens.
10. Next, we prove that any family of polynomials f_1, ..., f_m in variables x_1, ..., x_n defines a continuous map C^n → C^m by the properties of the topology on C we mentioned above.
11. For X in Aff/C choose a closed immersion X → A^n_C. Then you show that X(C) ⊂ C^n is closed in the usual topology because it is the vanishing set of some polynomials and {0} is a closed subset of C.
12. In the situation above, we can try to define the topology on X(C) to be the one induced from C^n. Let's show this is independent of the choice of closed immersion.
13. For X in Aff/C suppose we have two closed immersions X → A^n_C and X → A^m_C. Then we can consider the product closed immersion X → A^{n + m}_C. Hence we need only show that the topology τ induced from the map X(C) → C^{n + m} and the topology τ' induced from the map X(C) → C^n are the same. Clearly, the identity map of X(C) is a continuous map (X(C), τ) → (X(C), τ'). To show that (X(C), τ') → (X(C), τ) is also continuous, you can use that the additional coordinate functions x_{n + 1}, ..., x_{n + m} on A^{n + m}_C restrict to polynomials in x_1, ..., x_n on X because C[x_1, ..., x_n] surjects on to H^0(X, O_X). Hence these functions (which are continuous in the τ topology by fiat) are continuous in the τ' topology which proves what we want. (This is slightly better than what I said in the lecture.)
14. OK, so now we have a well defined topology on X(C) for any object X of Aff/C. An easy argument using suitable commutative diagrams show that this topology is functorial in other words a morphism X → Y of Aff/C is turned into a continuous map X(C) → Y(C). Similarly, it is easy to show that property b and c above hold.
15. Suppose that X = Spec(A) and U = D(h) = Spec(A_h) is a principal open. Property d says that U(C) should be identified with an open subspace in X(C). This is more or less clear: the map h : X → A^1_C turns into a continuous map h : X(C) → C. Then U(C) is the inverse image of C - {0} which is open (as the topology on C is Hausdorff). Thus U(C) → X(C) is a continuous bijection onto an open subset. This isn't quite enough to show that it is a homeomorphism; to see it you can use that U is a closed subscheme of X x A^1_C defined by the vanishing of ht - 1 where t is the coordinate on A^1_C. Then U(C) ⊂ X(C) x C is closed (by what we've already shown about the topology so far) and you show that using t = h^{-1} on the complement of (h = 0) in X(C) gives a continuous inverse. (This is slightly better than in the lecture as we argue directly on X and we don't use an embedding in affine n space.)
16. Now there is a theoretical argument that shows in order to esthablish d for all open immersions in Aff/C, it suffices to prove d for the open immersions of principal opens. This can be a confusing and long winded thing and is better done on a napkin than here in these notes.
20. Lecture XX: The question that motivated this lecture was: Have we proved in the lectures that Pic(P^1 x P^1) = Z ⊕ Z? The answer is that we haven't proved it or rather we haven't discussed it; see the end of this lecture for one possible proof. Motivated by this question we discussed the following:
1. Fact: given a variety X we have Pic(X x P^1) = Pic(X) ⊕ Z. This is not actually that easy to show with the tools we have developed so far (it is also not that hard). The case of X = P^1 is discussed at the end.
2. Let X and Y be schemes over a field k. Denote X x Y the fibre product over Spec(k). Then pulling back by the projection maps X x Y → X and X x Y → Y we obtain a map Pic(X) ⊕ Pic(Y) → Pic(X x Y).
3. If in the situation above X has a k-rational point x_0 and Y has a k-rational point y_0, then the inclusion morphisms X = X x {y_0} → X x Y and Y = {x_0} x Y → X x Y define a map Pic(X x Y) → Pic(X) ⊕ Pic(Y) which is inverse to the map in the previous point. Hence in this case Pic(X x Y) = Pic(X) ⊕ Pic(Y) ⊕ EXTRA for some abelian group EXTRA.
4. Here during the lecture we had a short discussion of how Pic is a quadratic functor on the category of smooth projective varieties and how the same is true for H^2( - , Z) on the category of connected pointed (reasonable) topological spaces.
5. The discussion above in particular applies with X and Y are varieties and k is algebraically closed (because then we always have k-rational points).
6. Warning: EXTRA can be nonzero even if X and Y are smooth projective varieties over an algebraically closed field.
7. Question: what is an example?
8. Answer: Let X = Y = C be a smooth projective curve over an algebraically closed field k of genus g_C > 0. Then EXTRA is nonzero.
9. Consider the diagonal Δ ⊂ C x C. This is an effective Cartier divisor, i.e., its sheaf of ideals I ⊂ O_{C x C} is an invertible module on C x C. Denote O_{C x C}(Δ) the dual invertible module, i.e., the inverse in Pic(C x C) of I.
10. Please read enough about (effective) Cartier divisors in order to understand the discussion above and below about the relationship between divisors on (smooth) varieties and their associated invertible modules. For example Hartshorne has a section on this (and you don't need to read all of it). In the Stacks project you can look at (parts of) 01WQ 0C4S 0B3Q
11. Claim: O_{C x C}(Δ) is an element of Pic(C x C) which is not contained in the summand Pic(C) ⊕ Pic(C) constructed above provided g_C > 0.
12. To prove the claim choose a base point c_0 in C. (In other words, let c_0 be a k-rational point on C.) Then using c_0 in the construction of the inverse to the inclusion map Pic(C) ⊕ Pic(C) → Pic(C x C) given above, we obtain: If the claim is wrong, then O_{C x C}(Δ) must be isomorphic to pr_1^*O_C(c_0) ⊗ pr_2^*O_C(c_0).
13. The proof of the previous fact was done using a picture: namely the diagonal Δ intersections the curve C x {c_0} in the point (c_0, c_0). Hence the pullback of the divisor Δ on the surface C x C via the inclusion C = C x {c_0} → C x C is the divisor c_0 on C. This then implies the corresponding relation in the Picard group (see for example Definition 01WV and Lemmas 02OO and 0C4U).
14. Sublemma: \dim H^0(C, O_C(c_0) = 1 if g_C > 0 and \dim H^0(C, O_C(c_0) = 2 if g_C = 0.
15. Assuming the sublemma we conclude using Kunneth that H^0(C x C, pr_1^*O_C(c_0) ⊗ pr_2^*O_C(c_0)) is one dimensional when g_C > 0. Thus there is a unique (up to scale) nonvanishing section σ of this invertible module. Namely, σ is the product of the pullback of the unique (up to scale) nonvanishing section of O_C(c_0) via the two projections. The vanishing locus of σ is therefore {c_0} x C ∪ C x {c_0}. Since O_{C x C}(Δ) has a canonical section vanishing exactly along Δ we conclude that the claim is true.
16. Proof of the sublemma. A global section of O_C(c_0) is an element f of k(C) wich has no poles except possibly at c_0 where the pole order is at most 1. An example is f = 1. Now if the dimension of H^0(C, O_C(c_0) is > 1, then we can find a nonconstant f with this property. Such an f would correspond to a nonconstant morphism f : C → P^1_k by the equivalence of categories we talked about in Lecture XVII. The condition on the pole orders implies that f^*(∞) = c_0 where here we are using pullbacks of divisors on curves that we will discuss below. This then finally implies that f has degree 1, in other words f is a birational maps, in other words, C has genus 0 as desired.
17. Discussion of facts about pullbacks of divisors via a nonconstant morphism f : X → Y of smooth projective curves over an algebraically closed field k; you should really try to know these facts and look up their proofs in Hartshorne for example.
1. f^* : Div(Y) → Div(X) is defined by sending D = ∑ n_i[y_i] to f^*D = ∑ n_if^*[y_i]
2. f^*[y] = ∑ e_x [x] where the sum is over the points x mapping to y
3. e_x is the ramification index of f at x which is defined as the ramification index of the map of dvrs O_{Y, y} → O_{X, x}
4. if A → B is an extension of dvrs then the ramification index of B over A is the integer e ≥ 0 such that π_A = (unit) π_B^e, see 09E4
5. FACT (sum e_i f_i = n): for any point y of Y we have ∑ e_x = [k(X) : k(Y)] where the sum is over the points x in X mapping to y.
6. Definition: the degree of f : X → Y is [k(X) : k(Y)]
7. We conclude that deg(f^*D) = deg(f) deg(D) for any divisor D on Y
8. Fact: f^*div_Y(g) = div_X(g) where g is an element of k(Y) which we may also consider as an element of k(X)
9. Cor: f^* induces a map f^* : Cl(Y) → Cl(X)
10. Fact: f^* on Weil divisor class groups agrees with f^* on Picard groups via the identification of those we've seen in a previous lecture. All this is saying is the fact that pullback of (effective) Cartier divisors and taking the associated invertible modules commute which we saw above in a more general setting.
18. Proof of Pic(P^1 x P^1) = Z ⊕ Z. We've seen that Pic(P^1_k) = Z and hence the discussion above gives a direct sum decomposition Pic(P^1 x P^1) = Z ⊕ Z ⊕ Extra. Thus it suffices to show that Pic(P^1 x P^1) has at most two generators. For this the easiest thing is to prove the following two things
1. Show that for a smooth variety X, such as P^1 x P^1, we have Pic(X) = Cl(X) where Cl(X) is the Weil divisor class group. This you prove in exactly the same way as we proved this fact for nonsingular curves; see for example 0BE9 and use that the local rings of a smooth variety are regular rings and in particular UFDs.
2. Show that if X has an open subvariety U with Pic(U) = Cl(U) = 0 then Cl(X) is generated as an abelian group by the classes of the irreducible components Z of the complement of U which have codimension 1 in X. Namely, if D is a Weil divisor, then we can write D ∩ U = div_U(f) for some f in k(U) = k(X) and then we see that D - div_X(f) is a Weil divisor rationally equivalent to D supported on X - U and hence D is rationally equivalent to a Weil divisor which is a sum of the irreducible components of the complement of U in X.
Now you apply this to the open U = A^1 x A^1 of P^1 x P^1. The same argument proves more generally that Pic(P^{n_1} x P^{n_2} x ... x P^{n_r}) = Z^r.
21. Lecture XXI: In this lecture we circled back to the de Rham complex.
1. First there was a question: how does one recognize an effective Cartier divisor?
2. Answer: the easiest thing to remember is that if X is a smooth variety, then any closed subvariety of codimension 1 is an effective Cartier divisor.
3. The reason for the answer: local rings of a smooth variety are UFDs and this is what locally gives you the single equation for the codimension 1 subvariety...
4. Example: the diagonal of a smooth projective curve over an algebraically closed field is an effective Cartier divisor.
5. The de Rham complex of a scheme over another scheme.
1. Let X → S be a morphism of schemes.
2. The module of differentials of X/S is a quasi-coherent O_X-module Ω_{X/S} which comes equipped with an S-derivation d_{X/S} : O_X → Ω_{X/S}
3. This S-derivation is the universal S-derivation, see 01UM if you're interested.
4. You can construct Ω_{X/S} and d_{X/S} by using affine locally the construction in algebra and then glueing. If you want to do this you have to show the following two things:
1. S^{-1} Ω_{A/R} = Ω_{S^{-1}A/R} if S is a multiplicative subset of A, and
2. if R → A factors as R → S^{-1} R → A for some multiplicative subset S of R, then Ω_{A/R} = Ω_{A/S^{-1}R}
5. Another option for constructing Ω_{X/S} is to use Ω_{X/S} = Δ^*(I) where I ⊂ O_{X x_S X} is the ideal sheaf of the diagonal (which is a closed immersion if X is separated over S so let's assume that here; you can extend this easily to the general case by working with a suitable open of X x_S X).
6. Once you have constructed Ω_{X/S} and d_{X/S} it is a simple matter to construct the de Rham complex of X/S by taking exterior powers and exterior derivation.
6. We computed that for X = P^1_k and S = Spec(k) where k is any ring, then Ω_{P^1_k/k} is isomorphic to O(-2). We did this in two ways
1. If x is a coordinate on P^1, then by the glueing method we have d(x) is a generator of Ω_{P^1_k/k} over the affine open with coordinate x but d(x) = -y^{-2}d(y) on the other affine open with coordinate y = x^{-1}. Hence d(x) is a meromorphic section of the invertible module Ω_{P^1_k/k} which has a pole of order exactly 2 along the point at infinity. Thus we get the result.
2. Using the description Ω_{X/S} = Δ^*(I) and using that O_{P^1 x P^1}(Δ) = O_{P^1 x P^1}(1, 1) we get I = O_{P^1 x P^1}(-1, -1) and hence the pullback of this givens O(-2). [Here we noted that the diagonal Δ in P^1 x P^1 is the zero locus of the section X_0Y_1 - X_1Y_0 of O_{P^1 x P^1}(1, 1) to get the correct invertible module.]
7. For a smooth projective curve E of genus 1 over an algebraically closed field k we have that Ω_{E/k} is isomorphic to O_E. We didn't prove this completely but we computed that Ω_{A/k} is free with basis element (1/y)d(x) when A = k[x, y]/(y^2 - x^3 + 1) and the characteristic of k is not 2 or 3.
22. Lecture XXII: In this lecture we tried to compute the de Rham cohomology of a couple of smooth projective curves.
1. For any morphism of schemes X → S there is a de Rham complex Ω_{X/S}.
2. For any topological space X and bounded below complex of abelian sheaves F^* on X there is a way to define the cohomology of F^* on X:
1. Choose an injective resolution F^* → I^* as defined in Definition 013I
2. Set H^n(X, F^*) = H^n(Γ(X, I^*)).
There is a way to do this for complexes of abelian sheaves which are not bounded below, but we'll not discuss this in these lectures.
3. Question: Shouldn't you resolve F^* by a complex of complexes of injective sheaves?
4. No, what was said above is the definition, please understand this first. For example, read about it in Section 013G.
5. Yes, in some sense you are correct. Namely, suppose we choose for each p an injective resolution F^p → I^{p, *}, then for each p we may choose a map of complexes I^{p, *} → I^{p + 1, *} compatible with the map F^p → F^{p + 1, *}. It turns out that you can do this in such a way that the compositions I^{p, *} → I^{p + 1, *} → I^{p + 2, *} are zero (as maps of complexes). Then I^{*, *} is a double complex (a complex of complexes as you were asking for above) and as our injective resolution (as defined above) of F^* we can take
F^* → Tot(I^{*, *})
where Tot(I^{*, *}) is the totalization of the double complex: it is the complex which in degree n has the direct sum of I^{p, q} with p + q = n.
6. The discussion above immediately gives that there exists a spectral sequence with E_1^{p, q} = H^q(X, F^p) with differentials d_1^{p, q} : E_1^{p, q} → E_1^{p + 1, q} converging to H^{p + q(X, F^*).
7. Going back to the de Rham complex Ω_{X/S} on our scheme X we define the de Rham cohomology of X/S H^n_{dR}(X/S) as the cohomology of the complex Ω_{X/S} on X
8. In the case of the de Rham complex, the spectral sequence above is the Hodge to de Rham spectral sequence and it looks like E_1^{p, q} = H^q(X, Ω_{X/S}^p) converging to H^n_{dR}(X/S).
9. We worked out what this spectral sequence does for P^1_k/k and we found that it degenerates at E_1 because the only nonzero terms are E_1^{0, 0} = H^0(P^1, O) and E_1^{1, 1} = H^1(P^1, Ω^1) which are both equal to k. Thus we see that H^n_{dR}(P^1/k) is k in degrees 0, 2 and zero else.
10. Let X be a smooth projective curve of genus 1 over an algebraically closed field k and assume we have proven that Ω_{X/k} is isomorphic to O_X (this is always true, but our arguments from last time didn't completely show this). Then we found that the Hodge to de Rham spectral sequence has nonvanishing terms E_1^{p, q} for (p, q) in {(0, 0), (0, 1), (1, 0), (1, 1)}. The differential d_1^{0, 0} is zero because the (exterior) derivative of a constant function is zero. However, it is not completely obvious that the other differential d : H^1(X, O_X) → H^1(X, Ω^1_{X/k}) is zero.
11. More generally, for a smooth projective curve of genus g over an algebraically closed field k, the Hodge to de Rham spectral sequence has nonvanishing terms E_1^{p, q} for (p, q) in {(0, 0), (0, 1), (1, 0), (1, 1)} having dimensions 1, g, ?, ?. To see that we get betti numbers 1, 2g, 1 and Poincare duality for de Rham cohomology (as we expect from the known picture over the complex numbers) we discussed the following:
1. There should be a duality on the de Rham complex itself which should be a consequence of coherent duality on X.
2. For a general smooth proper variety X/k of dimension n this coherent duality should give a duality between H^q(X, Ω^p_{X/k}) and H^{n - q}(X, Ω^{n - p}_{X/k}
3. Once we have this for curves we find the Hodge to de Rham spectral sequence has nonvanishing terms E_1^{p, q} for (p, q) in {(0, 0), (0, 1), (1, 0), (1, 1)} having dimensions 1, g, g, 1. Then we'll still need to show that the differential d : H^1(X, O_X) → H^1(X, Ω^1_{X/k}) is zero.
4. The vanishing of the differentials on the E_1 page of the Hodge to de Rham spectral sequence does not always hold. When we say degeneration of Hodge-to-de-Rham'' we are considering the statement that it does hold. In characteristic zero (i.e., when k contains the rational numbers), then it always does hold for smooth and proper X. The first proof of this fact used Hodge theory. A marvelous proof of this fact by Deligne and Illusie uses characteristic p methods to prove it in characteristic zero (which is strange because the result doesn't always hold in characteristic p).
12. The Hodge to de Rham spectral sequence for a smooth affine variety X = Spec(A) over k shows that there is a canonical isomorphism H^n_{dR}(X/k) = H^n(Ω^*_{A/k}) thereby linking back the global construction in this lecture to the algebraic one earlier in the lectures. In particular, our earlier calculations show that de Rham cohomology of the affine line in characteristic p is infinite dimensional! It turns out that for affine singular varieties in characteristic zero the de Rham cohomology can also be infinite dimensional.
13. Recall that Grothendieck proved that for a smooth variety X (not necessarily projective or proper) over the complex numbers C the algebraic de Rham cohomology H^n_{dR}(X/C) is the same as the singular cohomology with C coefficients of X(C) endowed with the usal topology. In particular, H^n_{dR}(X/C) is finite dimensional. The same is true for any smooth variety over any field of characteristic zero.
23. Lecture XXIII: In this lecture we talked about various and sundry.
1. First we talked a bit more about injective resolutions as defined in the lecture XXII. The question asked was: why given a left exact functor F : A → B of abelian categories and a quasi-isomorphism α: I^* → J^* of bounded below complexes of injective objects of A, is it true that F(I^*) → F(J^*) is a quasi-isomorphism?
2. Answer 1: consider the cone C(α) on α which fits into a triangle I^* → J^* → C(α) → I^*[1], see Section 014D. Then you show: (a) the triangle gives rise to a long exact sequence of cohomology objects in A, (b) the triangle F(I^*) → F(J^*) → F(C(α)) → F(I^*)[1] gives rise to a long exact sequence of cohomology objects in B, (c) the cohomology objects of C(α) are zero as α is a quasi-isomorphism, (d) C(α) is a bounded below complex of injectives, (e) F(C(α)) = C(F(α)). Having said all of this it remains to show that for a bounded below acyclic (all coh is 0) complex of injectives I^* we have F(I^*) = 0. To see this last fact you decompose any complex I^* like this into a sequence of split short exact sequence for which the statement is obvious.
3. Answer 2: Try to show that α: I^* → J^* as above has an inverse up to homotopy. The the same will be true for F(α) and we get what we want.
4. Then we talked about blowing up. Please take a look at
1. Harthorne's discussion of blow ups or Ravi's notes, or Shavarevich, etc
2. Affine blow up algebras as discussed in Section 052P
3. Blowing up as discussed in Section 01OF
5. Specifically, we tried to blow up the spectrum of k[x, y, z]/(xy - z^2) in the ideal (x, z) and we found that we got a nonsingular (smooth) scheme over k.
6. I suggest doing some blowing ups yourself. You can do standard blowing ups such as blowing up the spectrum of k[x, y] in the ideal (x, y) but you can also do crazy things such as blowing up the ideal generated by ε in the spectrum of the dual numbers.
24. Lectures XXIV and XXV: In these lectures we talked about
1. the Euler characteristic of a coherent sheaf on a proper varierty over an field,
2. the relationship between Hilbert polynomials of finite graded modules over graded polynomial rings and Hilbert polynomials of the associated coherent sheaves on projective space (in terms of Euler characteristics),
3. Riemann-Roch on smooth projective curves over algebraically closed fields formulated as a relationship between the Euler characteristic of an invertible sheaf, its degree, and the Euler characteristic of the structure sheaf,
4. Riemann-Roch on a smooth projective curve where we plug in everything we now know about the genus, duality, etc,
5. Riemann-Hurewitz for a nonconstant and separable morphism between smooth projective curves over an algebraically closed field where we worked out what the local structure of the morphism implies about the local vanishing of the induced map on Ωs...
25. That's all folks!

Homework:

1. Due Thursday, Jan 31. Choose and do 2 exercises from 027A and do 028P 028Q 028R 028W
2. Due Thursday, Feb 7. Do 02CJ, 0E9D, 0E9E, 0E9F, and give an example of a real valued function on the real line which is a quotient of polynomials but not a polynomial. I'm choosing these particular problems in the hope that these will generate confusion and therefore questions in the lectures.
3. Due Thursday, Feb 14. Prove the algebra statements (*) and (**) from Lecture VI above. Do exercises 078S and 02DU
4. Due Thursday, Feb 21. Do the following
1. Let K/k be a finitely generated extension of fields. Show that there exists a variety X over k such that k(X) is isomorphic to K.
2. Let A ⊂ B be an extension of domains such that (i) B is a finitely generated A-algebra and (ii) the incusion induces an isomorphism of fraction fields. Show that there exists some nonzero f in A such that A_f = B_f.
3. Let X and Y be varieties over the same ground field k. Show that if X and Y are birational, i.e., if k(X) and k(Y) are isomorphic as extensions of k, then there exist nonempty affine open subschemes U ⊂ X and V &subl; Y which are isomorphic as varieties over k. (Hint: use the previous exercise twice.)
4. Let k be your favorite algebraically closed field. Find a surface X over k such that there does not exist any nonconstant morphism from the affine line over k into X. Here the affine line over k is just A^1_k = Spec(k[t]) as usual.
5. Find a surjective morphism from A^1_k to P^1_k. (This is one of my favorite questions; it'll be a little bit hard to answer for you, because you'll have to think or read about what it means to map into projective space.)
5. Due Thursday, Feb 28. Do the following
1. Let X be a scheme over a field k. Show that k-rationl points of X correspond 1-to-1 with points x in X such that the map from k to the residue field of x is an isomorphism. Moreover, show that such points are closed points of X.
2. Do exercise 0CYH
6. Due Thursday, March 7. Do 5 of the exercises of 0D8P
7. Due Thursday, March 14. Do 5 of the exercises of 0DAI
8. Due Thursday, March 28. Explicitly find a smooth curve of genus 6 over the rational numbers. Explicitly find a smooth surface of degree 4 in P^3 over the field with 2 elements.
9. Due Thursday, April 4.
1. Let X be a scheme over a field k. Let x be a point of X whose residue field is equal to k as a k-algebra. Show that x is a closed point.
2. Do two exercises from Section 029C
10. Due Thursday, April 13.
1. Do three exercised from 0293
2. Do exercise 02EO
11. The exercises for Thursday, April 18.
1. Prepare a specific algebraic goemetry question to ask during the lecture on Thursday.
2. If you are an undergraduate: think about a topic to write about for your final paper and email me about it. I would prefer the topic to be very close to what we talked about during the lectures -- the best is probably if you just carefully work out a bit of theory and/or examples that came up. I intend to have a meeting with you about this, so please suggest dates/times when you are free.
3. If you are a graduate student and registered for this course: please email me with dates/times where we can have our oral exam about the material in this course. If you have questions about how this will work, then please ask during the lectures.

References:

Algebraic Geometry, by Robin Hartshorne springer link

Basic Algebraic Geometry I, II, by Igor Shavarevich springer link springer link