Exercises

Algebra

This first section just contains some assorted questions.

Exercise

Let \(A\) be a ring, and \({\mathfrak m}\) a maximal ideal. In \(A[X]\) let \(\tilde {\mathfrak m}_1 = ({\mathfrak m}, X)\) and \(\tilde {\mathfrak m}_2 = ({\mathfrak m}, X-1)\). Show that \[A[X]_{\tilde {\mathfrak m}_1} \cong A[X]_{\tilde {\mathfrak m}_2}.\]

Exercise

Find an example of a non Noetherian ring \(R\) such that every finitely generated ideal of \(R\) is finitely presented as an \(R\)-module. (A ring is said to be coherent if the last property holds.)

Exercise

Suppose that \((A, {\mathfrak m}, k)\) is a Noetherian local ring. For any finite \(A\)-module \(M\) define \(r(M)\) to be the minimum number of generators of \(M\) as an \(A\)-module. This number equals \(\dim_k M/{\mathfrak m} M = \dim_k M \otimes_A k\) by NAK.

1. Show that \(r(M \otimes_A N) = r(M)r(N)\).

2. Let \(I\subset A\) be an ideal with \(r(I) > 1\). Show that \(r(I^2) < r(I)^2\).

3. Conclude that if every ideal in \(A\) is a flat module, then \(A\) is a PID (or a field).

Exercise

Let \(k\) be a field. Show that the following pairs of \(k\)-algebras are not isomorphic:

1. \(k[x_1, \ldots, x_n]\) and \(k[x_1, \ldots, x_{n + 1}]\) for any \(n\geq 1\).

2. \(k[a, b, c, d, e, f]/(ab + cd + ef)\) and \(k[x_1, \ldots, x_n]\) for \(n = 5\).

3. \(k[a, b, c, d, e, f]/(ab + cd + ef)\) and \(k[x_1, \ldots, x_n]\) for \(n = 6\).

Remark

Of course the idea of this exercise is to find a simple argument in each case rather than applying a “big” theorem. Nonetheless it is good to be guided by general principles.

Exercise

Algebra. (Silly and should be easy.)

1. Give an example of a ring \(A\) and a nonsplit short exact sequence of \(A\)-modules \[0 \to M_1 \to M_2 \to M_3 \to 0.\]

2. Give an example of a nonsplit sequence of \(A\)-modules as above and a faithfully flat \(A \to B\) such that \[0 \to M_1\otimes_A B \to M_2\otimes_A B \to M_3\otimes_A B \to 0.\] is split as a sequence of \(B\)-modules.

Exercise

Suppose that \(k\) is a field having a primitive \(n\)th root of unity \(\zeta\). This means that \(\zeta^n = 1\), but \(\zeta^m\not = 1\) for \(0 < m < n\).

1. Show that the characteristic of \(k\) is prime to \(n\).

2. Suppose that \(a \in k\) is an element of \(k\) which is not an \(d\)th power in \(k\) for any divisor \(d\) of \(n\) for \(n \geq d > 1\). Show that \(k[x]/(x^n-a)\) is a field. (Hint: Consider a splitting field for \(x^n-a\) and use Galois theory.)

Exercise

Let \(\nu : k[x]\setminus \{0\} \to {\mathbf Z}\) be a map with the following properties: \(\nu(fg) = \nu(f) + \nu(g)\) whenever \(f\), \(g\) not zero, and \(\nu(f + g) \geq min(\nu(f), \nu(g))\) whenever \(f\), \(g\), \(f + g\) are not zero, and \(\nu(c) = 0\) for all \(c\in k^*\).

1. Show that if \(f\), \(g\), and \(f + g\) are nonzero and \(\nu(f) \not = \nu(g)\) then we have equality \(\nu(f + g) = min(\nu(f), \nu(g))\).

2. Show that if \(f = \sum a_i x^i\), \(f\not = 0\), then \(\nu(f) \geq min(\{i\nu(x)\}_{a_i\not = 0})\). When does equality hold?

3. Show that if \(\nu\) attains a negative value then \(\nu(f) = -n \deg(f)\) for some \(n\in {\mathbf N}\).

4. Suppose \(\nu(x) \geq 0\). Show that \(\{f \mid f = 0, \ or\ \nu(f) > 0\}\) is a prime ideal of \(k[x]\).

5. Describe all possible \(\nu\).

Let \(A\) be a ring. An idempotent is an element \(e \in A\) such that \(e^2 = e\). The elements \(1\) and \(0\) are always idempotent. A nontrivial idempotent is an idempotent which is not equal to zero. Two idempotents \(e, e' \in A\) are called orthogonal if \(ee' = 0\).

Exercise

Let \(A\) be a ring. Show that \(A\) is a product of two nonzero rings if and only if \(A\) has a nontrivial idempotent.

Exercise

Let \(A\) be a ring and let \(I \subset A\) be a locally nilpotent ideal. Show that the map \(A \to A/I\) induces a bijection on idempotents. (Hint: It may be easier to prove this when \(I\) is nilpotent. Do this first. Then use “absolute Noetherian reduction” to reduce to the nilpotent case.)

Colimits

Definition

A directed set is a nonempty set \(I\) endowed with a preorder \(\leq\) such that given any pair \(i, j \in I\) there exists a \(k \in I\) such that \(i \leq k\) and \(j \leq k\). A system of rings over \(I\) is given by a ring \(A_i\) for each \(i \in I\) and a map of rings \(\varphi_{ij} : A_i \to A_j\) whenever \(i \leq j\) such that the composition \(A_i \to A_j \to A_k\) is equal to \(A_i \to A_k\) whenever \(i \leq j \leq k\).

One similarly defines systems of groups, modules over a fixed ring, vector spaces over a field, etc.

Exercise

Let \(I\) be a directed set and let \((A_i, \varphi_{ij})\) be a system of rings over \(I\). Show that there exists a ring \(A\) and maps \(\varphi_i : A_i \to A\) such that \(\varphi_j \circ \varphi_{ij} = \varphi_i\) for all \(i \leq j\) with the following universal property: Given any ring \(B\) and maps \(\psi_i : A_i \to B\) such that \(\psi_j \circ \varphi_{ij} = \psi_i\) for all \(i \leq j\), then there exists a unique ring map \(\psi : A \to B\) such that \(\psi_i = \psi \circ \varphi_i\).

Definition

The ring \(A\) constructed in Exercise 078J is called the colimit of the system. Notation \(\colim A_i\).

Exercise

Let \((I, \geq)\) be a directed set and let \((A_i, \varphi_{ij})\) be a system of rings over \(I\) with colimit \(A\). Prove that there is a bijection \[\Spec(A) = \{(\mathfrak p_i)_{i \in I} \mid \mathfrak p_i \subset A_i \text{ and } \mathfrak p_i = \varphi_{ij}^{-1}(\mathfrak p_j)\ \forall i \leq j\} \subset \prod\nolimits_{i \in I} \Spec(A_i)\] The set on the right hand side of the equality is the limit of the sets \(\Spec(A_i)\). Notation \(\lim \Spec(A_i)\).

Exercise

Let \((I, \geq)\) be a directed set and let \((A_i, \varphi_{ij})\) be a system of rings over \(I\) with colimit \(A\). Suppose that \(\Spec(A_j) \to \Spec(A_i)\) is surjective for all \(i \leq j\). Show that \(\Spec(A) \to \Spec(A_i)\) is surjective for all \(i\). (Hint: You can try to use Tychonoff, but there is also a basically trivial direct algebraic proof based on Algebra, Lemma 00E7.)

Exercise

Let \(A \subset B\) be an integral ring extension. Prove that \(\Spec(B) \to \Spec(A)\) is surjective. Use the exercises above, the fact that this holds for a finite ring extension (proved in the lectures), and by proving that \(B = \colim B_i\) is a directed colimit of finite extensions \(A \subset B_i\).

Exercise

Let \((I, \geq)\) be a directed set. Let \(A\) be a ring and let \((N_i, \varphi_{i, i'})\) be a directed system of \(A\)-modules indexed by \(I\). Suppose that \(M\) is another \(A\)-module. Prove that \[\colim_{i\in I} M \otimes_A N_i\cong M \otimes_A \Big( \colim_{i\in I} N_i\Big).\]

Definition

A module \(M\) over \(R\) is said to be of finite presentation over \(R\) if it is isomorphic to the cokernel of a map of finite free modules \(R^{\oplus n} \to R^{\oplus m}\).

Exercise

Prove that any module over any ring is

1. the colimit of its finitely generated submodules, and

2. in some way a colimit of finitely presented modules.

Additive and abelian categories

Exercise

Let \(k\) be a field. Let \(\mathcal{C}\) be the category of filtered vector spaces over \(k\), see Homology, Definition 0121 for the definition of a filtered object of any category.

1. Show that this is an additive category (explain carefully what the direct sum of two objects is).

2. Let \(f : (V, F) \to (W, F)\) be a morphism of \(\mathcal{C}\). Show that \(f\) has a kernel and cokernel (explain precisely what the kernel and cokernel of \(f\) are).

3. Give an example of a map of \(\mathcal{C}\) such that the canonical map \(\Coim(f) \to \Im(f)\) is not an isomorphism.

Exercise

Let \(R\) be a Noetherian domain. Let \(\mathcal{C}\) be the category of finitely generated torsion free \(R\)-modules.

1. Show that this is an additive category.

2. Let \(f : N \to M\) be a morphism of \(\mathcal{C}\). Show that \(f\) has a kernel and cokernel (make sure you define precisely what the kernel and cokernel of \(f\) are).

3. Give an example of a Noetherian domain \(R\) and a map of \(\mathcal{C}\) such that the canonical map \(\Coim(f) \to \Im(f)\) is not an isomorphism.

Exercise

Give an example of a category which is additive and has kernels and cokernels but which is not as in Exercises 057Y and 057Z.

Tensor product

Tensor products are introduced in Algebra, Section 00CV. Let \(R\) be a ring. Let \(\text{Mod}_R\) be the category of \(R\)-modules. We will say that a functor \(F : \text{Mod}_R \to \text{Mod}_R\)

1. is additive if \(F : \Hom_R(M, N) \to \Hom_R(F(M), F(N))\) is a homomorphism of abelian groups for any \(R\)-modules \(M, N\), see Homology, Definition 00ZY.

2. \(R\)-linear if \(F : \Hom_R(M, N) \to \Hom_R(F(M), F(N))\) is \(R\)-linear for any \(R\)-modules \(M, N\),

3. right exact if for any short exact sequence \(0 \to M_1 \to M_2 \to M_3 \to 0\) the sequence \(F(M_1) \to F(M_2) \to F(M_3) \to 0\) is exact,

4. left exact if for any short exact sequence \(0 \to M_1 \to M_2 \to M_3 \to 0\) the sequence \(0 \to F(M_1) \to F(M_2) \to F(M_3)\) is exact,

5. commutes with direct sums, if given a set \(I\) and \(R\)-modules \(M_i\) the maps \(F(M_i) \to F(\bigoplus M_i)\) induce an isomorphism \(\bigoplus F(M_i) = F(\bigoplus M_i)\).

Exercise

Let \(R\) be a ring. With notation as above.

1. Give an example of a ring \(R\) and an additive functor \(F : \text{Mod}_R \to \text{Mod}_R\) which is not \(R\)-linear.

2. Let \(N\) be an \(R\)-module. Show that the functor \(F(M) = M \otimes_R N\) is \(R\)-linear, right exact, and commutes with direct sums,

3. Conversely, show that any functor \(F : \text{Mod}_R \to \text{Mod}_R\) which is \(R\)-linear, right exact, and commutes with direct sums is of the form \(F(M) = M \otimes_R N\) for some \(R\)-module \(N\).

4. Show that if in (3) we drop the assumption that \(F\) commutes with direct sums, then the conclusion no longer holds.

Flat ring maps

Exercise

Let \(S\) be a multiplicative subset of the ring \(A\).

1. For an \(A\)-module \(M\) show that \(S^{-1}M = S^{-1}A \otimes_A M\).

2. Show that \(S^{-1}A\) is flat over \(A\).

Exercise

Find an injection \(M_1 \to M_2\) of \(A\)-modules such that \(M_1\otimes N \to M_2 \otimes N\) is not injective in the following cases:

1. \(A = k[x, y]\) and \(N = (x, y) \subset A\). (Here and below \(k\) is a field.)

2. \(A = k[x, y]\) and \(N = A/(x, y)\).

Exercise

Give an example of a ring \(A\) and a finite \(A\)-module \(M\) which is a flat but not a projective \(A\)-module.

Remark

If \(M\) is of finite presentation and flat over \(A\), then \(M\) is projective over \(A\). Thus your example will have to involve a ring \(A\) which is not Noetherian. I know of an example where \(A\) is the ring of \({\mathcal C}^\infty\)-functions on \({\mathbf R}\).

Exercise

Find a flat but not free module over \({\mathbf Z}_{(2)}\).

Exercise

Flat deformations.

1. Suppose that \(k\) is a field and \(k[\epsilon]\) is the ring of dual numbers \(k[\epsilon] = k[x]/(x^2)\) and \(\epsilon = \bar x\). Show that for any \(k\)-algebra \(A\) there is a flat \(k[\epsilon]\)-algebra \(B\) such that \(A\) is isomorphic to \(B/\epsilon B\).

2. Suppose that \(k = {\mathbf F}_p = {\mathbf Z}/p{\mathbf Z}\) and \[A = k[x_1, x_2, x_3, x_4, x_5, x_6]/ (x_1^p, x_2^p, x_3^p, x_4^p, x_5^p, x_6^p).\] Show that there exists a flat \({\mathbf Z}/p^2{\mathbf Z}\)-algebra \(B\) such that \(B/pB\) is isomorphic to \(A\). (So here \(p\) plays the role of \(\epsilon\).)

3. Now let \(p = 2\) and consider the same question for \(k = {\mathbf F}_2 = {\mathbf Z}/2{\mathbf Z}\) and \[A = k[x_1, x_2, x_3, x_4, x_5, x_6]/ (x_1^2, x_2^2, x_3^2, x_4^2, x_5^2, x_6^2, x_1x_2 + x_3x_4 + x_5x_6).\] However, in this case show that there does not exist a flat \({\mathbf Z}/4{\mathbf Z}\)-algebra \(B\) such that \(B/2B\) is isomorphic to \(A\). (Find the trick! The same example works in arbitrary characteristic \(p > 0\), except that the computation is more difficult.)

Exercise

Let \((A, {\mathfrak m}, k)\) be a local ring and let \(k'/k\) be a finite field extension. Show there exists a flat, local map of local rings \(A \to B\) such that \({\mathfrak m}_B = {\mathfrak m} B\) and \(B/{\mathfrak m} B\) is isomorphic to \(k'\) as \(k\)-algebra. (Hint: first do the case where \(k \subset k'\) is generated by a single element.)

Remark

The same result holds for arbitrary field extensions \(K/k\).

The Spectrum of a ring

Exercise

Compute \(\Spec(\mathbf{Z})\) as a set and describe its topology.

Exercise

Let \(A\) be any ring. For \(f\in A\) we define \(D(f):= \{\mathfrak p \subset A \mid f \not \in \mathfrak p\}\). Prove that the open subsets \(D(f)\) form a basis of the topology of \(\Spec(A)\).

Exercise

Prove that the map \(I\mapsto V(I)\) defines a natural bijection \[\{I\subset A\text{ with }I = \sqrt{I}\} \longrightarrow \{T\subset \Spec(A)\text{ closed}\}\]

Definition

A topological space \(X\) is called quasi-compact if for any open covering \(X = \bigcup_{i\in I} U_i\) there is a finite subset \(\{i_1, \ldots, i_n\}\subset I\) such that \(X = U_{i_1}\cup\ldots U_{i_n}\).

Exercise

Prove that \(\Spec(A)\) is quasi-compact for any ring \(A\).

Definition

A topological space \(X\) is said to verify the separation axiom \(T_0\) if for any pair of points \(x, y\in X\), \(x\not = y\) there is an open subset of \(X\) containing one but not the other. We say that \(X\) is Hausdorff if for any pair \(x, y\in X\), \(x\not = y\) there are disjoint open subsets \(U, V\) such that \(x\in U\) and \(y\in V\).

Exercise

Show that \(\Spec(A)\) is not Hausdorff in general. Prove that \(\Spec(A)\) is \(T_0\). Give an example of a topological space \(X\) that is not \(T_0\).

Remark

Usually the word compact is reserved for quasi-compact and Hausdorff spaces.

Definition

A topological space \(X\) is called irreducible if \(X\) is not empty and if \(X = Z_1\cup Z_2\) with \(Z_1, Z_2\subset X\) closed, then either \(Z_1 = X\) or \(Z_2 = X\). A subset \(T\subset X\) of a topological space is called irreducible if it is an irreducible topological space with the topology induced from \(X\). This definition implies \(T\) is irreducible if and only if the closure \(\bar T\) of \(T\) in \(X\) is irreducible.

Exercise

Prove that \(\Spec(A)\) is irreducible if and only if \(Nil(A)\) is a prime ideal and that in this case it is the unique minimal prime ideal of \(A\).

Exercise

Prove that a closed subset \(T\subset \Spec(A)\) is irreducible if and only if it is of the form \(T = V({\mathfrak p})\) for some prime ideal \({\mathfrak p}\subset A\).

Definition

A point \(x\) of an irreducible topological space \(X\) is called a generic point of \(X\) if \(X\) is equal to the closure of the subset \(\{x\}\).

Exercise

Show that in a \(T_0\) space \(X\) every irreducible closed subset has at most one generic point.

Exercise

Prove that in \(\Spec(A)\) every irreducible closed subset does have a generic point. In fact show that the map \({\mathfrak p} \mapsto \overline{\{{\mathfrak p}\}}\) is a bijection of \(\Spec(A)\) with the set of irreducible closed subsets of \(X\).

Exercise

Give an example to show that an irreducible subset of \(\Spec(\mathbf{Z})\) does not necessarily have a generic point.

Definition

A topological space \(X\) is called Noetherian if any decreasing sequence \(Z_1\supset Z_2 \supset Z_3\supset \ldots\) of closed subsets of \(X\) stabilizes. (It is called Artinian if any increasing sequence of closed subsets stabilizes.)

Exercise

Show that if the ring \(A\) is Noetherian then the topological space \(\Spec(A)\) is Noetherian. Give an example to show that the converse is false. (The same for Artinian if you like.)

Definition

A maximal irreducible subset \(T\subset X\) is called an irreducible component of the space \(X\). Such an irreducible component of \(X\) is automatically a closed subset of \(X\).

Exercise

Prove that any irreducible subset of \(X\) is contained in an irreducible component of \(X\).

Exercise

Prove that a Noetherian topological space \(X\) has only finitely many irreducible components, say \(X_1, \ldots, X_n\), and that \(X = X_1\cup X_2\cup\ldots\cup X_n\). (Note that any \(X\) is always the union of its irreducible components, but that if \(X = {\mathbf R}\) with its usual topology for instance then the irreducible components of \(X\) are the one point subsets. This is not terribly interesting.)

Exercise

Show that irreducible components of \(\Spec(A)\) correspond to minimal primes of \(A\).

Definition

A point \(x\in X\) is called closed if \(\overline{\{x\}} = \{ x\}\). Let \(x, y\) be points of \(X\). We say that \(x\) is a specialization of \(y\), or that \(y\) is a generalization of \(x\) if \(x\in \overline{\{y\}}\).

Exercise

Show that closed points of \(\Spec(A)\) correspond to maximal ideals of \(A\).

Exercise

Show that \({\mathfrak p}\) is a generalization of \({\mathfrak q}\) in \(\Spec(A)\) if and only if \({\mathfrak p}\subset {\mathfrak q}\). Characterize closed points, maximal ideals, generic points and minimal prime ideals in terms of generalization and specialization. (Here we use the terminology that a point of a possibly reducible topological space \(X\) is called a generic point if it is a generic points of one of the irreducible components of \(X\).)

Exercise

Let \(I\) and \(J\) be ideals of \(A\). What is the condition for \(V(I)\) and \(V(J)\) to be disjoint?

Definition

A topological space \(X\) is called connected if it is nonempty and not the union of two nonempty disjoint open subsets. A connected component of \(X\) is a maximal connected subset. Any point of \(X\) is contained in a connected component of \(X\) and any connected component of \(X\) is closed in \(X\). (But in general a connected component need not be open in \(X\).)

Exercise

Let \(A\) be a nonzero ring. Show that \(\Spec(A)\) is disconnected iff \(A\cong B \times C\) for certain nonzero rings \(B, C\).

Exercise

Let \(T\) be a connected component of \(\Spec(A)\). Prove that \(T\) is stable under generalization. Prove that \(T\) is an open subset of \(\Spec(A)\) if \(A\) is Noetherian. (Remark: This is wrong when \(A\) is an infinite product of copies of \({\mathbf F}_2\) for example. The spectrum of this ring consists of infinitely many closed points.)

Exercise

Compute \(\Spec(k[x])\), i.e., describe the prime ideals in this ring, describe the possible specializations, and describe the topology. (Work this out when \(k\) is algebraically closed but also when \(k\) is not.)

Exercise

Compute \(\Spec(k[x, y])\), where \(k\) is algebraically closed. [Hint: use the morphism \(\varphi : \Spec(k[x, y]) \to \Spec(k[x])\); if \(\varphi({\mathfrak p}) = (0)\) then localize with respect to \(S = \{f\in k[x] \mid f \not = 0\}\) and use result of lecture on localization and \(\Spec\).] (Why do you think algebraic geometers call this affine 2-space?)

Exercise

Compute \(\Spec(\mathbf{Z}[y])\). [Hint: as above.] (Affine 1-space over \(\mathbf{Z}\).)

Localization

Exercise

Let \(A\) be a ring. Let \(S \subset A\) be a multiplicative subset. Let \(M\) be an \(A\)-module. Let \(N \subset S^{-1}M\) be an \(S^{-1}A\)-submodule. Show that there exists an \(A\)-submodule \(N' \subset M\) such that \(N = S^{-1}N'\). (This useful result applies in particular to ideals of \(S^{-1}A\).)

Exercise

Let \(A\) be a ring. Let \(M\) be an \(A\)-module. Let \(m \in M\).

1. Show that \(I = \{a \in A \mid am = 0\}\) is an ideal of \(A\).

2. For a prime \(\mathfrak p\) of \(A\) show that the image of \(m\) in \(M_\mathfrak p\) is zero if and only if \(I \not \subset \mathfrak p\).

3. Show that \(m\) is zero if and only if the image of \(m\) is zero in \(M_\mathfrak p\) for all primes \(\mathfrak p\) of \(A\).

4. Show that \(m\) is zero if and only if the image of \(m\) is zero in \(M_\mathfrak m\) for all maximal ideals \(\mathfrak m\) of \(A\).

5. Show that \(M = 0\) if and only if \(M_{\mathfrak m}\) is zero for all maximal ideals \(\mathfrak m\).

Exercise

Find a pair \((A, f)\) where \(A\) is a domain with three or more pairwise distinct primes and \(f \in A\) is an element such that the principal localization \(A_f = \{1, f, f^2, \ldots \}^{-1}A\) is a field.

Exercise

Let \(A\) be a ring. Let \(M\) be a finite \(A\)-module. Let \(S \subset A\) be a multiplicative set. Assume that \(S^{-1}M = 0\). Show that there exists an \(f \in S\) such that the principal localization \(M_f = \{1, f, f^2, \ldots \}^{-1}M\) is zero.

Exercise

Give an example of a triple \((A, I, S)\) where \(A\) is a ring, \(0 \not = I \not = A\) is a proper nonzero ideal, and \(S \subset A\) is a multiplicative subset such that \(A/I \cong S^{-1}A\) as \(A\)-algebras.

Nakayama’s Lemma

Exercise

Let \(A\) be a ring. Let \(I\) be an ideal of \(A\). Let \(M\) be an \(A\)-module. Let \(x_1, \ldots, x_n \in M\). Assume that

1. \(M/IM\) is generated by \(x_1, \ldots, x_n\),

2. \(M\) is a finite \(A\)-module,

3. \(I\) is contained in every maximal ideal of \(A\).

Show that \(x_1, \ldots, x_n\) generate \(M\). (Suggested solution: Reduce to a localization at a maximal ideal of \(A\) using Exercise 0768 and exactness of localization. Then reduce to the statement of Nakayama’s lemma in the lectures by looking at the quotient of \(M\) by the submodule generated by \(x_1, \ldots, x_n\).)

Length

Definition

Let \(A\) be a ring. Let \(M\) be an \(A\)-module. The length of \(M\) as an \(R\)-module is \[\text{length}_A(M) = \sup \{ n \mid \exists\ 0 = M_0 \subset M_1 \subset \ldots \subset M_n = M, \text{ }M_i \not = M_{i + 1} \}.\] In other words, the supremum of the lengths of chains of submodules.

Exercise

Show that a module \(M\) over a ring \(A\) has length \(1\) if and only if it is isomorphic to \(A/\mathfrak m\) for some maximal ideal \(\mathfrak m\) in \(A\).

Exercise

Compute the length of the following modules over the following rings. Briefly(!) explain your answer. (Please feel free to use additivity of the length function in short exact sequences, see Algebra, Lemma 00IV).

1. The length of \(\mathbf{Z}/120\mathbf{Z}\) over \(\mathbf{Z}\).

2. The length of \(\mathbf{C}[x]/(x^{100} + x + 1)\) over \(\mathbf{C}[x]\).

3. The length of \(\mathbf{R}[x]/(x^4 + 2x^2 + 1)\) over \(\mathbf{R}[x]\).

Exercise

Let \(A = k[x, y]_{(x, y)}\) be the local ring of the affine plane at the origin. Make any assumption you like about the field \(k\). Suppose that \(f = x^3 + x^2y^2 + y^{100}\) and \(g = y^3 - x^{999}\). What is the length of \(A/(f, g)\) as an \(A\)-module? (Possible way to proceed: think about the ideal that \(f\) and \(g\) generate in quotients of the form \(A/{\mathfrak m}_A^n= k[x, y]/(x, y)^n\) for varying \(n\). Try to find \(n\) such that \(A/(f, g)+{\mathfrak m}_A^n \cong A/(f, g)+{\mathfrak m}_A^{n + 1}\) and use NAK.)

Associated primes

Associated primes are discussed in Algebra, Section 00L9

Exercise

Compute the set of associated primes for each of the following modules.

1. \(R = k[x, y]\) and \(M = R/(xy(x + y))\),

2. \(R = \mathbf{Z}[x]\) and \(M = R/(300x + 75)\), and

3. \(R = k[x, y, z]\) and \(M = R/(x^3, x^2y, xz)\).

Here as usual \(k\) is a field.

Exercise

Give an example of a Noetherian ring \(R\) and a prime ideal \(\mathfrak p\) such that \(\mathfrak p\) is not the only associated prime of \(R/\mathfrak p^2\).

Exercise

Let \(R\) be a Noetherian ring with incomparable prime ideals \(\mathfrak p\), \(\mathfrak q\), i.e., \(\mathfrak p \not \subset \mathfrak q\) and \(\mathfrak q \not \subset \mathfrak p\).

1. Show that for \(N = R/(\mathfrak p \cap \mathfrak q)\) we have \(\text{Ass}(N) = \{\mathfrak p, \mathfrak q\}\).

2. Show by an example that the module \(M = R/\mathfrak p \mathfrak q\) can have an associated prime not equal to \(\mathfrak p\) or \(\mathfrak q\).

Ext groups

Ext groups are defined in Algebra, Section 00LO.

Exercise

Compute all the Ext groups \(\Ext^i(M, N)\) of the given modules in the category of \(\mathbf{Z}\)-modules (also known as the category of abelian groups).

1. \(M = \mathbf{Z}\) and \(N = \mathbf{Z}\),

2. \(M = \mathbf{Z}/4\mathbf{Z}\) and \(N = \mathbf{Z}/8\mathbf{Z}\),

3. \(M = \mathbf{Q}\) and \(N = \mathbf{Z}/2\mathbf{Z}\), and

4. \(M = \mathbf{Z}/2\mathbf{Z}\) and \(N = \mathbf{Q}/\mathbf{Z}\).

Exercise

Let \(R = k[x, y]\) where \(k\) is a field.

1. Show by hand that the Koszul complex \[0 \to R \xrightarrow{ \left( \begin{matrix} y \\ -x \end{matrix} \right) } R^{\oplus 2} \xrightarrow{(x, y)} R \xrightarrow{f \mapsto f(0, 0)} k \to 0\] is exact.

2. Compute \(\Ext^i_R(k, k)\) where \(k = R/(x, y)\) as an \(R\)-module.

Exercise

Give an example of a Noetherian ring \(R\) and finite modules \(M\), \(N\) such that \(\Ext^i_R(M, N)\) is nonzero for all \(i \geq 0\).

Exercise

Give an example of a ring \(R\) and ideal \(I\) such that \(\Ext^1_R(R/I, R/I)\) is not a finite \(R\)-module. (We know this cannot happen if \(R\) is Noetherian by Algebra, Lemma 08YR.)

Depth

Depth is defined in Algebra, Section 00LE and further studied in Dualizing Complexes, Section 0AVY.

Exercise

Let \(R\) be a ring, \(I \subset R\) an ideal, and \(M\) an \(R\)-module. Compute \(\text{depth}_I(M)\) in the following cases.

1. \(R = \mathbf{Z}\), \(I = (30)\), \(M = \mathbf{Z}\),

2. \(R = \mathbf{Z}\), \(I = (30)\), \(M = \mathbf{Z}/(300)\),

3. \(R = \mathbf{Z}\), \(I = (30)\), \(M = \mathbf{Z}/(7)\),

4. \(R = k[x, y, z]/(x^2 + y^2 + z^2)\), \(I = (x, y, z)\), \(M = R\),

5. \(R = k[x, y, z, w]/(xz, xw, yz, yw)\), \(I = (x, y, z, w)\), \(M = R\).

Here \(k\) is a field. In the last two cases feel free to localize at the maximal ideal \(I\).

Exercise

Give an example of a Noetherian local ring \((R, \mathfrak m, \kappa)\) of depth \(\geq 1\) and a prime ideal \(\mathfrak p\) such that

1. \(\text{depth}_\mathfrak m(R) \geq 1\),

2. \(\text{depth}_\mathfrak p(R_\mathfrak p) = 0\), and

3. \(\dim(R_\mathfrak p) \geq 1\).

If we don’t ask for (3) then the exercise is too easy. Why?

Exercise

Let \((R, \mathfrak m)\) be a local Noetherian domain. Let \(M\) be a finite \(R\)-module.

1. If \(M\) is torsion free, show that \(M\) has depth at least \(1\) over \(R\).

2. Give an example with depth equal to \(1\).

Exercise

For every \(m \geq n \geq 0\) give an example of a Noetherian local ring \(R\) with \(\dim(R) = m\) and \(\text{depth}(R) = n\).

Exercise

Let \((R, \mathfrak m)\) be a Noetherian local ring. Let \(M\) be a finite \(R\)-module. Show that there exists a canonical short exact sequence \[0 \to K \to M \to Q \to 0\] such that the following are true

1. \(\text{depth}(Q) \geq 1\),

2. \(K\) is zero or \(\text{Supp}(K) = \{\mathfrak m\}\), and

3. \(\text{length}_R(K) < \infty\).

Hint: using the Noetherian property show that there exists a maximal submodule \(K\) as in (2) and then show that \(Q = M/K\) satisfies (1) and \(K\) satisfies (3).

Exercise

Let \((R, \mathfrak m)\) be a Noetherian local ring. Let \(M\) be a finite \(R\)-module of depth \(\geq 2\). Let \(N \subset M\) be a nonzero submodule.

1. Show that \(\text{depth}(N) \geq 1\).

2. Show that \(\text{depth}(N) = 1\) if and only if the quotient module \(M/N\) has \(\text{depth}(M/N) = 0\).

3. Show there exists a submodule \(N' \subset M\) with \(N \subset N'\) of finite colength, i.e., \(\text{length}_R(N'/N) < \infty\), such that \(N'\) has depth \(\geq 2\). Hint: Apply Exercise 0CSZ to \(M/N\) and choose \(N'\) to be the inverse image of \(K\).

Exercise

Let \((R, \mathfrak m)\) be a Noetherian local ring. Assume that \(R\) is reduced, i.e., \(R\) has no nonzero nilpotent elements. Assume moreover that \(R\) has two distinct minimal primes \(\mathfrak p\) and \(\mathfrak q\).

1. Show that the sequence of \(R\)-modules \[0 \to R \to R/\mathfrak p \oplus R/\mathfrak q \to R/\mathfrak p + \mathfrak q \to 0\] is exact (check at all the spots). The maps are \(x \mapsto (x \bmod \mathfrak p, x \bmod \mathfrak q)\) and \((y \bmod \mathfrak p, z \bmod \mathfrak q) \mapsto (y - z \bmod \mathfrak p + \mathfrak q)\).

2. Show that if \(\text{depth}(R) \geq 2\), then \(\dim(R/\mathfrak p + \mathfrak q) \geq 1\).

3. Show that if \(\text{depth}(R) \geq 2\), then \(U = \Spec(R) \setminus \{\mathfrak m\}\) is a connected topological space.

This proves a very special case of Hartshorne’s connectedness theorem which says that the punctured spectrum \(U\) of a local Noetherian ring of \(\text{depth} \geq 2\) is connected.

Exercise

Let \((R, \mathfrak m)\) be a Noetherian local ring. Let \(x, y \in \mathfrak m\) be a regular sequence of length \(2\). For any \(n \geq 2\) show that there do not exist \(a, b \in R\) with \[x^{n - 1}y^{n - 1} = a x^n + b y^n\] Suggestion: First try for \(n = 2\) to see how to argue. Remark: There is a vast generalization of this result called the monomial conjecture.

Cohen-Macaulay modules and rings

Cohen-Macaulay modules are studied in Algebra, Section 00N2 and Cohen-Macaulay rings are studied in Algebra, Section 00N7.

Exercise

In the following cases, please answer yes or no. No explanation or proof necessary.

1. Let \(p\) be a prime number. Is the local ring \(\mathbf{Z}_{(p)}\) a Cohen-Macaulay local ring?

2. Let \(p\) be a prime number. Is the local ring \(\mathbf{Z}_{(p)}\) a regular local ring?

3. Let \(k\) be a field. Is the local ring \(k[x]_{(x)}\) a Cohen-Macaulay local ring?

4. Let \(k\) be a field. Is the local ring \(k[x]_{(x)}\) a regular local ring?

5. Let \(k\) be a field. Is the local ring \((k[x, y]/(y^2 - x^3))_{(x, y)} = k[x, y]_{(x, y)}/(y^2 - x^3)\) a Cohen-Macaulay local ring?

6. Let \(k\) be a field. Is the local ring \((k[x, y]/(y^2, xy))_{(x, y)} = k[x, y]_{(x, y)}/(y^2, xy)\) a Cohen-Macaulay local ring?

Singularities

Exercise

Let \(k\) be any field. Suppose that \(A = k[[x, y]]/(f)\) and \(B = k[[u, v]]/(g)\), where \(f = xy\) and \(g = uv + \delta\) with \(\delta \in (u, v)^3\). Show that \(A\) and \(B\) are isomorphic rings.

Remark

A singularity on a curve over a field \(k\) is called an ordinary double point if the complete local ring of the curve at the point is of the form \(k'[[x, y]]/(f)\), where (a) \(k'\) is a finite separable extension of \(k\), (b) the initial term of \(f\) has degree two, i.e., it looks like \(q = ax^2 + bxy + cy^2\) for some \(a, b, c\in k'\) not all zero, and (c) \(q\) is a nondegenerate quadratic form over \(k'\) (in char 2 this means that \(b\) is not zero). In general there is one isomorphism class of such rings for each isomorphism class of pairs \((k', q)\).

Exercise

Let \(R\) be a ring. Let \(n \geq 1\). Let \(A\), \(B\) be \(n \times n\) matrices with coefficients in \(R\) such that \(AB = f 1_{n \times n}\) for some nonzerodivisor \(f\) in \(R\). Set \(S = R/(f)\). Show that \[\ldots \to S^{\oplus n} \xrightarrow{B} S^{\oplus n} \xrightarrow{A} S^{\oplus n} \xrightarrow{B} S^{\oplus n} \to \ldots\] is exact.

Constructible sets

Let \(k\) be an algebraically closed field, for example the field \(\mathbf{C}\) of complex numbers. Let \(n \geq 0\). A polynomial \(f \in k[x_1, \ldots, x_n]\) gives a function \(f : k^n \to k\) by evaluation. A subset \(Z \subset k^n\) is called an algebraic set if it is the common vanishing set of a collection of polynomials.

Exercise

Prove that an algebraic set can always be written as the zero locus of finitely many polynomials.

With notation as above a subset \(E \subset k^n\) is called constructible if it is a finite union of sets of the form \(Z \cap \{f \not = 0\}\) where \(f\) is a polynomial.

Exercise

Show the following

1. the complement of a constructible set is a constructible set,

2. a finite union of constructible sets is a constructible set,

3. a finite intersection of constructible sets is a constructible set, and

4. any constructible set \(E\) can be written as a finite disjoint union \(E = \coprod E_i\) with each \(E_i\) of the form \(Z \cap \{f \not = 0\}\) where \(Z\) is an algebraic set and \(f\) is a polynomial.

Exercise

Let \(R\) be a ring. Let \(f = a_d x^d + a_{d - 1} x^{d - 1} + \ldots + a_0 \in R[x]\). (As usual this notation means \(a_0, \ldots, a_d \in R\).) Let \(g \in R[x]\). Prove that we can find \(N \geq 0\) and \(r, q \in R[x]\) such that \[a_d^N g = q f + r\] with \(\deg(r) < d\), i.e., for some \(c_i \in R\) we have \(r = c_0 + c_1 x + \ldots + c_{d - 1}x^{d - 1}\).

Hilbert Nullstellensatz

Exercise

A silly argument using the complex numbers! Let \({\mathbf C}\) be the complex number field. Let \(V\) be a vector space over \({\mathbf C}\). The spectrum of a linear operator \(T : V \to V\) is the set of complex numbers \(\lambda \in {\mathbf C}\) such that the operator \(T - \lambda \text{id}_V\) is not invertible.

1. Show that \(\mathbf{C}(X)\) has uncountable dimension over \({\mathbf C}\).

2. Show that any linear operator on \(V\) has a nonempty spectrum if the dimension of \(V\) is finite or countable.

3. Show that if a finitely generated \({\mathbf C}\)-algebra \(R\) is a field, then the map \({\mathbf C}\to R\) is an isomorphism.

4. Show that any maximal ideal \({\mathfrak m}\) of \({\mathbf C}[x_1, \ldots, x_n]\) is of the form \((x_1-\alpha_1, \ldots, x_n-\alpha_n)\) for some \(\alpha_i \in {\mathbf C}\).

Remark

Let \(k\) be a field. Then for every integer \(n\in {\mathbf N}\) and every maximal ideal \({\mathfrak m} \subset k[x_1, \ldots, x_n]\) the quotient \(k[x_1, \ldots, x_n]/{\mathfrak m}\) is a finite field extension of \(k\). This will be shown later in the course. Of course (please check this) it implies a similar statement for maximal ideals of finitely generated \(k\)-algebras. The exercise above proves it in the case \(k = {\mathbf C}\).

Exercise

Let \(k\) be a field. Please use Remark 027M.

1. Let \(R\) be a \(k\)-algebra. Suppose that \(\dim_k R < \infty\) and that \(R\) is a domain. Show that \(R\) is a field.

2. Suppose that \(R\) is a finitely generated \(k\)-algebra, and \(f\in R\) not nilpotent. Show that there exists a maximal ideal \({\mathfrak m} \subset R\) with \(f\not\in {\mathfrak m}\).

3. Show by an example that this statement fails when \(R\) is not of finite type over a field.

4. Show that any radical ideal \(I \subset {\mathbf C}[x_1, \ldots, x_n]\) is the intersection of the maximal ideals containing it.

Remark

This is the Hilbert Nullstellensatz. Namely it says that the closed subsets of \(\Spec(k[x_1, \ldots, x_n])\) (which correspond to radical ideals by a previous exercise) are determined by the closed points contained in them.

Exercise

Let \(A = {\mathbf C}[x_{11}, x_{12}, x_{21}, x_{22}, y_{11}, y_{12}, y_{21}, y_{22}]\). Let \(I\) be the ideal of \(A\) generated by the entries of the matrix \(XY\), with \[X = \left( \begin{matrix} x_{11} & x_{12}\\ x_{21} & x_{22} \end{matrix} \right) \quad\text{and}\quad Y = \left( \begin{matrix} y_{11} & y_{12}\\ y_{21} & y_{22} \end{matrix} \right).\] Find the irreducible components of the closed subset \(V(I)\) of \(\Spec(A)\). (I mean describe them and give equations for each of them. You do not have to prove that the equations you write down define prime ideals.) Hints:

1. You may use the Hilbert Nullstellensatz, and it suffices to find irreducible locally closed subsets which cover the set of closed points of \(V(I)\).

2. There are two easy components.

3. An image of an irreducible set under a continuous map is irreducible.

Dimension

Exercise

Construct a ring \(A\) with finitely many prime ideals having dimension \(> 1\).

Exercise

Let \(f \in \mathbf{C}[x, y]\) be a nonconstant polynomial. Show that \(\mathbf{C}[x, y]/(f)\) has dimension \(1\).

Exercise

Let \((R, \mathfrak m)\) be a Noetherian local ring. Let \(n \geq 1\). Let \(\mathfrak m' = (\mathfrak m, x_1, \ldots, x_n)\) in the polynomial ring \(R[x_1, \ldots, x_n]\). Show that \[\dim(R[x_1, \ldots, x_n]_{\mathfrak m'}) = \dim(R) + n.\]

Catenary rings

Definition

A Noetherian ring \(A\) is said to be catenary if for any triple of prime ideals \({\mathfrak p}_1 \subset {\mathfrak p}_2 \subset {\mathfrak p}_3\) we have \[ht({\mathfrak p}_3 / {\mathfrak p}_1) = ht({\mathfrak p}_3/{\mathfrak p}_2) + ht({\mathfrak p}_2/{\mathfrak p}_1).\] Here \(ht(\mathfrak p/\mathfrak q)\) means the height of \(\mathfrak p/\mathfrak q\) in the ring \(A/\mathfrak q\). In a formula \[ht(\mathfrak p/\mathfrak q) = \dim(A_\mathfrak p/\mathfrak qA_\mathfrak p) = \dim((A/\mathfrak q)_\mathfrak p) = \dim((A/\mathfrak q)_{\mathfrak p/\mathfrak q})\] A topological space \(X\) is catenary, if given \(T \subset T' \subset X\) with \(T\) and \(T'\) closed and irreducible, then there exists a maximal chain of irreducible closed subsets \[T = T_0 \subset T_1 \subset \ldots \subset T_n = T'\] and every such chain has the same (finite) length.

Exercise

Show that the notion of catenary defined in Algebra, Definition 00NI agrees with the notion of Definition 027O for Noetherian rings.

Exercise

Show that a Noetherian local domain of dimension \(2\) is catenary.

Exercise

Let \(k\) be a field. Show that a finite type \(k\)-algebra is catenary.

Exercise

Give an example of a finite, sober, catenary topological space \(X\) which does not have a dimension function \(\delta : X \to \mathbf{Z}\). Here \(\delta : X \to \mathbf{Z}\) is a dimension function if for \(x, y \in X\) we have

1. \(x \leadsto y\) and \(x \not = y\) implies \(\delta(x) > \delta(y)\),

2. \(x \leadsto y\) and \(\delta(x) \geq \delta(y) + 2\) implies there exists a \(z \in X\) with \(x \leadsto z \leadsto y\) and \(\delta(x) > \delta(z) > \delta(y)\).

Describe your space clearly and succinctly explain why there cannot be a dimension function.

Fraction fields

Exercise

Consider the domain \[{\mathbf Q}[r, s, t]/(s^2-(r-1)(r-2)(r-3), t^2-(r + 1)(r + 2)(r + 3)).\] Find a domain of the form \({\mathbf Q}[x, y]/(f)\) with isomorphic field of fractions.

Transcendence degree

Exercise

Let \(K'/K/k\) be field extensions with \(K'\) algebraic over \(K\). Prove that \(\text{trdeg}_k(K) = \text{trdeg}_k(K')\). (Hint: Show that if \(x_1, \ldots, x_d \in K\) are algebraically independent over \(k\) and \(d < \text{trdeg}_k(K')\) then \(k(x_1, \ldots, x_d) \subset K\) cannot be algebraic.)

Exercise

Let \(k\) be a field. Let \(K/k\) be a finitely generated extension of transcendence degree \(d\). If \(V, W \subset K\) are finite dimensional \(k\)-subvector spaces denote \[VW = \{f \in K \mid f = \sum\nolimits_{i = 1, \ldots, n} v_i w_i \text{ for some }n\text{ and }v_i \in V, w_i \in W\}\] This is a finite dimensional \(k\)-subvector space. Set \(V^2 = VV\), \(V^3 = V V^2\), etc.

1. Show you can find \(V \subset K\) and \(\epsilon > 0\) such that \(\dim V^n \geq \epsilon n^d\) for all \(n \geq 1\).

2. Conversely, show that for every finite dimensional \(V \subset K\) there exists a \(C > 0\) such that \(\dim V^n \leq C n^d\) for all \(n \geq 1\). (One possible way to proceed: First do this for subvector spaces of \(k[x_1, \ldots, x_d]\). Then do this for subvector spaces of \(k(x_1, \ldots, x_d)\). Finally, if \(K/k(x_1, \ldots, x_d)\) is a finite extension choose a basis of \(K\) over \(k(x_1, \ldots, x_d)\) and argue using expansion in terms of this basis.)

3. Conclude that you can redefine the transcendence degree in terms of growth of powers of finite dimensional subvector spaces of \(K\).

This is related to Gelfand-Kirillov dimension of (noncommutative) algebras over \(k\).

Dimension of fibres

Some questions related to the dimension formula, see Algebra, Section 02II.

Exercise

Let \(k\) be your favorite algebraically closed field. Below \(k[x]\) and \(k[x, y]\) denote the polynomial rings.

1. For every integer \(n \geq 0\) find a finite type extension \(k[x] \subset A\) of domains such that the spectrum of \(A/xA\) has exactly \(n\) irreducible components.

2. Make an example of a finite type extension \(k[x] \subset A\) of domains such that the spectrum of \(A/(x - \alpha)A\) is nonempty and reducible for every \(\alpha \in k\).

3. Make an example of a finite type extension \(k[x, y] \subset A\) of domains such that the spectrum of \(A/(x - \alpha, y - \beta)A\) is irreducible¹ for all \((\alpha, \beta) \in k^2 \setminus \{(0, 0)\}\) and the spectrum of \(A/(x, y)A\) is nonempty and reducible.

Exercise

Let \(k\) be your favorite algebraically closed field. Let \(n \geq 1\). Let \(k[x_1, \ldots, x_n]\) be the polynomial ring. Set \(\mathfrak m = (x_1, \ldots, x_n)\). Let \(k[x_1, \ldots, x_n] \subset A\) be a finite type extension of domains. Set \(d = \dim(A)\).

1. Show that \(d - 1 \geq \dim(A/\mathfrak m A) \geq d - n\) if \(A/\mathfrak mA \not = 0\).

2. Show by example that every value can occur.

3. Show by example that \(\Spec(A/\mathfrak m A)\) can have irreducible components of different dimensions.

Finite locally free modules

Definition

Let \(A\) be a ring. Recall that a finite locally free \(A\)-module \(M\) is a module such that for every \({\mathfrak p} \in \Spec(A)\) there exists an \(f\in A\), \(f \not \in {\mathfrak p}\) such that \(M_f\) is a finite free \(A_f\)-module. We say \(M\) is an invertible module if \(M\) is finite locally free of rank \(1\), i.e., for every \({\mathfrak p} \in \Spec(A)\) there exists an \(f\in A\), \(f \not \in \mathfrak p\) such that \(M_f \cong A_f\) as an \(A_f\)-module.

Exercise

Prove that the tensor product of finite locally free modules is finite locally free. Prove that the tensor product of two invertible modules is invertible.

Definition

Let \(A\) be a ring. The class group of \(A\), sometimes called the Picard group of \(A\) is the set \(\Pic(A)\) of isomorphism classes of invertible \(A\)-modules endowed with a group operation defined by tensor product (see Exercise 078P).

Note that the class group of \(A\) is trivial exactly when every invertible module is isomorphic to a free module of rank 1.

Exercise

Show that the class groups of the following rings are trivial

1. a polynomial ring \(A = k[x]\) where \(k\) is a field,

2. the integers \(A = \mathbf{Z}\),

3. a polynomial ring \(A = k[x, y]\) where \(k\) is a field, and

4. the quotient \(k[x, y]/(xy)\) where \(k\) is a field.

Exercise

Show that the class group of the ring \(A = k[x, y]/(y^2 - f(x))\) where \(k\) is a field of characteristic not \(2\) and where \(f(x) = (x - t_1) \ldots (x - t_n)\) with \(t_1, \ldots, t_n \in k\) distinct and \(n \geq 3\) an odd integer is not trivial. (Hint: Show that the ideal \((y, x - t_1)\) defines a nontrivial element of \(\Pic(A)\).)

Exercise

Let \(A\) be a ring.

1. Suppose that \(M\) is a finite locally free \(A\)-module, and suppose that \(\varphi : M \to M\) is an endomorphism. Define/construct the trace and determinant of \(\varphi\) and prove that your construction is “functorial in the triple \((A, M, \varphi)\)”.

2. Show that if \(M, N\) are finite locally free \(A\)-modules, and if \(\varphi : M \to N\) and \(\psi : N \to M\) then \(\text{Trace}(\varphi \circ \psi) = \text{Trace}(\psi \circ \varphi)\) and \(\det(\varphi \circ \psi) = \det(\psi \circ \varphi)\).

3. In case \(M\) is finite locally free show that \(\text{Trace}\) defines an \(A\)-linear map \(\text{End}_A(M) \to A\) and \(\det\) defines a multiplicative map \(\text{End}_A(M) \to A\).

Exercise

Now suppose that \(B\) is an \(A\)-algebra which is finite locally free as an \(A\)-module, in other words \(B\) is a finite locally free \(A\)-algebra.

1. Define \(\text{Trace}_{B/A}\) and \(\text{Norm}_{B/A}\) using \(\text{Trace}\) and \(\det\) from Exercise 02DU.

2. Let \(b\in B\) and let \(\pi : \Spec(B) \to \Spec(A)\) be the induced morphism. Show that \(\pi(V(b)) = V(\text{Norm}_{B/A}(b))\). (Recall that \(V(f) = \{ {\mathfrak p} \mid f \in {\mathfrak p}\}\).)

3. (Base change.) Suppose that \(i : A \to A'\) is a ring map. Set \(B' = B \otimes_A A'\). Indicate why \(i(\text{Norm}_{B/A}(b))\) equals \(\text{Norm}_{B'/A'}(b \otimes 1)\).

4. Compute \(\text{Norm}_{B/A}(b)\) when \(B = A \times A \times A \times \ldots \times A\) and \(b = (a_1, \ldots, a_n)\).

5. Compute the norm of \(y-y^3\) under the finite flat map \({\mathbf Q}[x] \to {\mathbf Q}[y]\), \(x \to y^n\). (Hint: use the “base change” \(A = {\mathbf Q}[x] \subset A' = {\mathbf Q}(\zeta_n)(x^{1/n})\).)

Glueing

Exercise

Suppose that \(A\) is a ring and \(M\) is an \(A\)-module. Let \(f_i\), \(i \in I\) be a collection of elements of \(A\) such that \[\Spec(A) = \bigcup D(f_i).\]

1. Show that if \(M_{f_i}\) is a finite \(A_{f_i}\)-module, then \(M\) is a finite \(A\)-module.

2. Show that if \(M_{f_i}\) is a flat \(A_{f_i}\)-module, then \(M\) is a flat \(A\)-module. (This is kind of silly if you think about it right.)

Remark

In algebraic geometric language this means that the property of “being finitely generated” or “being flat” is local for the Zariski topology (in a suitable sense). You can also show this for the property “being of finite presentation”.

Exercise

Suppose that \(A \to B\) is a ring map. Let \(f_i \in A\), \(i \in I\) and \(g_j \in B\), \(j \in J\) be collections of elements such that \[\Spec(A) = \bigcup D(f_i) \quad\text{and}\quad \Spec(B) = \bigcup D(g_j).\] Show that if \(A_{f_i} \to B_{f_ig_j}\) is of finite type for all \(i, j\) then \(A \to B\) is of finite type.

Going up and going down

Definition

Let \(\phi : A \to B\) be a homomorphism of rings. We say that the going-up theorem holds for \(\phi\) if the following condition is satisfied:

• for any \({\mathfrak p}, {\mathfrak p}' \in \Spec(A)\) such that \({\mathfrak p} \subset {\mathfrak p}'\), and for any \(P \in \Spec(B)\) lying over \({\mathfrak p}\), there exists \(P'\in \Spec(B)\) lying over \({\mathfrak p}'\) such that \(P \subset P'\).

Similarly, we say that the going-down theorem holds for \(\phi\) if the following condition is satisfied:

• for any \({\mathfrak p}, {\mathfrak p}' \in \Spec(A)\) such that \({\mathfrak p} \subset {\mathfrak p}'\), and for any \(P' \in \Spec(B)\) lying over \({\mathfrak p}'\), there exists \(P\in \Spec(B)\) lying over \({\mathfrak p}\) such that \(P \subset P'\).

Exercise

In each of the following cases determine whether (GU), (GD) holds, and explain why. (Use any Prop/Thm/Lemma you can find, but check the hypotheses in each case.)

1. \(k\) is a field, \(A = k\), \(B = k[x]\).

2. \(k\) is a field, \(A = k[x]\), \(B = k[x, y]\).

3. \(A = {\mathbf Z}\), \(B = {\mathbf Z}[1/11]\).

4. \(k\) is an algebraically closed field, \(A = k[x, y]\), \(B = k[x, y, z]/(x^2-y, z^2-x)\).

5. \(A = {\mathbf Z}\), \(B = {\mathbf Z}[i, 1/(2 + i)]\).

6. \(A = {\mathbf Z}\), \(B = {\mathbf Z}[i, 1/(14 + 7i)]\).

7. \(k\) is an algebraically closed field, \(A = k[x]\), \(B = k[x, y, 1/(xy-1)]/(y^2-y)\).

Exercise

Let \(A\) be a ring. Let \(B = A[x]\) be the polynomial algebra in one variable over \(A\). Let \(f = a_0 + a_1 x + \ldots + a_r x^r \in B = A[x]\). Prove carefully that the image of \(D(f)\) in \(\Spec(A)\) is equal to \(D(a_0) \cup \ldots \cup D(a_r)\).

Exercise

Let \(k\) be an algebraically closed field. Compute the image in \(\Spec(k[x, y])\) of the following maps:

1. \(\Spec(k[x, yx^{-1}]) \to \Spec(k[x, y])\), where \(k[x, y] \subset k[x, yx^{-1}] \subset k[x, y, x^{-1}]\). (Hint: To avoid confusion, give the element \(yx^{-1}\) another name.)

2. \(\Spec(k[x, y, a, b]/(ax-by-1))\to \Spec(k[x, y])\).

3. \(\Spec(k[t, 1/(t-1)]) \to \Spec(k[x, y])\), induced by \(x \mapsto t^2\), and \(y \mapsto t^3\).

4. \(k = {\mathbf C}\) (complex numbers), \(\Spec(k[s, t]/(s^3 + t^3-1)) \to \Spec(k[x, y])\), where \(x\mapsto s^2\), \(y \mapsto t^2\).

Remark

Finding the image as above usually is done by using elimination theory.

Fitting ideals

Exercise

Let \(R\) be a ring and let \(M\) be a finite \(R\)-module. Choose a presentation \[\bigoplus\nolimits_{j \in J} R \longrightarrow R^{\oplus n} \longrightarrow M \longrightarrow 0.\] of \(M\). Note that the map \(R^{\oplus n} \to M\) is given by a sequence of elements \(x_1, \ldots, x_n\) of \(M\). The elements \(x_i\) are generators of \(M\). The map \(\bigoplus_{j \in J} R \to R^{\oplus n}\) is given by a \(n \times J\) matrix \(A\) with coefficients in \(R\). In other words, \(A = (a_{ij})_{i = 1, \ldots, n, j \in J}\). The columns \((a_{1j}, \ldots, a_{nj})\), \(j \in J\) of \(A\) are said to be the relations. Any vector \((r_i) \in R^{\oplus n}\) such that \(\sum r_i x_i = 0\) is a linear combination of the columns of \(A\). Of course any finite \(R\)-module has a lot of different presentations.

1. Show that the ideal generated by the \((n - k) \times (n - k)\) minors of \(A\) is independent of the choice of the presentation. This ideal is the \(k\)th Fitting ideal of \(M\). Notation \(Fit_k(M)\).

2. Show that \(Fit_0(M) \subset Fit_1(M) \subset Fit_2(M) \subset \ldots\). (Hint: Use that a determinant can be computed by expanding along a column.)

3. Show that the following are equivalent:

   1. \(Fit_{r - 1}(M) = (0)\) and \(Fit_r(M) = R\), and

   2. \(M\) is locally free of rank \(r\).

Hilbert functions

Definition

A numerical polynomial is a polynomial \(f(x) \in {\mathbf Q}[x]\) such that \(f(n) \in {\mathbf Z}\) for every integer \(n\).

Definition

A graded module \(M\) over a ring \(A\) is an \(A\)-module \(M\) endowed with a direct sum decomposition \(\bigoplus\nolimits_{n \in {\mathbf Z}} M_n\) into \(A\)-submodules. We will say that \(M\) is locally finite if all of the \(M_n\) are finite \(A\)-modules. Suppose that \(A\) is a Noetherian ring and that \(\varphi\) is a Euler-Poincaré function on finite \(A\)-modules. This means that for every finitely generated \(A\)-module \(M\) we are given an integer \(\varphi(M) \in {\mathbf Z}\) and for every short exact sequence \[0 \longrightarrow M' \longrightarrow M \longrightarrow M'' \longrightarrow 0\] we have \(\varphi(M) = \varphi(M') + \varphi(M'')\). The Hilbert function of a locally finite graded module \(M\) (with respect to \(\varphi\)) is the function \(\chi_\varphi(M, n) = \varphi(M_n)\). We say that \(M\) has a Hilbert polynomial if there is some numerical polynomial \(P_\varphi\) such that \(\chi_\varphi(M, n) = P_\varphi(n)\) for all sufficiently large integers \(n\).

Definition

A graded \(A\)-algebra is a graded \(A\)-module \(B = \bigoplus_{n \geq 0} B_n\) together with an \(A\)-bilinear map \[B \times B \longrightarrow B, \ (b, b') \longmapsto bb'\] that turns \(B\) into an \(A\)-algebra so that \(B_n \cdot B_m \subset B_{n + m}\). Finally, a graded module \(M\) over a graded \(A\)-algebra \(B\) is given by a graded \(A\)-module \(M\) together with a (compatible) \(B\)-module structure such that \(B_n \cdot M_d \subset M_{n + d}\). Now you can define homomorphisms of graded modules/rings, graded submodules, graded ideals, exact sequences of graded modules, etc, etc.

Exercise

Let \(A = k\) a field. What are all possible Euler-Poincaré functions on finite \(A\)-modules in this case?

Exercise

Let \(A ={\mathbf Z}\). What are all possible Euler-Poincaré functions on finite \(A\)-modules in this case?

Exercise

Let \(A = k[x, y]/(xy)\) with \(k\) algebraically closed. What are all possible Euler-Poincaré functions on finite \(A\)-modules in this case?

Exercise

Suppose that \(A\) is Noetherian. Show that the kernel of a map of locally finite graded \(A\)-modules is locally finite.

Exercise

Let \(k\) be a field and let \(A = k\) and \(B = k[x, y]\) with grading determined by \(\deg(x) = 2\) and \(\deg(y) = 3\). Let \(\varphi(M) = \dim_k(M)\). Compute the Hilbert function of \(B\) as a graded \(k\)-module. Is there a Hilbert polynomial in this case?

Exercise

Let \(k\) be a field and let \(A = k\) and \(B = k[x, y]/(x^2, xy)\) with grading determined by \(\deg(x) = 2\) and \(\deg(y) = 3\). Let \(\varphi(M) = \dim_k(M)\). Compute the Hilbert function of \(B\) as a graded \(k\)-module. Is there a Hilbert polynomial in this case?

Exercise

Let \(k\) be a field and let \(A = k\). Let \(\varphi(M) = \dim_k(M)\). Fix \(d\in {\mathbf N}\). Consider the graded \(A\)-algebra \(B = k[x, y, z]/(x^d + y^d + z^d)\), where \(x, y, z\) each have degree \(1\). Compute the Hilbert function of \(B\). Is there a Hilbert polynomial in this case?

Proj of a ring

Definition

Let \(R\) be a graded ring. A homogeneous ideal is simply an ideal \(I \subset R\) which is also a graded submodule of \(R\). Equivalently, it is an ideal generated by homogeneous elements. Equivalently, if \(f \in I\) and \[f = f_0 + f_1 + \ldots + f_n\] is the decomposition of \(f\) into homogeneous pieces in \(R\) then \(f_i \in I\) for each \(i\).

Definition

We define the homogeneous spectrum \(\text{Proj}(R)\) of the graded ring \(R\) to be the set of homogeneous, prime ideals \({\mathfrak p}\) of \(R\) such that \(R_{+} \not \subset {\mathfrak p}\). Note that \(\text{Proj}(R)\) is a subset of \(\Spec(R)\) and hence has a natural induced topology.

Definition

Let \(R = \oplus_{d \geq 0} R_d\) be a graded ring, let \(f\in R_d\) and assume that \(d \geq 1\). We define \(R_{(f)}\) to be the subring of \(R_f\) consisting of elements of the form \(r/f^n\) with \(r\) homogeneous and \(\deg(r) = nd\). Furthermore, we define \[D_{+}(f) = \{ {\mathfrak p} \in \text{Proj}(R) | f \not\in {\mathfrak p} \}.\] Finally, for a homogeneous ideal \(I \subset R\) we define \(V_{+}(I) = V(I) \cap \text{Proj}(R)\).

Exercise

On the topology on \(\text{Proj}(R)\). With definitions and notation as above prove the following statements.

1. Show that \(D_{+}(f)\) is open in \(\text{Proj}(R)\).

2. Show that \(D_{+}(ff') = D_{+}(f) \cap D_{+}(f')\).

3. Let \(g = g_0 + \ldots + g_m\) be an element of \(R\) with \(g_i \in R_i\). Express \(D(g) \cap \text{Proj}(R)\) in terms of \(D_{+}(g_i)\), \(i \geq 1\) and \(D(g_0) \cap \text{Proj}(R)\). No proof necessary.

4. Let \(g\in R_0\) be a homogeneous element of degree \(0\). Express \(D(g) \cap \text{Proj}(R)\) in terms of \(D_{+}(f_\alpha)\) for a suitable family \(f_\alpha \in R\) of homogeneous elements of positive degree.

5. Show that the collection \(\{D_{+}(f)\}\) of opens forms a basis for the topology of \(\text{Proj}(R)\).

6. Show that there is a canonical bijection \(D_{+}(f) \to \Spec(R_{(f)})\). (Hint: Imitate the proof for \(\Spec\) but at some point thrown in the radical of an ideal.)

7. Show that the map from (078U) is a homeomorphism.

8. Give an example of an \(R\) such that \(\text{Proj}(R)\) is not quasi-compact. No proof necessary.

9. Show that any closed subset \(T \subset \text{Proj}(R)\) is of the form \(V_{+}(I)\) for some homogeneous ideal \(I \subset R\).

Remark

There is a continuous map \(\text{Proj}(R) \longrightarrow \Spec(R_0)\).

Exercise

If \(R = A[X]\) with \(\deg(X) = 1\), show that the natural map \(\text{Proj}(R) \to \Spec(A)\) is a bijection and in fact a homeomorphism.

Exercise

Blowing up: part I. In this exercise \(R = Bl_I(A) = A \oplus I \oplus I^2 \oplus \ldots\). Consider the natural map \(b : \text{Proj}(R) \to \Spec(A)\). Set \(U = \Spec(A) - V(I)\). Show that \[b : b^{-1}(U) \longrightarrow U\] is a homeomorphism. Thus we may think of \(U\) as an open subset of \(\text{Proj}(R)\). Let \(Z \subset \Spec(A)\) be an irreducible closed subscheme with generic point \(\xi \in Z\). Assume that \(\xi \not\in V(I)\), in other words \(Z \not\subset V(I)\), in other words \(\xi \in U\), in other words \(Z\cap U \not = \emptyset\). We define the strict transform \(Z'\) of \(Z\) to be the closure of the unique point \(\xi'\) lying above \(\xi\). Another way to say this is that \(Z'\) is the closure in \(\text{Proj}(R)\) of the locally closed subset \(Z\cap U \subset U \subset \text{Proj}(R)\).

Exercise

Blowing up: Part II. Let \(A = k[x, y]\) where \(k\) is a field, and let \(I = (x, y)\). Let \(R\) be the blowup algebra for \(A\) and \(I\).

1. Show that the strict transforms of \(Z_1 = V(\{x\})\) and \(Z_2 = V(\{y\})\) are disjoint.

2. Show that the strict transforms of \(Z_1 = V(\{x\})\) and \(Z_2 = V(\{x-y^2\})\) are not disjoint.

3. Find an ideal \(J \subset A\) such that \(V(J) = V(I)\) and such that the strict transforms of \(Z_1 = V(\{x\})\) and \(Z_2 = V(\{x-y^2\})\) in the blowup along \(J\) are disjoint.

Exercise

Let \(R\) be a graded ring.

1. Show that \(\text{Proj}(R)\) is empty if \(R_n = (0)\) for all \(n >> 0\).

2. Show that \(\text{Proj}(R)\) is an irreducible topological space if \(R\) is a domain and \(R_{+}\) is not zero. (Recall that the empty topological space is not irreducible.)

Exercise

Blowing up: Part III. Consider \(A\), \(I\) and \(U\), \(Z\) as in the definition of strict transform. Let \(Z = V({\mathfrak p})\) for some prime ideal \({\mathfrak p}\). Let \(\bar A = A/{\mathfrak p}\) and let \(\bar I\) be the image of \(I\) in \(\bar A\).

1. Show that there exists a surjective ring map \(R: = Bl_I(A) \to \bar R: = Bl_{\bar I}(\bar A)\).

2. Show that the ring map above induces a bijective map from \(\text{Proj}(\bar R)\) onto the strict transform \(Z'\) of \(Z\). (This is not so easy. Hint: Use 5(b) above.)

3. Conclude that the strict transform \(Z' = V_{+}(P)\) where \(P \subset R\) is the homogeneous ideal defined by \(P_d = I^d \cap {\mathfrak p}\).

4. Suppose that \(Z_1 = V({\mathfrak p})\) and \(Z_2 = V({\mathfrak q})\) are irreducible closed subsets defined by prime ideals such that \(Z_1 \not \subset Z_2\), and \(Z_2 \not \subset Z_1\). Show that blowing up the ideal \(I = {\mathfrak p} + {\mathfrak q}\) separates the strict transforms of \(Z_1\) and \(Z_2\), i.e., \(Z_1' \cap Z_2' = \emptyset\). (Hint: Consider the homogeneous ideal \(P\) and \(Q\) from part (c) and consider \(V(P + Q)\).)

Cohen-Macaulay rings of dimension 1

Definition

A Noetherian local ring \(A\) is said to be Cohen-Macaulay of dimension \(d\) if it has dimension \(d\) and there exists a system of parameters \(x_1, \ldots, x_d\) for \(A\) such that \(x_i\) is a nonzerodivisor in \(A/(x_1, \ldots, x_{i-1})\) for \(i = 1, \ldots, d\).

Exercise

Cohen-Macaulay rings of dimension 1. Part I: Theory.

1. Let \((A, {\mathfrak m})\) be a local Noetherian with \(\dim A = 1\). Show that if \(x\in {\mathfrak m}\) is not a zerodivisor then

   1. \(\dim A/xA = 0\), in other words \(A/xA\) is Artinian, in other words \(\{x\}\) is a system of parameters for \(A\).

   2. \(A\) is has no embedded prime.

2. Conversely, let \((A, {\mathfrak m})\) be a local Noetherian ring of dimension \(1\). Show that if \(A\) has no embedded prime then there exists a nonzerodivisor in \({\mathfrak m}\).

Exercise

Cohen-Macaulay rings of dimension 1. Part II: Examples.

1. Let \(A\) be the local ring at \((x, y)\) of \(k[x, y]/(x^2, xy)\).

   1. Show that \(A\) has dimension 1.

   2. Prove that every element of \({\mathfrak m}\subset A\) is a zerodivisor.

   3. Find \(z\in {\mathfrak m}\) such that \(\dim A/zA = 0\) (no proof required).

2. Let \(A\) be the local ring at \((x, y)\) of \(k[x, y]/(x^2)\). Find a nonzerodivisor in \({\mathfrak m}\) (no proof required).

Exercise

Local rings of embedding dimension \(1\). Suppose that \((A, {\mathfrak m}, k)\) is a Noetherian local ring of embedding dimension \(1\), i.e., \[\dim_k {\mathfrak m}/{\mathfrak m}^2 = 1.\] Show that the function \(f(n) = \dim_k {\mathfrak m}^n/{\mathfrak m}^{n + 1}\) is either constant with value \(1\), or its values are \[1, 1, \ldots, 1, 0, 0, 0, 0, 0, \ldots\]

Exercise

Regular local rings of dimension \(1\). Suppose that \((A, {\mathfrak m}, k)\) is a regular Noetherian local ring of dimension \(1\). Recall that this means that \(A\) has dimension \(1\) and embedding dimension \(1\), i.e., \[\dim_k {\mathfrak m}/{\mathfrak m}^2 = 1.\] Let \(x\in{\mathfrak m}\) be any element whose class in \({\mathfrak m}/{\mathfrak m}^2\) is not zero.

1. Show that for every element \(y\) of \({\mathfrak m}\) there exists an integer \(n\) such that \(y\) can be written as \(y = ux^n\) with \(u\in A^\ast\) a unit.

2. Show that \(x\) is a nonzerodivisor in \(A\).

3. Conclude that \(A\) is a domain.

Exercise

Let \((A, {\mathfrak m}, k)\) be a Noetherian local ring with associated graded \(Gr_{\mathfrak m}(A)\).

1. Suppose that \(x\in {\mathfrak m}^d\) maps to a nonzerodivisor \(\bar x \in {\mathfrak m}^d/{\mathfrak m}^{d + 1}\) in degree \(d\) of \(Gr_{\mathfrak m}(A)\). Show that \(x\) is a nonzerodivisor.

2. Suppose the depth of \(A\) is at least \(1\). Namely, suppose that there exists a nonzerodivisor \(y \in {\mathfrak m}\). In this case we can do better: assume just that \(x\in {\mathfrak m}^d\) maps to the element \(\bar x \in {\mathfrak m}^d/{\mathfrak m}^{d + 1}\) in degree \(d\) of \(Gr_{\mathfrak m}(A)\) which is a nonzerodivisor on sufficiently high degrees: \(\exists N\) such that for all \(n \geq N\) the map of multiplication by \(\bar x\) \[{\mathfrak m}^n/{\mathfrak m}^{n + 1} \longrightarrow {\mathfrak m}^{n + d}/{\mathfrak m}^{n + d + 1}\] is injective. Then show that \(x\) is a nonzerodivisor.

Exercise

Suppose that \((A, {\mathfrak m}, k)\) is a Noetherian local ring of dimension \(1\). Assume also that the embedding dimension of \(A\) is \(2\), i.e., assume that \[\dim_k {\mathfrak m}/{\mathfrak m}^2 = 2.\] Notation: \(f(n) = \dim_k {\mathfrak m}^n/{\mathfrak m}^{n + 1}\). Pick generators \(x, y \in {\mathfrak m}\) and write \(Gr_{\mathfrak m}(A) = k[\bar x, \bar y]/I\) for some homogeneous ideal \(I\).

1. Show that there exists a homogeneous element \(F\in k[\bar x, \bar y]\) such that \(I \subset (F)\) with equality in all sufficiently high degrees.

2. Show that \(f(n) \leq n + 1\).

3. Show that if \(f(n) < n + 1\) then \(n \geq \deg(F)\).

4. Show that if \(f(n) < n + 1\), then \(f(n + 1) \leq f(n)\).

5. Show that \(f(n) = \deg(F)\) for all \(n >> 0\).

Exercise

Cohen-Macaulay rings of dimension 1 and embedding dimension 2. Suppose that \((A, {\mathfrak m}, k)\) is a Noetherian local ring which is Cohen-Macaulay of dimension \(1\). Assume also that the embedding dimension of \(A\) is \(2\), i.e., assume that \[\dim_k {\mathfrak m}/{\mathfrak m}^2 = 2.\] Notations: \(f\), \(F\), \(x, y\in {\mathfrak m}\), \(I\) as in Exercise 02EL. Please use any results from the problems above.

1. Suppose that \(z\in {\mathfrak m}\) is an element whose class in \({\mathfrak m}/{\mathfrak m}^2\) is a linear form \(\alpha \bar x + \beta \bar y \in k[\bar x, \bar y]\) which is coprime with \(F\).

   1. Show that \(z\) is a nonzerodivisor on \(A\).

   2. Let \(d = \deg(F)\). Show that \({\mathfrak m}^n = z^{n + 1-d}{\mathfrak m}^{d-1}\) for all sufficiently large \(n\). (Hint: First show \(z^{n + 1-d}{\mathfrak m}^{d-1} \to {\mathfrak m}^n/{\mathfrak m}^{n + 1}\) is surjective by what you know about \(Gr_{\mathfrak m}(A)\). Then use NAK.)

2. What condition on \(k\) guarantees the existence of such a \(z\)? (No proof required; it’s too easy.)

   Now we are going to assume there exists a \(z\) as above. This turns out to be a harmless assumption (in the sense that you can reduce to the situation where it holds in order to obtain the results in parts (d) and (e) below).

3. Now show that \({\mathfrak m}^\ell = z^{\ell - d + 1} {\mathfrak m}^{d-1}\) for all \(\ell \geq d\).

4. Conclude that \(I = (F)\).

5. Conclude that the function \(f\) has values \[2, 3, 4, \ldots, d-1, d, d, d, d, d, d, d, \ldots\]

Remark

This suggests that a local Noetherian Cohen-Macaulay ring of dimension 1 and embedding dimension 2 is of the form \(B/FB\), where \(B\) is a 2-dimensional regular local ring. This is more or less true (under suitable “niceness” properties of the ring).

Infinitely many primes

A section with a collection of strange questions on rings where infinitely many primes are not invertible.

Exercise

Give an example of a finite type \({\mathbf Z}\)-algebra \(R\) with the following two properties:

1. There is no ring map \(R \to {\mathbf Q}\).

2. For every prime \(p\) there exists a maximal ideal \({\mathfrak m} \subset R\) such that \(R/{\mathfrak m} \cong {\mathbf F}_p\).

Exercise

For \(f \in {\mathbf Z}[x, u]\) we define \(f_p(x) = f(x, x^p) \bmod p \in {\mathbf F}_p[x]\). Give an example of an \(f \in {\mathbf Z}[x, u]\) such that the following two properties hold:

1. There exist infinitely many \(p\) such that \(f_p\) does not have a zero in \({\mathbf F}_p\).

2. For all \(p >> 0\) the polynomial \(f_p\) either has a linear or a quadratic factor.

Exercise

For \(f \in {\mathbf Z}[x, y, u, v]\) we define \(f_p(x, y) = f(x, y, x^p, y^p) \bmod p \in {\mathbf F}_p[x, y]\). Give an “interesting” example of an \(f\) such that \(f_p\) is reducible for all \(p >> 0\). For example, \(f = xv-yu\) with \(f_p = xy^p-x^py = xy(x^{p-1}-y^{p-1})\) is “uninteresting”; any \(f\) depending only on \(x, u\) is “uninteresting”, etc.

Remark

Let \(h \in {\mathbf Z}[y]\) be a monic polynomial of degree \(d\). Then:

1. The map \(A = {\mathbf Z}[x] \to B ={\mathbf Z}[y]\), \(x \mapsto h\) is finite locally free of rank \(d\).

2. For all primes \(p\) the map \(A_p = {\mathbf F}_p[x]\to B_p = {\mathbf F}_p[y]\), \(y \mapsto h(y) \bmod p\) is finite locally free of rank \(d\).

Exercise

Let \(h, A, B, A_p, B_p\) be as in the remark. For \(f \in {\mathbf Z}[x, u]\) we define \(f_p(x) = f(x, x^p) \bmod p \in {\mathbf F}_p[x]\). For \(g \in {\mathbf Z}[y, v]\) we define \(g_p(y) = g(y, y^p) \bmod p \in {\mathbf F}_p[y]\).

1. Give an example of a \(h\) and \(g\) such that there does not exist a \(f\) with the property \[f_p = Norm_{B_p/A_p}(g_p).\]

2. Show that for any choice of \(h\) and \(g\) as above there exists a nonzero \(f\) such that for all \(p\) we have \[Norm_{B_p/A_p}(g_p)\quad\text{divides}\quad f_p .\] If you want you can restrict to the case \(h = y^n\), even with \(n = 2\), but it is true in general.

3. Discuss the relevance of this to Exercises 6 and 7 of the previous set.

Exercise

Unsolved problems. They may be really hard or they may be easy. I don’t know.

1. Is there any \(f \in {\mathbf Z}[x, u]\) such that \(f_p\) is irreducible for an infinite number of \(p\)? (Hint: Yes, this happens for \(f(x, u) = u - x - 1\) and also for \(f(x, u) = u^2 - x^2 + 1\).)

2. Let \(f \in {\mathbf Z}[x, u]\) nonzero, and suppose \(\deg_x(f_p) = dp + d'\) for all large \(p\). (In other words \(\deg_u(f) = d\) and the coefficient \(c\) of \(u^d\) in \(f\) has \(\deg_x(c) = d'\).) Suppose we can write \(d = d_1 + d_2\) and \(d' = d'_1 + d'_2\) with \(d_1, d_2 > 0\) and \(d'_1, d'_2 \geq 0\) such that for all sufficiently large \(p\) there exists a factorization \[f_p = f_{1, p} f_{2, p}\] with \(\deg_x(f_{1, p}) = d_1p + d'_1\). Is it true that \(f\) comes about via a norm construction as in Exercise 4? (More precisely, are there a \(h\) and \(g\) such that \(Norm_{B_p/A_p}(g_p)\) divides \(f_p\) for all \(p >> 0\).)

3. Analogous question to the one in (b) but now with \(f \in {\mathbf Z}[x_1, x_2, u_1, u_2]\) irreducible and just assuming that \(f_p(x_1, x_2) = f(x_1, x_2, x_1^p, x_2^p) \bmod p\) factors for all \(p >> 0\).

Filtered derived category

In order to do the exercises in this section, please read the material in Homology, Section 0120. We will say \(A\) is a filtered object of \(\mathcal{A}\), to mean that \(A\) comes endowed with a filtration \(F\) which we omit from the notation.

Exercise

Let \(\mathcal{A}\) be an abelian category. Let \(I\) be a filtered object of \(\mathcal{A}\). Assume that the filtration on \(I\) is finite and that each \(\text{gr}^p(I)\) is an injective object of \(\mathcal{A}\). Show that there exists an isomorphism \(I \cong \bigoplus \text{gr}^p(I)\) with filtration \(F^p(I)\) corresponding to \(\bigoplus_{p' \geq p} \text{gr}^p(I)\).

Exercise

Let \(\mathcal{A}\) be an abelian category. Let \(I\) be a filtered object of \(\mathcal{A}\). Assume that the filtration on \(I\) is finite. Show the following are equivalent:

1. For any solid diagram \[\xymatrix{ A \ar[r]_\alpha \ar[d] & B \ar@{-->}[ld] \\ I & }\] of filtered objects with (i) the filtrations on \(A\) and \(B\) are finite, and (ii) \(\text{gr}(\alpha)\) injective the dotted arrow exists making the diagram commute.

2. Each \(\text{gr}^p I\) is injective.

Note that given a morphism \(\alpha : A \to B\) of filtered objects with finite filtrations to say that \(\text{gr}(\alpha)\) injective is the same thing as saying that \(\alpha\) is a strict monomorphism in the category \(\text{Fil}(\mathcal{A})\). Namely, being a monomorphism means \(\Ker(\alpha) = 0\) and strict means that this also implies \(\Ker(\text{gr}(\alpha)) = 0\). See Homology, Lemma 0127. (We only use the term “injective” for a morphism in an abelian category, although it makes sense in any additive category having kernels.) The exercises above justifies the following definition.

Definition

Let \(\mathcal{A}\) be an abelian category. Let \(I\) be a filtered object of \(\mathcal{A}\). Assume the filtration on \(I\) is finite. We say \(I\) is filtered injective if each \(\text{gr}^p(I)\) is an injective object of \(\mathcal{A}\).

We make the following definition to avoid having to keep saying “with a finite filtration” everywhere.

Definition

Let \(\mathcal{A}\) be an abelian category. We denote \(\text{Fil}^f(\mathcal{A})\) the full subcategory of \(\text{Fil}(\mathcal{A})\) whose objects consist of those \(A \in \Ob(\text{Fil}(\mathcal{A}))\) whose filtration is finite.

Exercise

Let \(\mathcal{A}\) be an abelian category. Assume \(\mathcal{A}\) has enough injectives. Let \(A\) be an object of \(\text{Fil}^f(\mathcal{A})\). Show that there exists a strict monomorphism \(\alpha : A \to I\) of \(A\) into a filtered injective object \(I\) of \(\text{Fil}^f(\mathcal{A})\).

Definition

Let \(\mathcal{A}\) be an abelian category. Let \(\alpha : K^\bullet \to L^\bullet\) be a morphism of complexes of \(\text{Fil}(\mathcal{A})\). We say that \(\alpha\) is a filtered quasi-isomorphism if for each \(p \in \mathbf{Z}\) the morphism \(\text{gr}^p(K^\bullet) \to \text{gr}^p(L^\bullet)\) is a quasi-isomorphism.

Definition

Let \(\mathcal{A}\) be an abelian category. Let \(K^\bullet\) be a complex of \(\text{Fil}^f(\mathcal{A})\). We say that \(K^\bullet\) is filtered acyclic if for each \(p \in \mathbf{Z}\) the complex \(\text{gr}^p(K^\bullet)\) is acyclic.

Exercise

Let \(\mathcal{A}\) be an abelian category. Let \(\alpha : K^\bullet \to L^\bullet\) be a morphism of bounded below complexes of \(\text{Fil}^f(\mathcal{A})\). (Note the superscript \(f\).) Show that the following are equivalent:

1. \(\alpha\) is a filtered quasi-isomorphism,

2. for each \(p \in \mathbf{Z}\) the map \(\alpha : F^pK^\bullet \to F^pL^\bullet\) is a quasi-isomorphism,

3. for each \(p \in \mathbf{Z}\) the map \(\alpha : K^\bullet/F^pK^\bullet \to L^\bullet/F^pL^\bullet\) is a quasi-isomorphism, and

4. the cone of \(\alpha\) (see Derived Categories, Definition 014E) is a filtered acyclic complex.

Moreover, show that if \(\alpha\) is a filtered quasi-isomorphism then \(\alpha\) is also a usual quasi-isomorphism.

Exercise

Let \(\mathcal{A}\) be an abelian category. Assume \(\mathcal{A}\) has enough injectives. Let \(A\) be an object of \(\text{Fil}^f(\mathcal{A})\). Show there exists a complex \(I^\bullet\) of \(\text{Fil}^f(\mathcal{A})\), and a morphism \(A[0] \to I^\bullet\) such that

1. each \(I^p\) is filtered injective,

2. \(I^p = 0\) for \(p < 0\), and

3. \(A[0] \to I^\bullet\) is a filtered quasi-isomorphism.

Exercise

Let \(\mathcal{A}\) be an abelian category. Assume \(\mathcal{A}\) has enough injectives. Let \(K^\bullet\) be a bounded below complex of objects of \(\text{Fil}^f(\mathcal{A})\). Show there exists a filtered quasi-isomorphism \(\alpha : K^\bullet \to I^\bullet\) with \(I^\bullet\) a complex of \(\text{Fil}^f(\mathcal{A})\) having filtered injective terms \(I^n\), and bounded below. In fact, we may choose \(\alpha\) such that each \(\alpha^n\) is a strict monomorphism.

Exercise

Let \(\mathcal{A}\) be an abelian category. Consider a solid diagram \[\xymatrix{ K^\bullet \ar[r]_\alpha \ar[d]_\gamma & L^\bullet \ar@{-->}[dl]^\beta \\ I^\bullet }\] of complexes of \(\text{Fil}^f(\mathcal{A})\). Assume \(K^\bullet\), \(L^\bullet\) and \(I^\bullet\) are bounded below and assume each \(I^n\) is a filtered injective object. Also assume that \(\alpha\) is a filtered quasi-isomorphism.

1. There exists a map of complexes \(\beta\) making the diagram commute up to homotopy.

2. If \(\alpha\) is a strict monomorphism in every degree then we can find a \(\beta\) which makes the diagram commute.

Exercise

Let \(\mathcal{A}\) be an abelian category. Let \(K^\bullet\), \(K^\bullet\) be complexes of \(\text{Fil}^f(\mathcal{A})\). Assume

1. \(K^\bullet\) bounded below and filtered acyclic, and

2. \(I^\bullet\) bounded below and consisting of filtered injective objects.

Then any morphism \(K^\bullet \to I^\bullet\) is homotopic to zero.

Exercise

Let \(\mathcal{A}\) be an abelian category. Consider a solid diagram \[\xymatrix{ K^\bullet \ar[r]_\alpha \ar[d]_\gamma & L^\bullet \ar@{-->}[dl]^{\beta_i} \\ I^\bullet }\] of complexes of \(\text{Fil}^f(\mathcal{A})\). Assume \(K^\bullet\), \(L^\bullet\) and \(I^\bullet\) bounded below and each \(I^n\) a filtered injective object. Also assume \(\alpha\) a filtered quasi-isomorphism. Any two morphisms \(\beta_1, \beta_2\) making the diagram commute up to homotopy are homotopic.

Regular functions

Exercise

Consider the affine curve \(X\) given by the equation \(t^2 = s^5 + 8\) in \(\mathbf{C}^2\) with coordinates \(s, t\). Let \(x \in X\) be the point with coordinates \((1, 3)\). Let \(U = X \setminus \{x\}\). Prove that there is a regular function on \(U\) which is not the restriction of a regular function on \(\mathbf{C}^2\), i.e., is not the restriction of a polynomial in \(s\) and \(t\) to \(U\).

Exercise

Let \(n \geq 2\). Let \(E \subset \mathbf{C}^n\) be a finite subset. Show that any regular function on \(\mathbf{C}^n \setminus E\) is a polynomial.

Exercise

Let \(X \subset \mathbf{C}^n\) be an affine variety. Let us say \(X\) is a cone if \(x = (a_1, \ldots, a_n) \in X\) and \(\lambda \in \mathbf{C}\) implies \((\lambda a_1, \ldots, \lambda a_n) \in X\). Of course, if \(\mathfrak p \subset \mathbf{C}[x_1, \ldots, x_n]\) is a prime ideal generated by homogeneous polynomials in \(x_1, \ldots, x_n\), then the affine variety \(X = V(\mathfrak p) \subset \mathbf{C}^n\) is a cone. Show that conversely the prime ideal \(\mathfrak p \subset \mathbf{C}[x_1, \ldots, x_n]\) corresponding to a cone can be generated by homogeneous polynomials in \(x_1, \ldots, x_n\).

Exercise

Give an example of an affine variety \(X \subset \mathbf{C}^n\) which is a cone (see Exercise 0E9F) and a regular function \(f\) on \(U = X \setminus \{(0, \ldots, 0)\}\) which is not the restriction of a polynomial function on \(\mathbf{C}^n\).

Exercise

In this exercise we try to see what happens with regular functions over non-algebraically closed fields. Let \(k\) be a field. Let \(Z \subset k^n\) be a Zariski locally closed subset, i.e., there exist ideals \(I \subset J \subset k[x_1, \ldots, x_n]\) such that \[Z = \{a \in k^n \mid f(a) = 0\ \forall\ f \in I,\ \exists\ g \in J,\ g(a) \not = 0\}.\] A function \(\varphi : Z \to k\) is said to be regular if for every \(z \in Z\) there exists a Zariski open neighbourhood \(z \in U \subset Z\) and polynomials \(f, g \in k[x_1, \ldots, x_n]\) such that \(g(u) \not = 0\) for all \(u \in U\) and such that \(\varphi(u) = f(u)/g(u)\) for all \(u \in U\).

1. If \(k = \bar k\) and \(Z = k^n\) show that regular functions are given by polynomials. (Only do this if you haven’t seen this argument before.)

2. If \(k\) is finite show that (a) every function \(\varphi\) is regular, (b) the ring of regular functions is finite dimensional over \(k\). (If you like you can take \(Z = k^n\) and even \(n = 1\).)

3. If \(k = \mathbf{R}\) give an example of a regular function on \(Z = \mathbf{R}\) which is not given by a polynomial.

4. If \(k = \mathbf{Q}_p\) give an example of a regular function on \(Z = \mathbf{Q}_p\) which is not given by a polynomial.

Sheaves

A morphism \(f : X \to Y\) of a category \(\mathcal{C}\) is an monomorphism if for every pair of morphisms \(a, b : W \to X\) we have \(f \circ a = f \circ b \Rightarrow a = b\). A monomorphism in the category of sets is an injective map of sets.

Exercise

Carefully prove that a map of sheaves of sets is an monomorphism (in the category of sheaves of sets) if and only if the induced maps on all the stalks are injective.

A morphism \(f : X \to Y\) of a category \(\mathcal{C}\) is an isomorphism if there exists a morphism \(g : Y \to X\) such that \(f \circ g = \text{id}_Y\) and \(g \circ f = \text{id}_X\). An isomorphism in the category of sets is a bijective map of sets.

Exercise

Carefully prove that a map of sheaves of sets is an isomorphism (in the category of sheaves of sets) if and only if the induced maps on all the stalks are bijective.

A morphism \(f : X \to Y\) of a category \(\mathcal{C}\) is an epimorphism if for every pair of morphisms \(a, b : Y \to Z\) we have \(a \circ f = b \circ f \Rightarrow a = b\). An epimorphism in the category of sets is a surjective map of sets.

Exercise

Carefully prove that a map of sheaves of sets is an epimorphism (in the category of sheaves of sets) if and only if the induced maps on all the stalks are surjective.

Exercise

Let \(f : X \to Y\) be a map of topological spaces. Prove pushforward \(f_\ast\) and pullback \(f^{-1}\) for sheaves of sets form an adjoint pair of functors.

Exercise

Let \(j : U \to X\) be an open immersion. Show that

1. Pullback \(j^{-1} : \Sh(X) \to \Sh(U)\) has a left adjoint \(j_{!} : \Sh(U) \to \Sh(X)\) called extension by the empty set.

2. Characterize the stalks of \(j_{!}({\mathcal G})\) for \(\mathcal{G} \in \Sh(U)\).

3. Pullback \(j^{-1} : \textit{Ab}(X) \to \textit{Ab}(U)\) has a left adjoint \(j_{!} : \textit{Ab}(U) \to \textit{Ab}(X)\) called extension by zero.

4. Characterize the stalks of \(j_{!}({\mathcal G})\) for \(\mathcal{G} \in \textit{Ab}(U)\).

Observe that extension by zero differs from extension by the empty set!

Exercise

Let \(X = \mathbf{R}\) with the usual topology. Let \(\mathcal{O}_X = \underline{\mathbf{Z}/2\mathbf{Z}}_X\). Let \(i : Z = \{0\} \to X\) be the inclusion and let \(\mathcal{O}_Z = \underline{\mathbf{Z}/2\mathbf{Z}}_Z\). Prove the following (the first three follow from the definitions but if you are not clear on the definitions you should elucidate them):

1. \(i_*\mathcal{O}_Z\) is a skyscraper sheaf.

2. There is a canonical surjective map from \(\underline{\mathbf{Z}/2\mathbf{Z}}_X \to i_*\underline{\mathbf{Z}/2\mathbf{Z}}_Z\). Denote the kernel \(\mathcal{I} \subset \mathcal{O}_X\).

3. \(\mathcal{I}\) is an ideal sheaf of \(\mathcal{O}_X\).

4. The sheaf \(\mathcal{I}\) on \(X\) cannot be locally generated by sections (as in Modules, Definition 01B2.)

Exercise

Let \(X\) be a topological space. Let \({\mathcal F}\) be an abelian sheaf on \(X\). Show that \({\mathcal F}\) is the quotient of a (possibly very large) direct sum of sheaves all of whose terms are of the form \[j_{!}(\underline{{\mathbf Z}}_U)\] where \(U \subset X\) is open and \(\underline{{\mathbf Z}}_U\) denotes the constant sheaf with value \({\mathbf Z}\) on \(U\).

Remark

Let \(X\) be a topological space. In the category of abelian sheaves the direct sum of a family of sheaves \(\{{\mathcal F}_i\}_{i\in I}\) is the sheaf associated to the presheaf \(U \mapsto \oplus {\mathcal F}_i(U)\). Consequently the stalk of the direct sum at a point \(x\) is the direct sum of the stalks of the \({\mathcal F}_i\) at \(x\).

Exercise

Let \(X\) be a topological space. Suppose we are given a collection of abelian groups \(A_x\) indexed by \(x \in X\). Show that the rule \(U \mapsto \prod_{x \in U} A_x\) with obvious restriction mappings defines a sheaf \(\mathcal{G}\) of abelian groups. Show, by an example, that usually it is not the case that \(\mathcal{G}_x = A_x\) for \(x \in X\).

Exercise

Let \(X\), \(A_x\), \(\mathcal{G}\) be as in Exercise 078Z. Let \(\mathcal{B}\) be a basis for the topology of \(X\), see Topology, Definition 004P. For \(U \in \mathcal{B}\) let \(A_U\) be a subgroup \(A_U \subset \mathcal{G}(U) = \prod_{x \in U} A_x\). Assume that for \(U \subset V\) with \(U, V \in \mathcal{B}\) the restriction maps \(A_V\) into \(A_U\). For \(U \subset X\) open set \[\mathcal{F}(U) = \left\{ (s_x)_{x \in U} \middle| \begin{matrix} \text{ for every }x\text{ in }U\text{ there exists } V \in \mathcal{B} \\ x \in V \subset U\text{ such that } (s_y)_{y \in V} \in A_V \end{matrix} \right\}\] Show that \(\mathcal{F}\) defines a sheaf of abelian groups on \(X\). Show, by an example, that it is usually not the case that \(\mathcal{F}(U) = A_U\) for \(U \in \mathcal{B}\).

Exercise

Give an example of a topological space \(X\) and a functor \[F : \Sh(X) \longrightarrow \textit{Sets}\] which is exact (commutes with finite products and equalizers and commutes with finite coproducts and coequalizers, see Categories, Section 0033), but there is no point \(x \in X\) such that \(F\) is isomorphic to the stalk functor \(\mathcal{F} \mapsto \mathcal{F}_x\).

Schemes

Let \(LRS\) be the category of locally ringed spaces. An affine scheme is an object in \(LRS\) isomorphic in \(LRS\) to a pair of the form \((\Spec(A), {\mathcal O}_{\Spec(A)})\). A scheme is an object \((X, {\mathcal O}_X)\) of \(LRS\) such that every point \(x\in X\) has an open neighbourhood \(U \subset X\) such that the pair \((U, {\mathcal O}_X|_U)\) is an affine scheme.

Exercise

Find a \(1\)-point locally ringed space which is not a scheme.

Exercise

Suppose that \(X\) is a scheme whose underlying topological space has 2 points. Show that \(X\) is an affine scheme.

Exercise

Suppose that \(X\) is a scheme whose underlying topological space is a finite discrete set. Show that \(X\) is an affine scheme.

Exercise

Show that there exists a non-affine scheme having three points.

Exercise

Suppose that \(X\) is a nonempty quasi-compact scheme. Show that \(X\) has a closed point.

Remark

When \((X, {\mathcal O}_X)\) is a ringed space and \(U \subset X\) is an open subset then \((U, {\mathcal O}_X|_U)\) is a ringed space. Notation: \({\mathcal O}_U = {\mathcal O}_X|_U\). There is a canonical morphisms of ringed spaces \[j : (U, {\mathcal O}_U) \longrightarrow (X, {\mathcal O}_X).\] If \((X, {\mathcal O}_X)\) is a locally ringed space, so is \((U, {\mathcal O}_U)\) and \(j\) is a morphism of locally ringed spaces. If \((X, {\mathcal O}_X)\) is a scheme so is \((U, {\mathcal O}_U)\) and \(j\) is a morphism of schemes. We say that \((U, {\mathcal O}_U)\) is an open subscheme of \((X, {\mathcal O}_X)\) and that \(j\) is an open immersion. More generally, any morphism \(j' : (V, {\mathcal O}_V) \to (X, {\mathcal O}_X)\) that is isomorphic to a morphism \(j : (U, {\mathcal O}_U) \to (X, {\mathcal O}_X)\) as above is called an open immersion.

Exercise

Give an example of an affine scheme \((X, {\mathcal O}_X)\) and an open \(U \subset X\) such that \((U, {\mathcal O}_X|U)\) is not an affine scheme.

Exercise

Given an example of a pair of affine schemes \((X, {\mathcal O}_X)\), \((Y, {\mathcal O}_Y)\), an open subscheme \((U, {\mathcal O}_X|_U)\) of \(X\) and a morphism of schemes \((U, {\mathcal O}_X|_U) \to (Y, {\mathcal O}_Y)\) that does not extend to a morphism of schemes \((X, {\mathcal O}_X) \to (Y, {\mathcal O}_Y)\).

Exercise

(This is pretty hard.) Given an example of a scheme \(X\), and open subscheme \(U \subset X\) and a closed subscheme \(Z \subset U\) such that \(Z\) does not extend to a closed subscheme of \(X\).

Exercise

Give an example of a scheme \(X\), a field \(K\), and a morphism of ringed spaces \(\Spec(K) \to X\) which is NOT a morphism of schemes.

Exercise

Do all the exercises in [H, Chapter II], Sections 1 and 2...  Just kidding!

Definition

A scheme \(X\) is called integral if \(X\) is nonempty and for every nonempty affine open \(U \subset X\) the ring \(\Gamma(U, \mathcal{O}_X) = \mathcal{O}_X(U)\) is a domain.

Exercise

Give an example of a morphism of integral schemes \(f : X \to Y\) such that the induced maps \({\mathcal O}_{Y, f(x)} \to {\mathcal O}_{X, x}\) are surjective for all \(x\in X\), but \(f\) is not a closed immersion.

Exercise

Give an example of a fibre product \(X \times_S Y\) such that \(X\) and \(Y\) are affine but \(X \times_S Y\) is not.

Remark

It turns out this cannot happen with \(S\) separated. Do you know why?

Exercise

Give an example of a scheme \(V\) which is integral 1-dimensional scheme of finite type over \({\mathbf Q}\) such that \(\Spec({\mathbf C}) \times_{\Spec({\mathbf Q})} V\) is not integral.

Exercise

Give an example of a scheme \(V\) which is integral 1-dimensional scheme of finite type over a field \(k\) such that \(\Spec(k') \times_{\Spec(k)} V\) is not reduced for some finite field extension \(k'/k\).

Remark

If your scheme is affine then dimension is the same as the Krull dimension of the underlying ring. So you can use last semesters results to compute dimension.

Morphisms

An important question is, given a morphism \(\pi : X \to S\), whether the morphism has a section or a rational section. Here are some example exercises.

Exercise

Consider the morphism of schemes \[\pi : X = \Spec(\mathbf{C}[x, t, 1/xt]) \longrightarrow S = \Spec(\mathbf{C}[t]).\]

1. Show there does not exist a morphism \(\sigma : S \to X\) such that \(\pi \circ \sigma = \text{id}_S\).

2. Show there does exist a nonempty open \(U \subset S\) and a morphism \(\sigma : U \to X\) such that \(\pi \circ \sigma = \text{id}_U\).

Exercise

Consider the morphism of schemes \[\pi : X = \Spec(\mathbf{C}[x, t]/(x^2 + t)) \longrightarrow S = \Spec(\mathbf{C}[t]).\] Show there does not exist a nonempty open \(U \subset S\) and a morphism \(\sigma : U \to X\) such that \(\pi \circ \sigma = \text{id}_U\).

Exercise

Let \(A, B, C \in \mathbf{C}[t]\) be nonzero polynomials. Consider the morphism of schemes \[\pi : X = \Spec(\mathbf{C}[x, y, t]/(A + Bx^2 + Cy^2)) \longrightarrow S = \Spec(\mathbf{C}[t]).\] Show there does exist a nonempty open \(U \subset S\) and a morphism \(\sigma : U \to X\) such that \(\pi \circ \sigma = \text{id}_U\). (Hint: Symbolically, write \(x = X/Z\), \(y = Y/Z\) for some \(X, Y, Z \in \mathbf{C}[t]\) of degree \(\leq d\) for some \(d\), and work out the condition that this solves the equation. Then show, using dimension theory, that if \(d >> 0\) you can find nonzero \(X, Y, Z\) solving the equation.)

Remark

Exercise 0296 is a special case of “Tsen’s theorem”. Exercise 0297 shows that the method is limited to low degree equations (conics when the base and fibre have dimension 1).

Exercise

Consider the morphism of schemes \[\pi : X = \Spec(\mathbf{C}[x, y, t] /(1 + t x^3 + t^2 y^3)) \longrightarrow S = \Spec(\mathbf{C}[t])\] Show there does not exist a nonempty open \(U \subset S\) and a morphism \(\sigma : U \to X\) such that \(\pi \circ \sigma = \text{id}_U\).

Exercise

Consider the schemes \[X = \Spec(\mathbf{C}[\{x_i\}_{i = 1}^{8}, s, t] /(1 + s x_1^3 + s^2 x_2^3 + t x_3^3 + st x_4^3 + s^2t x_5^3 + t^2 x_6^3 + st^2 x_7^3 + s^2t^2 x_8^3))\] and \[S = \Spec(\mathbf{C}[s, t])\] and the morphism of schemes \[\pi : X \longrightarrow S\] Show there does not exist a nonempty open \(U \subset S\) and a morphism \(\sigma : U \to X\) such that \(\pi \circ \sigma = \text{id}_U\).

Exercise

(For the number theorists.) Give an example of a closed subscheme \[Z \subset \Spec\left({\mathbf Z}[x, \frac{1 }{ x(x-1)(2x-1)}]\right)\] such that the morphism \(Z \to \Spec({\mathbf Z})\) is finite and surjective.

Exercise

If you do not like number theory, you can try the variant where you look at \[\Spec\left({\mathbf F}_p[t, x, \frac{1 }{ x(x-t)(tx-1)}]\right) \longrightarrow \Spec({\mathbf F}_p[t])\] and you try to find a closed subscheme of the top scheme which maps finite surjectively to the bottom one. (There is a theoretical reason for having a finite ground field here; although it may not be necessary in this particular case.)

Remark

In Exercises 0299 and 029A we are given a morphism \(f : X \to S\) where \(S\) is the spectrum of a Dedekind ring \(A\) with \(f\) flat and surjective with geometrically irreducible generic fibre. In both cases the outcome is that there exists a finite surjective morphism \(S' \to S\) such that the base change \(X_{S'} \to S'\) does have a section. It turns out this holds if \(A\) is excellent, its residue fields at maximal ideals are algebraic extensions of finite fields, and \(\Pic(A')\) is torsion when \(A'\) is the integral closure of \(A\) in a finite extension of its fraction field, see [MB1]. For example, if \(A = \mathbf{Z}\) or \(A = \mathbf{F}_p[x]\) or a finite extension of these. However, it turns out that there exists a Dedekind ring \(A\) with finite residue fields at maximal primes whose Picard group has a nontorsion element, see [Goldman], and the result is false for the spectrum of such a ring. Exercise 029B gives a geometric example where it doesn’t work.

Exercise

Prove there exist a \(f \in \mathbf{C}[x, t]\) which is not divisible by \(t - \alpha\) for any \(\alpha \in \mathbf{C}\) such that there does not exist any \(Z \subset \Spec(\mathbf{C}[x, t, 1/f])\) which maps finite surjectively to \(\Spec(\mathbf{C}[t])\). (I think that \(f(x, t) = (xt - 2)(x - t + 3)\) works. To show any candidate has the required property is not so easy I think.)

Exercise

Let \(A \to B\) be a finite type ring map. Suppose that \(\Spec(B) \to \Spec(A)\) factors through a closed immersion \(\Spec(B) \to \mathbf{P}^n_A\) for some \(n\). Prove that \(A \to B\) is a finite ring map, i.e., that \(B\) is finite as an \(A\)-module. Hint: if \(A\) is Noetherian (please just assume this) you can argue using that \(H^i(Z, \mathcal{O}_Z)\) for \(i \in \mathbf{Z}\) is a finite \(A\)-module for every closed subscheme \(Z \subset \mathbf{P}^n_A\).

Exercise

Let \(k\) be an algebraically closed field. Let \(f : X \to Y\) be a morphism of projective varieties such that \(f^{-1}(\{y\})\) is finite for every closed point \(y \in Y\). Prove that \(f\) is finite as a morphism of schemes. Hints: (a) being finite is a local property, (b) try to reduce to Exercise 0EG6, and (c) use a closed immersion \(X \to \mathbf{P}^n_k\) to get a closed immersion \(X \to \mathbf{P}^n_Y\) over \(Y\).

Tangent Spaces

Definition

For any ring \(R\) we denote \(R[\epsilon]\) the ring of dual numbers. As an \(R\)-module it is free with basis \(1\), \(\epsilon\). The ring structure comes from setting \(\epsilon^2 = 0\).

Exercise

Let \(f : X \to S\) be a morphism of schemes. Let \(x \in X\) be a point, let \(s = f(x)\). Consider the solid commutative diagram \[\xymatrix{ \Spec(\kappa(x)) \ar[r] \ar[dr] \ar@/^1pc/[rr] & \Spec(\kappa(x)[\epsilon]) \ar@{.>}[r] \ar[d]& X \ar[d] \\ & \Spec(\kappa(s)) \ar[r] & S }\] with the curved arrow being the canonical morphism of \(\Spec(\kappa(x))\) into \(X\). If \(\kappa(x) = \kappa(s)\) show that the set of dotted arrows which make the diagram commute are in one to one correspondence with the set of linear maps \[\Hom_{\kappa(x)}( \frac{\mathfrak m_x}{\mathfrak m_x^2 + \mathfrak m_s\mathcal{O}_{X, x}}, \kappa(x))\] In other words: describe such a bijection. (This works more generally if \(\kappa(x) \supset \kappa(s)\) is a separable algebraic extension.)

Definition

Let \(f : X \to S\) be a morphism of schemes. Let \(x \in X\). We dub the set of dotted arrows of Exercise 029E the tangent space of \(X\) over \(S\) and we denote it \(T_{X/S, x}\). An element of this space is called a tangent vector of \(X/S\) at \(x\).

Exercise

For any field \(K\) prove that the diagram \[\xymatrix{ \Spec(K) \ar[r] \ar[d] & \Spec(K[\epsilon_1]) \ar[d] \\ \Spec(K[\epsilon_2]) \ar[r] & \Spec(K[\epsilon_1, \epsilon_2]/(\epsilon_1\epsilon_2)) }\] is a pushout diagram in the category of schemes. (Here \(\epsilon_i^2 = 0\) as before.)

Exercise

Let \(f : X \to S\) be a morphism of schemes. Let \(x \in X\). Define addition of tangent vectors, using Exercise 029G and a suitable morphism \[\Spec(K[\epsilon]) \longrightarrow \Spec(K[\epsilon_1, \epsilon_2]/(\epsilon_1\epsilon_2)).\] Similarly, define scalar multiplication of tangent vectors (this is easier). Show that \(T_{X/S, x}\) becomes a \(\kappa(x)\)-vector space with your constructions.

Exercise

Let \(k\) be a field. Consider the structure morphism \(f : X = \mathbf{A}^1_k \to \Spec(k) = S\).

1. Let \(x \in X\) be a closed point. What is the dimension of \(T_{X/S, x}\)?

2. Let \(\eta \in X\) be the generic point. What is the dimension of \(T_{X/S, \eta}\)?

3. Consider now \(X\) as a scheme over \(\Spec(\mathbf{Z})\). What are the dimensions of \(T_{X/\mathbf{Z}, x}\) and \(T_{X/\mathbf{Z}, \eta}\)?

Remark

Exercise 029I explains why it is necessary to consider the tangent space of \(X\) over \(S\) to get a good notion.

Exercise

Consider the morphism of schemes \[f : X = \Spec(\mathbf{F}_p(t)) \longrightarrow \Spec(\mathbf{F}_p(t^p)) = S\] Compute the tangent space of \(X/S\) at the unique point of \(X\). Isn’t that weird? What do you think happens if you take the morphism of schemes corresponding to \(\mathbf{F}_p[t^p] \to \mathbf{F}_p[t]\)?

Exercise

Let \(k\) be a field. Compute the tangent space of \(X/k\) at the point \(x = (0, 0)\) where \(X = \Spec(k[x, y]/(x^2 - y^3))\).

Exercise

Let \(f : X \to Y\) be a morphism of schemes over \(S\). Let \(x \in X\) be a point. Set \(y = f(x)\). Assume that the natural map \(\kappa(y) \to \kappa(x)\) is bijective. Show, using the definition, that \(f\) induces a natural linear map \[\text{d}f : T_{X/S, x} \longrightarrow T_{Y/S, y}.\] Match it with what happens on local rings via Exercise 029E in case \(\kappa(x) = \kappa(s)\).

Exercise

Let \(k\) be an algebraically closed field. Let \[\begin{eqnarray*} f : \mathbf{A}_k^n & \longrightarrow & \mathbf{A}^m_k \\ (x_1, \ldots, x_n) & \longmapsto & (f_1(x_i), \ldots, f_m(x_i)) \end{eqnarray*}\] be a morphism of schemes over \(k\). This is given by \(m\) polynomials \(f_1, \ldots, f_m\) in \(n\) variables. Consider the matrix \[A = \left( \frac{\partial f_j}{\partial x_i} \right)\] Let \(x \in \mathbf{A}^n_k\) be a closed point. Set \(y = f(x)\). Show that the map on tangent spaces \(T_{\mathbf{A}^n_k/k, x} \to T_{\mathbf{A}^m_k/k, y}\) is given by the value of the matrix \(A\) at the point \(x\).

Quasi-coherent Sheaves

Definition

Let \(X\) be a scheme. A sheaf \(\mathcal{F}\) of \(\mathcal{O}_X\)-modules is quasi-coherent if for every affine open \(\Spec(R) = U \subset X\) the restriction \(\mathcal{F}|_U\) is of the form \(\widetilde M\) for some \(R\)-module \(M\).

It is enough to check this conditions on the members of an affine open covering of \(X\). See Schemes, Section 01LA for more results.

Definition

Let \(X\) be a topological space. Let \(x, x' \in X\). We say \(x\) is a specialization of \(x'\) if and only if \(x \in \overline{\{x'\}}\).

Exercise

Let \(X\) be a scheme. Let \(x, x' \in X\). Let \(\mathcal{F}\) be a quasi-coherent sheaf of \(\mathcal{O}_X\)-modules. Suppose that (a) \(x\) is a specialization of \(x'\) and (b) \(\mathcal{F}_{x'} \not = 0\). Show that \(\mathcal{F}_x \not = 0\).

Exercise

Find an example of a scheme \(X\), points \(x, x' \in X\), a sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}\) such that (a) \(x\) is a specialization of \(x'\) and (b) \(\mathcal{F}_{x'} \not = 0\) and \(\mathcal{F}_x = 0\).

Definition

A scheme \(X\) is called locally Noetherian if and only if for every point \(x \in X\) there exists an affine open \(\Spec(R) = U \subset X\) such that \(R\) is Noetherian. A scheme is Noetherian if it is locally Noetherian and quasi-compact.

If \(X\) is locally Noetherian then any affine open of \(X\) is the spectrum of a Noetherian ring, see Properties, Lemma 01OW.

Definition

Let \(X\) be a locally Noetherian scheme. Let \(\mathcal{F}\) be a quasi-coherent sheaf of \(\mathcal{O}_X\)-modules. We say \(\mathcal{F}\) is coherent if for every point \(x \in X\) there exists an affine open \(\Spec(R) = U \subset X\) such that \(\mathcal{F}|_U\) is isomorphic to \(\widetilde M\) for some finite \(R\)-module \(M\).

Exercise

Let \(X = \Spec(R)\) be an affine scheme.

1. Let \(f \in R\). Let \(\mathcal{G}\) be a quasi-coherent sheaf of \(\mathcal{O}_{D(f)}\)-modules on the open subscheme \(D(f)\). Show that \(\mathcal{G} = \mathcal{F}|_{D(f)}\) for some quasi-coherent sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}\).

2. Let \(I \subset R\) be an ideal. Let \(i : Z \to X\) be the closed subscheme of \(X\) corresponding to \(I\). Let \(\mathcal{G}\) be a quasi-coherent sheaf of \(\mathcal{O}_Z\)-modules on the closed subscheme \(Z\). Show that \(\mathcal{G} = i^*\mathcal{F}\) for some quasi-coherent sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}\). (Why is this silly?)

3. Assume that \(R\) is Noetherian. Let \(f \in R\). Let \(\mathcal{G}\) be a coherent sheaf of \(\mathcal{O}_{D(f)}\)-modules on the open subscheme \(D(f)\). Show that \(\mathcal{G} = \mathcal{F}|_{D(f)}\) for some coherent sheaf of \(\mathcal{O}_X\)-modules \(\mathcal{F}\).

Remark

If \(U \to X\) is a quasi-compact immersion then any quasi-coherent sheaf on \(U\) is the restriction of a quasi-coherent sheaf on \(X\). If \(X\) is a Noetherian scheme, and \(U \subset X\) is open, then any coherent sheaf on \(U\) is the restriction of a coherent sheaf on \(X\). Of course the exercise above is easier, and shouldn’t use these general facts.

Proj and projective schemes

Exercise

Give examples of graded rings \(S\) such that

1. \(\text{Proj}(S)\) is affine and nonempty, and

2. \(\text{Proj}(S)\) is integral, nonempty but not isomorphic to \({\mathbf P}^n_A\) for any \(n\geq 0\), any ring \(A\).

Exercise

Give an example of a nonconstant morphism of schemes \({\mathbf P}^1_{\mathbf C} \to {\mathbf P}^5_{\mathbf C}\) over \(\Spec({\mathbf C})\).

Exercise

Give an example of an isomorphism of schemes \[{\mathbf P}^1_{\mathbf C} \to \text{Proj}({\mathbf C}[X_0, X_1, X_2]/(X_0^2 + X_1^2 + X_2^2))\]

Exercise

Give an example of a morphism of schemes \(f : X \to {\mathbf A}^1_{\mathbf C} = \Spec({\mathbf C}[T])\) such that the (scheme theoretic) fibre \(X_t\) of \(f\) over \(t \in {\mathbf A}^1_{\mathbf C}\) is (a) isomorphic to \({\mathbf P}^1_{\mathbf C}\) when \(t\) is a closed point not equal to \(0\), and (b) not isomorphic to \({\mathbf P}^1_{\mathbf C}\) when \(t = 0\). We will call \(X_0\) the special fibre of the morphism. This can be done in many, many ways. Try to give examples that satisfy (each of) the following additional restraints (unless it isn’t possible):

1. Can you do it with special fibre projective?

2. Can you do it with special fibre irreducible and projective?

3. Can you do it with special fibre integral and projective?

4. Can you do it with special fibre smooth and projective?

5. Can you do it with \(f\) a flat morphism? This just means that for every affine open \(\Spec(A) \subset X\) the induced ring map \(\mathbf{C}[t] \to A\) is flat, which in this case means that any nonzero polynomial in \(t\) is a nonzerodivisor on \(A\).

6. Can you do it with \(f\) a flat and projective morphism?

7. Can you do it with \(f\) flat, projective and special fibre reduced?

8. Can you do it with \(f\) flat, projective and special fibre irreducible?

9. Can you do it with \(f\) flat, projective and special fibre integral?

What do you think happens when you replace \({\mathbf P}^1_{\mathbf C}\) with another variety over \({\mathbf C}\)? (This can get very hard depending on which of the variants above you ask for.)

Exercise

Let \(n \geq 1\) be any positive integer. Give an example of a surjective morphism \(X \to {\mathbf P}^n_{\mathbf C}\) with \(X\) affine.

Exercise

Maps of \(\text{Proj}\). Let \(R\) and \(S\) be graded rings. Suppose we have a ring map \[\psi : R \to S\] and an integer \(e \geq 1\) such that \(\psi(R_d) \subset S_{de}\) for all \(d \geq 0\). (By our conventions this is not a homomorphism of graded rings, unless \(e = 1\).)

1. For which elements \(\mathfrak p \in \text{Proj}(S)\) is there a well-defined corresponding point in \(\text{Proj}(R)\)? In other words, find a suitable open \(U \subset \text{Proj}(S)\) such that \(\psi\) defines a continuous map \(r_\psi : U \to \text{Proj}(R)\).

2. Give an example where \(U \not = \text{Proj}(S)\).

3. Give an example where \(U = \text{Proj}(S)\).

4. (Do not write this down.) Convince yourself that the continuous map \(U \to \text{Proj}(R)\) comes canonically with a map on sheaves so that \(r_\psi\) is a morphism of schemes: \[\text{Proj}(S) \supset U \longrightarrow \text{Proj}(R).\]

5. What can you say about this map if \(R = \bigoplus_{d \geq 0} S_{de}\) (as a graded ring with \(S_e\), \(S_{2e}\), etc in degree \(1\), \(2\), etc) and \(\psi\) is the inclusion mapping?

Notation. Let \(R\) be a graded ring as above and let \(n \geq 0\) be an integer. Let \(X = \text{Proj}(R)\). Then there is a unique quasi-coherent \({\mathcal O}_X\)-module \({\mathcal O}_X(n)\) on \(X\) such that for every homogeneous element \(f \in R\) of positive degree we have \({\mathcal O}_X |_{D_{+}(f)}\) is the quasi-coherent sheaf associated to the \(R_{(f)} = (R_f)_0\)-module \((R_f)_n\) (\(=\)elements homogeneous of degree \(n\) in \(R_f = R[1/f]\)). See [H, page 116+]. Note that there are natural maps \[{\mathcal O}_X(n_1) \otimes_{{\mathcal O}_X} {\mathcal O}_X(n_2) \longrightarrow {\mathcal O}_X(n_1 + n_2)\]

Exercise

Pathologies in \(\text{Proj}\). Give examples of \(R\) as above such that

1. \({\mathcal O}_X(1)\) is not an invertible \({\mathcal O}_X\)-module.

2. \({\mathcal O}_X(1)\) is invertible, but the natural map \({\mathcal O}_X(1) \otimes_{{\mathcal O}_X} {\mathcal O}_X(1) \to {\mathcal O}_X(2)\) is NOT an isomorphism.

Exercise

Let \(S\) be a graded ring. Let \(X = \text{Proj}(S)\). Show that any finite set of points of \(X\) is contained in a standard affine open.

Exercise

Let \(S\) be a graded ring. Let \(X = \text{Proj}(S)\). Let \(Z, Z' \subset X\) be two closed subschemes. Let \(\varphi : Z \to Z'\) be an isomorphism. Assume \(Z \cap Z' = \emptyset\). Show that for any \(z \in Z\) there exists an affine open \(U \subset X\) such that \(z \in U\), \(\varphi(z) \in U\) and \(\varphi(Z \cap U) = Z' \cap U\). (Hint: Use Exercise 02A4 and something akin to Schemes, Lemma 01IW.)

Morphisms from the projective line

In this section we study morphisms from \(\mathbf{P}^1\) to projective schemes.

Exercise

Let \(k\) be a field. Let \(k[t] \subset k(t)\) be the inclusion of the polynomial ring into its fraction field. Let \(X\) be a finite type scheme over \(k\). Show that for any morphism \[\varphi : \Spec(k(t)) \longrightarrow X\] over \(k\), there exist a nonzero \(f \in k[t]\) and a morphism \(\psi : \Spec(k[t, 1/f]) \to X\) over \(k\) such that \(\varphi\) is the composition \[\Spec(k(t)) \longrightarrow \Spec(k[t, 1/f]) \longrightarrow X\]

Exercise

Let \(k\) be a field. Let \(k[t] \subset k(t)\) be the inclusion of the polynomial ring into its fraction field. Show that for any morphism \[\varphi : \Spec(k(t)) \longrightarrow \mathbf{P}^n_k\] over \(k\), there exists a morphism \(\psi : \Spec(k[t]) \to \mathbf{P}^n_k\) over \(k\) such that \(\varphi\) is the composition \[\Spec(k(t)) \longrightarrow \Spec(k[t]) \longrightarrow \mathbf{P}^n_k\] Hint: the image of \(\varphi\) is in a standard open \(D_+(T_i)\) for some \(i\); then show that you can “clear denominators”.

Exercise

Let \(k\) be a field. Let \(k[t] \subset k(t)\) be the inclusion of the polynomial ring into its fraction field. Let \(X\) be a projective scheme over \(k\). Show that for any morphism \[\varphi : \Spec(k(t)) \longrightarrow X\] over \(k\), there exists a morphism \(\psi : \Spec(k[t]) \to X\) over \(k\) such that \(\varphi\) is the composition \[\Spec(k(t)) \longrightarrow \Spec(k[t]) \longrightarrow X\] Hint: use Exercise 0DJ2.

Exercise

Let \(k\) be a field. Let \(X\) be a projective scheme over \(k\). Let \(K\) be the function field of \(\mathbf{P}^1_k\) (see hint below). Show that for any morphism \[\varphi : \Spec(K) \longrightarrow X\] over \(k\), there exists a morphism \(\psi : \mathbf{P}^1_k \to X\) over \(k\) such that \(\varphi\) is the composition \[\Spec(k(t)) \longrightarrow \mathbf{P}^1_k \longrightarrow X\] Hint: use Exercise 0DJ3 for each of the two pieces of the affine open covering \(\mathbf{P}^1_k = D_+(T_0) \cup D_+(T_1)\), use that \(D_+(T_0)\) is the spectrum of a polynomial ring and that \(K\) is the fraction field of this polynomial ring.

Morphisms from surfaces to curves

Exercise

Let \(R\) be a ring. Let \(R \to k\) be a map from \(R\) to a field. Let \(n \geq 0\). Show that \[\Mor_{\Spec(R)}(\Spec(k), \mathbf{P}^n_R) = (k^{n + 1} \setminus \{0\})/k^*\] where \(k^*\) acts via scalar multiplication on \(k^{n + 1}\). From now on we denote \((x_0 : \ldots : x_n)\) the morphism \(\Spec(k) \to \mathbf{P}^n_k\) corresponding to the equivalence class of the element \((x_0, \ldots, x_n) \in k^{n + 1} \setminus \{0\}\).

Exercise

Let \(k\) be a field. Let \(Z \subset \mathbf{P}^2_k\) be an irreducible and reduced closed subscheme. Show that either (a) \(Z\) is a closed point, or (b) there exists an homogeneous irreducible \(F \in k[X_0, X_1, X_2]\) of degree \(> 0\) such that \(Z = V_{+}(F)\), or (c) \(Z = \mathbf{P}^2_k\). (Hint: Look on a standard affine open.)

Exercise

Let \(k\) be a field. Let \(Z_1, Z_2 \subset \mathbf{P}^2_k\) be irreducible closed subschemes of the form \(V_{+}(F)\) for some homogeneous irreducible \(F_i \in k[X_0, X_1, X_2]\) of degree \(> 0\). Show that \(Z_1 \cap Z_2\) is not empty. (Hint: Use dimension theory to estimate the dimension of the local ring of \(k[X_0, X_1, X_2]/(F_1, F_2)\) at \(0\).)

Exercise

Show there does not exist a nonconstant morphism of schemes \(\mathbf{P}^2_{\mathbf{C}} \to \mathbf{P}^1_{\mathbf{C}}\) over \(\Spec(\mathbf{C})\). Here a constant morphism is one whose image is a single point. (Hint: If the morphism is not constant consider the fibres over \(0\) and \(\infty\) and argue that they have to meet to get a contradiction.)

Exercise

Let \(k\) be a field. Suppose that \(X \subset \mathbf{P}^3_k\) is a closed subscheme given by a single homogeneous equation \(F \in k[X_0, X_1, X_2, X_3]\). In other words, \[X = \text{Proj}(k[X_0, X_1, X_2, X_3]/(F)) \subset \mathbf{P}^3_k\] as explained in the course. Assume that \[F = X_0 G + X_1 H\] for some homogeneous polynomials \(G, H \in k[X_0, X_1, X_2, X_3]\) of positive degree. Show that if \(X_0, X_1, G, H\) have no common zeros then there exists a nonconstant morphism \[X \longrightarrow \mathbf{P}^1_k\] of schemes over \(\Spec(k)\) which on field points (see Exercise 02A7) looks like \((x_0 : x_1 : x_2 : x_3) \mapsto (x_0 : x_1)\) whenever \(x_0\) or \(x_1\) is not zero.

Invertible sheaves

Definition

Let \(X\) be a locally ringed space. An invertible \({\mathcal O}_X\)-module on \(X\) is a sheaf of \({\mathcal O}_X\)-modules \({\mathcal L}\) such that every point has an open neighbourhood \(U \subset X\) such that \({\mathcal L}|_U\) is isomorphic to \({\mathcal O}_U\) as \({\mathcal O}_U\)-module. We say that \({\mathcal L}\) is trivial if it is isomorphic to \({\mathcal O}_X\) as a \({\mathcal O}_X\)-module.

Exercise

General facts.

1. Show that an invertible \({\mathcal O}_X\)-module on a scheme \(X\) is quasi-coherent.

2. Suppose \(X\to Y\) is a morphism of locally ringed spaces, and \({\mathcal L}\) an invertible \({\mathcal O}_Y\)-module. Show that \(f^\ast {\mathcal L}\) is an invertible \({\mathcal O}_X\) module.

Exercise

Algebra.

1. Show that an invertible \({\mathcal O}_X\)-module on an affine scheme \(\Spec(A)\) corresponds to an \(A\)-module \(M\) which is (i) finite, (ii) projective, (iii) locally free of rank 1, and hence (iv) flat, and (v) finitely presented. (Feel free to quote things from last semesters course; or from algebra books.)

2. Suppose that \(A\) is a domain and that \(M\) is a module as in (a). Show that \(M\) is isomorphic as an \(A\)-module to an ideal \(I \subset A\) such that \(IA_{\mathfrak p}\) is principal for every prime \({\mathfrak p}\).

Definition

Let \(R\) be a ring. An invertible module \(M\) is an \(R\)-module \(M\) such that \(\widetilde M\) is an invertible sheaf on the spectrum of \(R\). We say \(M\) is trivial if \(M \cong R\) as an \(R\)-module.

In other words, \(M\) is invertible if and only if it satisfies all of the following conditions: it is flat, of finite presentation, projective, and locally free of rank 1. (Of course it suffices for it to be locally free of rank 1).

Exercise

Simple examples.

1. Let \(k\) be a field. Let \(A = k[x]\). Show that \(X = \Spec(A)\) has only trivial invertible \({\mathcal O}_X\)-modules. In other words, show that every invertible \(A\)-module is free of rank 1.

2. Let \(A\) be the ring \[A = \{ f\in k[x] \mid f(0) = f(1) \}.\] Show there exists a nontrivial invertible \(A\)-module, unless \(k = {\mathbf F}_2\). (Hint: Think about \(\Spec(A)\) as identifying \(0\) and \(1\) in \({\mathbf A}^1_k = \Spec(k[x])\).)

3. Same question as in (02AJ) for the ring \(A = k[x^2, x^3] \subset k[x]\) (except now \(k = {\mathbf F}_2\) works as well).

Exercise

Higher dimensions.

1. Prove that every invertible sheaf on two dimensional affine space is trivial. More precisely, let \({\mathbf A}^2_k = \Spec(k[x, y])\) where \(k\) is a field. Show that every invertible sheaf on \({\mathbf A}^2_k\) is trivial. (Hint: One way to do this is to consider the corresponding module \(M\), to look at \(M \otimes_{k[x, y]} k(x)[y]\), and then use Exercise 02AH (02AI) to find a generator for this; then you still have to think. Another way to is to use Exercise 02AF and use what we know about ideals of the polynomial ring: primes of height one are generated by an irreducible polynomial; then you still have to think.)

2. Prove that every invertible sheaf on any open subscheme of two dimensional affine space is trivial. More precisely, let \(U \subset {\mathbf A}^2_k\) be an open subscheme where \(k\) is a field. Show that every invertible sheaf on \(U\) is trivial. Hint: Show that every invertible sheaf on \(U\) extends to one on \({\mathbf A}^2_k\). Not easy; but you can find it in [H].

3. Find an example of a nontrivial invertible sheaf on a punctured cone over a field. More precisely, let \(k\) be a field and let \(C = \Spec(k[x, y, z]/(xy-z^2))\). Let \(U = C \setminus \{ (x, y, z) \}\). Find a nontrivial invertible sheaf on \(U\). Hint: It may be easier to compute the group of isomorphism classes of invertible sheaves on \(U\) than to just find one. Note that \(U\) is covered by the opens \(\Spec(k[x, y, z, 1/x]/(xy-z^2))\) and \(\Spec(k[x, y, z, 1/y]/(xy-z^2))\) which are “easy” to deal with.

Definition

Let \(X\) be a locally ringed space. The Picard group of \(X\) is the set \(\Pic(X)\) of isomorphism classes of invertible \(\mathcal{O}_X\)-modules with addition given by tensor product. See Modules, Definition 01CX. For a ring \(R\) we set \(\Pic(R) = \Pic(\Spec(R))\).

Exercise

Let \(R\) be a ring.

1. Show that if \(R\) is a Noetherian normal domain, then \(\Pic(R) = \Pic(R[t])\). [Hint: There is a map \(R[t] \to R\), \(t \mapsto 0\) which is a left inverse to the map \(R \to R[t]\). Hence it suffices to show that any invertible \(R[t]\)-module \(M\) such that \(M/tM \cong R\) is free of rank \(1\). Let \(K\) be the fraction field of \(R\). Pick a trivialization \(K[t] \to M \otimes_{R[t]} K[t]\) which is possible by Exercise 02AH (02AI). Adjust it so it agrees with the trivialization of \(M/tM\) above. Show that it is in fact a trivialization of \(M\) over \(R[t]\) (this is where normality comes in).]

2. Let \(k\) be a field. Show that \(\Pic(k[x^2, x^3, t]) \not = \Pic(k[x^2, x^3])\).

Čech Cohomology

Exercise

Čech cohomology. Here \(k\) is a field.

1. Let \(X\) be a scheme with an open covering \({\mathcal U} : X = U_1 \cup U_2\), with \(U_1 = \Spec(k[x])\), \(U_2 = \Spec(k[y])\) with \(U_1 \cap U_2 = \Spec(k[z, 1/z])\) and with open immersions \(U_1 \cap U_2 \to U_1\) resp. \(U_1 \cap U_2 \to U_2\) determined by \(x \mapsto z\) resp. \(y \mapsto z\) (and I really mean this). (We’ve seen in the lectures that such an \(X\) exists; it is the affine line with zero doubled.) Compute \({\check H}^1({\mathcal U}, {\mathcal O})\); eg. give a basis for it as a \(k\)-vectorspace.

2. For each element in \({\check H}^1({\mathcal U}, {\mathcal O})\) construct an exact sequence of sheaves of \({\mathcal O}_X\)-modules \[0 \to {\mathcal O}_X \to E \to {\mathcal O}_X \to 0\] such that the boundary \(\delta(1) \in {\check H}^1({\mathcal U}, {\mathcal O})\) equals the given element. (Part of the problem is to make sense of this. See also below. It is also OK to show abstractly such a thing has to exist.)

Definition

(Definition of delta.) Suppose that \[0 \to {\mathcal F}_1 \to {\mathcal F}_2 \to {\mathcal F}_3 \to 0\] is a short exact sequence of abelian sheaves on any topological space \(X\). The boundary map \(\delta : H^0(X, {\mathcal F}_3) \to {\check H}^1(X, {\mathcal F}_1)\) is defined as follows. Take an element \(\tau \in H^0(X, {\mathcal F}_3)\). Choose an open covering \({\mathcal U} : X = \bigcup_{i\in I} U_i\) such that for each \(i\) there exists a section \(\tilde \tau_i \in {\mathcal F}_2\) lifting the restriction of \(\tau\) to \(U_i\). Then consider the assignment \[(i_0, i_1) \longmapsto \tilde \tau_{i_0}|_{U_{i_0i_1}} - \tilde \tau_{i_1}|_{U_{i_0i_1}}.\] This is clearly a 1-coboundary in the Čech complex \({\check C}^\ast({\mathcal U}, {\mathcal F}_2)\). But we observe that (thinking of \({\mathcal F}_1\) as a subsheaf of \({\mathcal F}_2\)) the RHS always is a section of \({\mathcal F}_1\) over \(U_{i_0i_1}\). Hence we see that the assignment defines a 1-cochain in the complex \({\check C}^\ast({\mathcal U}, {\mathcal F}_2)\). The cohomology class of this 1-cochain is by definition \(\delta(\tau)\).

Cohomology

Exercise

Let \(X = \mathbf{R}\) with the usual Euclidean topology. Using only formal properties of cohomology (functoriality and the long exact cohomology sequence) show that there exists a sheaf \(\mathcal{F}\) on \(X\) with nonzero \(H^1(X, \mathcal{F})\).

Exercise

Let \(X = U \cup V\) be a topological space written as the union of two opens. Then we have a long exact Mayer-Vietoris sequence \[0 \to H^0(X, \mathcal{F}) \to H^0(U, \mathcal{F}) \oplus H^0(V, \mathcal{F}) \to H^0(U \cap V, \mathcal{F}) \to H^1(X, \mathcal{F}) \to \ldots\] What property of injective sheaves is essential for the construction of the Mayer-Vietoris long exact sequence? Why does it hold?

Exercise

Let \(X\) be a topological space.

1. Show that \(H^i(X, \mathcal{F})\) is zero for \(i > 0\) if \(X\) has \(2\) or fewer points.

2. What if \(X\) has \(3\) points?

Exercise

Let \(X\) be the spectrum of a local ring. Show that \(H^i(X, \mathcal{F})\) is zero for \(i > 0\) and any sheaf of abelian groups \(\mathcal{F}\).

Exercise

Let \(f : X \to Y\) be an affine morphism of schemes. Prove that \(H^i(X, \mathcal{F}) = H^i(Y, f_*\mathcal{F})\) for any quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\). Feel free to impose some further conditions on \(X\) and \(Y\) and use the agreement of Čech cohomology with cohomology for quasi-coherent sheaves and affine open coverings of separated schemes.

Exercise

Let \(A\) be a ring. Let \(\mathbf{P}^n_A = \text{Proj}(A[T_0, \ldots, T_n])\) be projective space over \(A\). Let \(\mathbf{A}^{n + 1}_A = \Spec(A[T_0, \ldots, T_n])\) and let \[U = \bigcup\nolimits_{i = 0, \ldots, n} D(T_i) \subset \mathbf{A}^{n + 1}_A\] be the complement of the image of the closed immersion \(0 : \Spec(A) \to \mathbf{A}^{n + 1}_A\). Construct an affine surjective morphism \[f : U \longrightarrow \mathbf{P}^n_A\] and prove that \(f_*\mathcal{O}_U = \bigoplus_{d \in \mathbf{Z}} \mathcal{O}_{\mathbf{P}^n_A}(d)\). More generally, show that for a graded \(A[T_0, \ldots, T_n]\)-module \(M\) one has \[f_*(\widetilde{M}|_U) = \bigoplus\nolimits_{d \in \mathbf{Z}} \widetilde{M(d)}\] where on the left hand side we have the quasi-coherent sheaf \(\widetilde{M}\) associated to \(M\) on \(\mathbf{A}^{n + 1}_A\) and on the right we have the quasi-coherent sheaves \(\widetilde{M(d)}\) associated to the graded module \(M(d)\).

Exercise

Let \(A\) be a ring and let \(\mathbf{P}^n_A = \text{Proj}(A[T_0, \ldots, T_n])\) be projective space over \(A\). Carefully compute the cohomology of the Serre twists \(\mathcal{O}_{\mathbf{P}^n_A}(d)\) of the structure sheaf on \(\mathbf{P}^n_A\). Feel free to use Čech cohomology and the agreement of Čech cohomology with cohomology for quasi-coherent sheaves and affine open coverings of separated schemes.

Exercise

Let \(A\) be a ring and let \(\mathbf{P}^n_A = \text{Proj}(A[T_0, \ldots, T_n])\) be projective space over \(A\). Let \(F \in A[T_0, \ldots, T_n]\) be homogeneous of degree \(d\). Let \(X \subset \mathbf{P}^n_A\) be the closed subscheme corresponding to the graded ideal \((F)\) of \(A[T_0, \ldots, T_n]\). What can you say about \(H^i(X, \mathcal{O}_X)\)?

Exercise

Let \(R\) be a ring such that for any left exact functor \(F : \text{Mod}_R \to \textit{Ab}\) we have \(R^iF = 0\) for \(i > 0\). Show that \(R\) is a finite product of fields.

More cohomology

Exercise

Let \(k\) be a field. Let \(X \subset \mathbf{P}^n_k\) be the “coordinate cross”. Namely, let \(X\) be defined by the homogeneous equations \[T_i T_j = 0\text{ for }i > j > 0\] where as usual we write \(\mathbf{P}^n_k = \text{Proj}(k[T_0, \ldots, T_n])\). In other words, \(X\) is the closed subscheme corresponding to the quotient \(k[T_0, \ldots, T_n]/(T_iT_j; i > j > 0)\) of the polynomial ring. Compute \(H^i(X, \mathcal{O}_X)\) for all \(i\). Hint: use Čech cohomology.

Exercise

Let \(A\) be a ring. Let \(I = (f_1, \ldots, f_t)\) be a finitely generated ideal of \(A\). Let \(U \subset \Spec(A)\) be the complement of \(V(I)\). For any \(A\)-module \(M\) write down a complex of \(A\)-modules (in terms of \(A\), \(f_1, \ldots, f_t\), \(M\)) whose cohomology groups give \(H^n(U, \widetilde{M})\).

Exercise

Let \(k\) be a field. Let \(U \subset \mathbf{A}^d_k\) be the complement of the closed point \(0\) of \(\mathbf{A}^d_k\). Compute \(H^n(U, \mathcal{O}_U)\) for all \(n\).

Exercise

Let \(k\) be a field. Find explicitly a scheme \(X\) projective over \(k\) of dimension \(1\) with \(H^0(X, \mathcal{O}_X) = k\) and \(\dim_k H^1(X, \mathcal{O}_X) = 100\).

Exercise

Let \(f : X \to Y\) be a finite locally free morphism of degree \(2\). Assume that \(X\) and \(Y\) are integral schemes and that \(2\) is invertible in the structure sheaf of \(Y\), i.e., \(2 \in \Gamma(Y, \mathcal{O}_Y)\) is invertible. Show that the \(\mathcal{O}_Y\)-module map \[f^\sharp : \mathcal{O}_Y \longrightarrow f_*\mathcal{O}_X\] has a left inverse, i.e., there is an \(\mathcal{O}_Y\)-module map \(\tau : f_*\mathcal{O}_X \to \mathcal{O}_Y\) with \(\tau \circ f^\sharp = \text{id}\). Conclude that \(H^n(Y, \mathcal{O}_Y) \to H^n(X, \mathcal{O}_X)\) is injective².

Exercise

Let \(X\) be a scheme (or a locally ringed space). The rule \(U \mapsto \mathcal{O}_X(U)^*\) defines a sheaf of groups denoted \(\mathcal{O}_X^*\). Briefly explain why the Picard group of \(X\) (Definition 02AM) is equal to \(H^1(X, \mathcal{O}_X^*)\).

Exercise

Give an example of an affine scheme \(X\) with nontrivial \(\Pic(X)\). Conclude using Exercise 0DAP that \(H^1(X, -)\) is not the zero functor for any such \(X\).

Exercise

Let \(A\) be a ring. Let \(I = (f_1, \ldots, f_t)\) be a finitely generated ideal of \(A\). Let \(U \subset \Spec(A)\) be the complement of \(V(I)\). Given a quasi-coherent \(\mathcal{O}_{\Spec(A)}\)-module \(\mathcal{F}\) and \(\xi \in H^p(U, \mathcal{F})\) with \(p > 0\), show that there exists \(n > 0\) such that \(f_i^n \xi = 0\) for \(i = 1, \ldots, t\). Hint: One possible way to proceed is to use the complex you found in Exercise 0DAK.

Exercise

Let \(A\) be a ring. Let \(I = (f_1, \ldots, f_t)\) be a finitely generated ideal of \(A\). Let \(U \subset \Spec(A)\) be the complement of \(V(I)\). Let \(M\) be an \(A\)-module whose \(I\)-torsion is zero, i.e., \(0 = \Ker((f_1, \ldots, f_t) : M \to M^{\oplus t})\). Show that there is a canonical isomorphism \[H^0(U, \widetilde{M}) = \colim \Hom_A(I^n, M).\] Warning: this is not trivial.

Exercise

Let \(A\) be a Noetherian ring. Let \(I\) be an ideal of \(A\). Let \(M\) be an \(A\)-module. Let \(M[I^\infty]\) be the set of \(I\)-power torsion elements defined by \[M[I^\infty] = \{x \in M \mid \text{ there exists an }n \geq 1\text{ such that }I^nx = 0\}\] Set \(M' = M/M[I^\infty]\). Then the \(I\)-power torsion of \(M'\) is zero. Show that \[\colim \Hom_A(I^n, M) = \colim \Hom_A(I^n, M').\] Warning: this is not trivial. Hints: (1) try to reduce to \(M\) finite, (2) show any element of \(\Ext^1_A(I^n, N)\) maps to zero in \(\Ext^1_A(I^{n + m}, N)\) for some \(m > 0\) if \(N = M[I^\infty]\) and \(M\) finite, (3) show the same thing as in (2) for \(\Hom_A(I^n, N)\), (3) consider the long exact sequence \[0 \to \Hom_A(I^n, M[I^\infty]) \to \Hom_A(I^n, M) \to \Hom_A(I^n, M') \to \Ext^1_A(I^n, M[I^\infty])\] for \(M\) finite and compare with the sequence for \(I^{n + m}\) to conclude.

Cohomology revisited

Exercise

Make an example of a field \(k\), a curve \(X\) over \(k\), an invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\) and a cohomology class \(\xi \in H^1(X, \mathcal{L})\) with the following property: for every surjective finite morphism \(\pi : Y \to X\) of schemes the element \(\xi\) pulls back to a nonzero element of \(H^1(Y, \pi^*\mathcal{L})\). Hint: construct \(X\), \(k\), \(\mathcal{L}\) such that there is a short exact sequence \(0 \to \mathcal{L} \to \mathcal{O}_X \to i_*\mathcal{O}_Z \to 0\) where \(Z \subset X\) is a closed subscheme consisting of more than \(1\) closed point. Then look at what happens when you pullback.

Exercise

Let \(k\) be an algebraically closed field. Let \(X\) be a projective \(1\)-dimensional scheme. Suppose that \(X\) contains a cycle of curves, i.e., suppose there exist an \(n \geq 2\) and pairwise distinct \(1\)-dimensional integral closed subschemes \(C_1, \ldots, C_n\) and pairwise distinct closed points \(x_1, \ldots, x_n \in X\) such that \(x_n \in C_n \cap C_1\) and \(x_i \in C_i \cap C_{i + 1}\) for \(i = 1, \ldots, n - 1\). Prove that \(H^1(X, \mathcal{O}_X)\) is nonzero. Hint: Let \(\mathcal{F}\) be the image of the map \(\mathcal{O}_X \to \bigoplus \mathcal{O}_{C_i}\), and show \(H^1(X, \mathcal{F})\) is nonzero using that \(\kappa(x_i) = k\) and \(H^0(C_i, \mathcal{O}_{C_i}) = k\). Also use that \(H^2(X, -) = 0\) by Grothendieck’s theorem.

Exercise

Let \(X\) be a projective surface over an algebraically closed field \(k\). Prove there exists a proper closed subscheme \(Z \subset X\) such that \(H^1(Z, \mathcal{O}_Z)\) is nonzero. Hint: Use Exercise 0DB5.

Exercise

Let \(X\) be a projective surface over an algebraically closed field \(k\). Show that for every \(n \geq 0\) there exists a proper closed subscheme \(Z \subset X\) such that \(\dim_k H^1(Z, \mathcal{O}_Z) > n\). Only explain how to do this by modifying the arguments in Exercise 0DB6 and 0DB5; don’t give all the details.

Exercise

Let \(X\) be a projective surface over an algebraically closed field \(k\). Prove there exists a coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) such that \(H^2(X, \mathcal{F})\) is nonzero. Hint: Use the result of Exercise 0DB7 and a cleverly chosen exact sequence.

Exercise

Let \(X\) and \(Y\) be schemes over a field \(k\) (feel free to assume \(X\) and \(Y\) are nice, for example qcqs or projective over \(k\)). Set \(X \times Y = X \times_{\Spec(k)} Y\) with projections \(p : X \times Y \to X\) and \(q : X \times Y \to Y\). For a quasi-coherent \(\mathcal{O}_X\)-module \(\mathcal{F}\) and a quasi-coherent \(\mathcal{O}_Y\)-module \(\mathcal{G}\) prove that \[H^n(X \times Y, p^*\mathcal{F} \otimes_{\mathcal{O}_{X \times Y}} q^*\mathcal{G}) = \bigoplus\nolimits_{a + b = n} H^a(X, \mathcal{F}) \otimes_k H^b(Y, \mathcal{G})\] or just show that this holds when one takes dimensions. Extra points for “clean” solutions.

Exercise

Let \(k\) be a field. Let \(X = \mathbf{P}|^1 \times \mathbf{P}^1\) be the product of the projective line over \(k\) with itself with projections \(p : X \to \mathbf{P}^1_k\) and \(q : X \to \mathbf{P}^1_k\). Let \[\mathcal{O}(a, b) = p^*\mathcal{O}_{\mathbf{P}^1_k}(a) \otimes_{\mathcal{O}_X} q^*\mathcal{O}_{\mathbf{P}^1_k}(b)\] Compute the dimensions of \(H^i(X, \mathcal{O}(a, b))\) for all \(i, a, b\). Hint: Use Exercise 0DB9.

Cohomology and Hilbert polynomials

Situation

Let \(k\) be a field. Let \(X = \mathbf{P}^n_k\) be \(n\)-dimensional projective space. Let \(\mathcal{F}\) be a coherent \(\mathcal{O}_X\)-module. Recall that \[\chi(X, \mathcal{F}) = \sum\nolimits_{i = 0}^n (-1)^i \dim_k H^i(X, \mathcal{F})\] Recall that the Hilbert polynomial of \(\mathcal{F}\) is the function \[t \longmapsto \chi(X, \mathcal{F}(t))\] We also recall that \(\mathcal{F}(t) = \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{O}_X(t)\) where \(\mathcal{O}_X(t)\) is the \(t\)th twist of the structure sheaf as in Constructions, Definition 01MN. In Varieties, Subsection 08A9 we have proved the Hilbert polynomial is a polynomial in \(t\).

Exercise

In Situation 0DCE.

1. If \(P(t)\) is the Hilbert polynomial of \(\mathcal{F}\), what is the Hilbert polynomial of \(\mathcal{F}(-13)\).

2. If \(P_i\) is the Hilbert polynomial of \(\mathcal{F}_i\), what is the Hilbert polynomial of \(\mathcal{F}_1 \oplus \mathcal{F}_2\).

3. If \(P_i\) is the Hilbert polynomial of \(\mathcal{F}_i\) and \(\mathcal{F}\) is the kernel of a surjective map \(\mathcal{F}_1 \to \mathcal{F}_2\), what is the Hilbert polynomial of \(\mathcal{F}\)?

Exercise

In Situation 0DCE assume \(n \geq 1\). Find a coherent sheaf whose Hilbert polynomial is \(t - 101\).

Exercise

In Situation 0DCE assume \(n \geq 2\). Find a coherent sheaf whose Hilbert polynomial is \(t^2/2 + t/2 - 1\). (This is a bit tricky; it suffices if you just show there is such a coherent sheaf.)

Exercise

In Situation 0DCE assume \(n \geq 2\) and \(k\) algebraically closed. Let \(C \subset X\) be an integral closed subscheme of dimension \(1\). In other words, \(C\) is a projective curve. Let \(d t + e\) be the Hilbert polynomial of \(\mathcal{O}_C\) viewed as a coherent sheaf on \(X\).

1. Give an upper bound on \(e\). (Hints: Use that \(\mathcal{O}_C(t)\) only has cohomology in degrees \(0\) and \(1\) and study \(H^0(C, \mathcal{O}_C)\).)

2. Pick a global section \(s\) of \(\mathcal{O}_X(1)\) which intersects \(C\) transversally, i.e., such that there are pairwise distinct closed points \(c_1, \ldots, c_r \in C\) and a short exact sequence \[0 \to \mathcal{O}_C \xrightarrow{s} \mathcal{O}_C(1) \to \bigoplus\nolimits_{i = 1, \ldots, r} k_{c_i} \to 0\] where \(k_{c_i}\) is the skyscraper sheaf with value \(k\) in \(c_i\). (Such an \(s\) exists; please just use this.) Show that \(r = d\). (Hint: twist the sequence and see what you get.)

3. Twisting the short exact sequence gives a \(k\)-linear map \(\varphi_t : \Gamma(C, \mathcal{O}_C(t)) \to \bigoplus_{i = 1, \ldots, d} k\) for any \(t\). Show that if this map is surjective for \(t \geq d - 1\).

4. Give a lower bound on \(e\) in terms of \(d\). (Hint: show that \(H^1(C, \mathcal{O}_C(d - 2)) = 0\) using the result of (3) and use vanishing.)

Exercise

In Situation 0DCE assume \(n = 2\). Let \(s_1, s_2, s_3 \in \Gamma(X, \mathcal{O}_X(2))\) be three quadric equations. Consider the coherent sheaf \[\mathcal{F} = \Coker\left(\mathcal{O}_X(-2)^{\oplus 3} \xrightarrow{s_1, s_2, s_3} \mathcal{O}_X\right)\] List the possible Hilbert polynomials of such \(\mathcal{F}\). (Try to visualize intersections of quadrics in the projective plane.)

Curves

Exercise

Let \(k\) be an algebraically closed field. Let \(X\) be a projective curve over \(k\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(s_0, \ldots, s_n \in H^0(X, \mathcal{L})\) be global sections of \(\mathcal{L}\). Prove there is a natural closed subscheme \[Z \subset \mathbf{P}^n \times X\] such that the closed point \(((\lambda_0 : \ldots : \lambda_n), x)\) is in \(Z\) if and only if the section \(\lambda_0 s_0 + \ldots + \lambda_n s_n\) vanishes at \(x\). (Hint: describe \(Z\) affine locally.)

Exercise

Let \(k\) be an algebraically closed field. Let \(X\) be a smooth curve over \(k\). Let \(r \geq 1\). Show that the closed subset \[D \subset X \times X^r = X^{r + 1}\] whose closed points are the tuples \((x, x_1, \ldots, x_r)\) with \(x = x_i\) for some \(i\), has an invertible ideal sheaf. In other words, show that \(D\) is an effective Cartier divisor. Hints: reduce to \(r = 1\) and use that \(X\) is a smooth curves to say something about the diagonal (look in books for this).

Exercise

Let \(k\) be an algebraically closed field. Let \(X\) be a smooth projective curve over \(k\). Let \(T\) be a scheme of finite type over \(k\) and let \[D_1 \subset X \times T \quad\text{and}\quad D_2 \subset X \times T\] be two effective Cartier divisors such that for \(t \in T\) the fibres \(D_{i, t} \subset X_t\) are not dense (i.e., do not contain the generic point of the curve \(X_t\)). Prove that there is a canonical closed subscheme \(Z \subset T\) such that a closed point \(t \in T\) is in \(Z\) if and only if for the scheme theoretic fibres \(D_{1, t}\), \(D_{2, t}\) of \(D_1\), \(D_2\) we have \[D_{1, t} \subset D_{2, t}\] as closed subschemes of \(X_t\). Hints: Show that, possibly after shrinking \(T\), you may assume \(T = \Spec(A)\) is affine and there is an affine open \(U \subset X\) such that \(D_i \subset U \times T\). Then show that \(M_1 = \Gamma(D_1, \mathcal{O}_{D_1})\) is a finite locally free \(A\)-module (here you will need some nontrivial algebra — ask your friends). After shrinking \(T\) you may assume \(M_1\) is a free \(A\)-module. Then look at \[\Gamma(U \times T, \mathcal{I}_{D_2}) \to M_1 = A^{\oplus N}\] and you define \(Z\) as the closed subscheme cut out by the ideal generated by coefficients of vectors in the image of this map. Explain why this works (this will require perhaps a bit more commutative algebra).

Exercise

Let \(k\) be an algebraically closed field. Let \(X\) be a smooth projective curve over \(k\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Let \(s_0, \ldots, s_n \in H^0(X, \mathcal{L})\) be global sections of \(\mathcal{L}\). Let \(r \geq 1\). Prove there is a natural closed subscheme \[Z \subset \mathbf{P}^n \times X \times \ldots \times X = \mathbf{P}^n \times X^r\] such that the closed point \(((\lambda_0 : \ldots : \lambda_n), x_1, \ldots, x_r)\) is in \(Z\) if and only if the section \(s_\lambda = \lambda_0 s_0 + \ldots + \lambda_n s_n\) vanishes on the divisor \(D = x_1 + \ldots + x_r\), i.e., the section \(s_\lambda\) is in \(\mathcal{L}(-D)\). Hint: explain how this follows by combining then results of Exercises 0EGA and 0EGB.

Exercise

Let \(k\) be an algebraically closed field. Let \(X\) be a smooth projective curve over \(k\). Let \(\mathcal{L}\) be an invertible \(\mathcal{O}_X\)-module. Show that there is a natural closed subset \[Z \subset X^r\] such that a closed point \((x_1, \ldots, x_r)\) of \(X^r\) is in \(Z\) if and only if \(\mathcal{L}(-x_1 - \ldots -x_r)\) has a nonzero global section. Hint: use Exercise 0EGC.

Exercise

Let \(k\) be an algebraically closed field. Let \(X\) be a smooth projective curve over \(k\). Let \(r \geq s\) be integers. Show that there is a natural closed subset \[Z \subset X^r \times X^s\] such that a closed point \((x_1, \ldots, x_r, y_1, \ldots, y_s)\) of \(X^r \times X^s\) is in \(Z\) if and only if \(x_1 + \ldots + x_r - y_1 - \ldots - y_s\) is linearly equivalent to an effective divisor. Hint: Choose an auxiliary invertible module \(\mathcal{L}\) of very high degree so that \(\mathcal{L}(-D)\) has a nonvanshing section for any effective divisor \(D\) of degree \(r\). Then use the result of Exercise 0EGD twice.

Exercise

Choose your favorite algebraically closed field \(k\). As best as you can determine all possible \(\mathfrak g^r_d\) that can exist on some curve of genus \(7\). While doing this also try to

1. determine in which cases the \(\mathfrak g^r_d\) is base point free, and

2. determine in which cases the \(\mathfrak g^r_d\) gives a closed embedding in \(\mathbf{P}^r\).

Do the same thing if you assume your curve is “general” (make up your own notion of general – this may be easier than the question above). Do the same thing if you assume your curve is hyperelliptic. Do the same thing if you assume your curve is trigonal (and not hyperelliptic). Etc.

Moduli

In this section we consider some naive approaches to moduli of algebraic geometric objects.

Let \(k\) be an algebraically closed field. Suppose that \(M\) is a moduli problem over \(k\). We won’t define exactly what this means here, but in each exercise it should be clear what we mean. To understand the following it suffices to know what the objects of \(M\) over \(k\) are, what the isomorphisms between objects of \(M\) over \(k\) are, and what the families of object of \(M\) over a variety are. Then we say the number of moduli of \(M\) is \(d \geq 0\) if the following are true

1. there is a finite number of families \(X_i \to V_i\), \(i = 1, \ldots, n\) such that every object of \(M\) over \(k\) is isomorphic to a fibre of one of these and such that \(\max \dim(V_i) = d\), and

2. there is no way to do this with a smaller \(d\).

This is really just a very rough approximation of better notions in the literature.

Exercise

Let \(k\) be an algebraically closed field. Let \(d \geq 1\) and \(n \geq 1\). Let us say the moduli of hypersurfaces of degree \(d\) in \(P^n\) is given by

1. an object is a hypersurface \(X \subset \mathbf{P}^n_k\) of degree \(d\),

2. an isomorphism between two objects \(X \subset \mathbf{P}^n_k\) and \(Y \subset \mathbf{P}^n_k\) is an element \(g \in \text{PGL}_n(k)\) such that \(g(X) = Y\), and

3. a family of hypersurfaces over a variety \(V\) is a closed subscheme \(X \subset \mathbf{P}^n_V\) such that for all \(v \in V\) the scheme theoretic fibre \(X_v\) of \(X \to V\) is a hypersurfaces in \(\mathbf{P}^n_v\).

Compute (if you can – these get progressively harder)

1. the number of moduli when \(n = 1\) and \(d\) arbitrary,

2. the number of moduli when \(n = 2\) and \(d = 1\),

3. the number of moduli when \(n = 2\) and \(d = 2\),

4. the number of moduli when \(n \geq 1\) and \(d = 2\),

5. the number of moduli when \(n = 2\) and \(d = 3\),

6. the number of moduli when \(n = 3\) and \(d = 3\), and

7. the number of moduli when \(n = 2\) and \(d = 4\).

Exercise

Let \(k\) be an algebraically closed field. Let \(g \geq 2\). Let us say the moduli of hyperelliptic curves of genus \(g\) is given by

1. an object is a smooth projective hyperelliptic curve \(X\) of genus \(g\),

2. an isomorphism between two objects \(X\) and \(Y\) is an isomorphism \(X \to Y\) of schemes over \(k\), and

3. a family of hyperelliptic curves of genus \(g\) over a variety \(V\) is a proper flat³ morphism \(X \to Y\) such that all scheme theoretic fibres of \(X \to V\) are smooth projective hyperelliptic curves of genus \(g\).

Show that the number of moduli is \(2g - 1\).

Global Exts

Exercise

Let \(k\) be a field. Let \(X = \mathbf{P}^3_k\). Let \(L \subset X\) and \(P \subset X\) be a line and a plane, viewed as closed subschemes cut out by \(1\), resp., \(2\) linear equations. Compute the dimensions of \[\Ext^i_X(\mathcal{O}_L, \mathcal{O}_P)\] for all \(i\). Make sure to do both the case where \(L\) is contained in \(P\) and the case where \(L\) is not contained in \(P\).

Exercise

Let \(k\) be a field. Let \(X = \mathbf{P}^n_k\). Let \(Z \subset X\) be a closed \(k\)-rational point viewed as a closed subscheme. For example the point with homogeneous coordinates \((1 : 0 : \ldots : 0)\). Compute the dimensions of \[\Ext^i_X(\mathcal{O}_Z, \mathcal{O}_Z)\] for all \(i\).

Exercise

Let \(X\) be a ringed space. Define cup-product maps \[\Ext^i_X(\mathcal{G}, \mathcal{H}) \times \Ext^j_X(\mathcal{F}, \mathcal{G}) \longrightarrow \Ext^{i + j}_X(\mathcal{F}, \mathcal{H})\] for \(\mathcal{O}_X\)-modules \(\mathcal{F}, \mathcal{G}, \mathcal{H}\). (Hint: this is a super general thing.)

Exercise

Let \(X\) be a ringed space. Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_X\)-module with dual \(\mathcal{E}^\vee = \SheafHom_{\mathcal{O}_X}(\mathcal{E}, \mathcal{O}_X)\). Prove the following statements

1. \(\SheafExt^i_{\mathcal{O}_X}( \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{E}, \mathcal{G}) = \SheafExt^i_{\mathcal{O}_X}( \mathcal{F}, \mathcal{E}^\vee \otimes_{\mathcal{O}_X} \mathcal{G}) = \SheafExt^i_{\mathcal{O}_X}( \mathcal{F}, \mathcal{G}) \otimes_{\mathcal{O}_X} \mathcal{E}^\vee\), and

2. \(\Ext^i_X( \mathcal{F} \otimes_{\mathcal{O}_X} \mathcal{E}, \mathcal{G}) = \Ext^i_X(\mathcal{F}, \mathcal{E}^\vee \otimes_{\mathcal{O}_X} \mathcal{G})\).

Here \(\mathcal{F}\) and \(\mathcal{G}\) are \(\mathcal{O}_X\)-modules. Conclude that \[\Ext^i_X(\mathcal{E}, \mathcal{G}) = H^i(X, \mathcal{E}^\vee \otimes_{\mathcal{O}_X} \mathcal{G})\]

Exercise

Let \(X\) be a ringed space. Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_X\)-module. Construct a trace map \[\Ext^i_X(\mathcal{E}, \mathcal{E}) \to H^i(X, \mathcal{O}_X)\] for all \(i\). Generalize to a trace map \[\Ext^i_X(\mathcal{E}, \mathcal{E} \otimes_{\mathcal{O}_X} \mathcal{F}) \to H^i(X, \mathcal{F})\] for any \(\mathcal{O}_X\)-module \(\mathcal{F}\).

Exercise

Let \(k\) be a field. Let \(X = \mathbf{P}^d_k\). Set \(\omega_{X/k} = \mathcal{O}_X(-d - 1)\). Prove that for finite locally free modules \(\mathcal{E}\), \(\mathcal{F}\) the cup product on Ext combined with the trace map on Ext \[\Ext^i_X(\mathcal{E}, \mathcal{F} \otimes_{\mathcal{O}_X} \omega_{X/k}) \times \Ext^{d - i}_X(\mathcal{F}, \mathcal{E}) \to \Ext_X^d(\mathcal{F}, \mathcal{F} \otimes_{\mathcal{O}_X} \omega_{X/k}) \to H^d(X, \omega_{X/k}) = k\] produces a nondegenerate pairing. Hint: you can either reprove duality in this setting or you can reduce to cohomology of sheaves and apply the Serre duality theorem as proved in the lectures.

Divisors

We collect all relevant definitions here in one spot for convenience.

Definition

Throughout, let \(S\) be any scheme and let \(X\) be a Noetherian, integral scheme.

1. A Weil divisor on \(X\) is a formal linear combination \(\Sigma n_i[Z_i]\) of prime divisors \(Z_i\) with integer coefficients.

2. A prime divisor is a closed subscheme \(Z \subset X\), which is integral with generic point \(\xi \in Z\) such that \({\mathcal O}_{X, \xi}\) has dimension \(1\). We will use the notation \({\mathcal O}_{X, Z} = {\mathcal O}_{X, \xi}\) when \(\xi \in Z \subset X\) is as above. Note that \({\mathcal O}_{X, Z} \subset K(X)\) is a subring of the function field of \(X\).

3. The Weil divisor associated to a rational function \(f \in K(X)^\ast\) is the sum \(\Sigma v_Z(f)[Z]\). Here \(v_Z(f)\) is defined as follows

   1. If \(f \in {\mathcal O}_{X, Z}^\ast\) then \(v_Z(f) = 0\).

   2. If \(f \in {\mathcal O}_{X, Z}\) then \[v_Z(f) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(f)).\]

   3. If \(f = \frac{a}{b}\) with \(a, b \in {\mathcal O}_{X, Z}\) then \[v_Z(f) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(a)) - \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(b)).\]

4. An effective Cartier divisor on a scheme \(S\) is a closed subscheme \(D \subset S\) such that every point \(d\in D\) has an affine open neighbourhood \(\Spec(A) = U \subset S\) in \(S\) so that \(D \cap U = \Spec(A/(f))\) with \(f \in A\) a nonzerodivisor.

5. The Weil divisor \([D]\) associated to an effective Cartier divisor \(D \subset X\) of our Noetherian integral scheme \(X\) is defined as the sum \(\Sigma v_Z(D)[Z]\) where \(v_Z(D)\) is defined as follows

   1. If the generic point \(\xi\) of \(Z\) is not in \(D\) then \(v_Z(D) = 0\).

   2. If the generic point \(\xi\) of \(Z\) is in \(D\) then \[v_Z(D) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(f))\] where \(f \in {\mathcal O}_{X, Z} = {\mathcal O}_{X, \xi}\) is the nonzerodivisor which defines \(D\) in an affine neighbourhood of \(\xi\) (as in (4) above).

6. Let \(S\) be a scheme. The sheaf of total quotient rings \({\mathcal K}_S\) is the sheaf of \({\mathcal O}_S\)-algebras which is the sheafification of the pre-sheaf \({\mathcal K}'\) defined as follows. For \(U \subset S\) open we set \({\mathcal K}'(U) = S_U^{-1}{\mathcal O}_S(U)\) where \(S_U \subset {\mathcal O}_S(U)\) is the multiplicative subset consisting of sections \(f \in {\mathcal O}_S(U)\) such that the germ of \(f\) in \({\mathcal O}_{S, u}\) is a nonzerodivisor for every \(u\in U\). In particular the elements of \(S_U\) are all nonzerodivisors. Thus \({\mathcal O}_S\) is a subsheaf of \({\mathcal K}_S\), and we get a short exact sequence \[0 \to {\mathcal O}_S^\ast \to {\mathcal K}_S^\ast \to {\mathcal K}_S^\ast/{\mathcal O}_S^\ast \to 0.\]

7. A Cartier divisor on a scheme \(S\) is a global section of the quotient sheaf \({\mathcal K}_S^\ast/{\mathcal O}_S^\ast\).

8. The Weil divisor associated to a Cartier divisor \(\tau \in \Gamma(X, {\mathcal K}_X^\ast/{\mathcal O}_X^\ast)\) over our Noetherian integral scheme \(X\) is the sum \(\Sigma v_Z(\tau)[Z]\) where \(v_Z(\tau)\) is defined as by the following recipe

   1. If the germ of \(\tau\) at the generic point \(\xi\) of \(Z\) is zero – in other words the image of \(\tau\) in the stalk \(({\mathcal K}^\ast/{\mathcal O}^\ast)_\xi\) is “zero” – then \(v_Z(\tau) = 0\).

   2. Find an affine open neighbourhood \(\Spec(A) = U \subset X\) so that \(\tau|_U\) is the image of a section \(f \in {\mathcal K}(U)\) and moreover \(f = a/b\) with \(a, b \in A\). Then we set \[v_Z(f) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(a)) - \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(b)).\]

Remarks

Here are some trivial remarks.

1. On a Noetherian integral scheme \(X\) the sheaf \({\mathcal K}_X\) is constant with value the function field \(K(X)\).

2. To make sense out of the definitions above one needs to show that \[\text{length}_{\mathcal O}({\mathcal O}/(ab)) = \text{length}_{\mathcal O}({\mathcal O}/(a)) + \text{length}_{\mathcal O}({\mathcal O}/(b))\] for any pair \((a, b)\) of nonzero elements of a Noetherian 1-dimensional local domain \({\mathcal O}\). This will be done in the lectures.

Exercise

(On any scheme.) Describe how to assign a Cartier divisor to an effective Cartier divisor.

Exercise

(On an integral scheme.) Describe how to assign a Cartier divisor \(D\) to a rational function \(f\) such that the Weil divisor associated to \(D\) and to \(f\) agree. (This is silly.)

Exercise

Give an example of a Weil divisor on a variety which is not the Weil divisor associated to any Cartier divisor.

Exercise

Give an example of a Weil divisor \(D\) on a variety which is not the Weil divisor associated to any Cartier divisor but such that \(nD\) is the Weil divisor associated to a Cartier divisor for some \(n > 1\).

Exercise

Give an example of a Weil divisor \(D\) on a variety which is not the Weil divisor associated to any Cartier divisor and such that \(nD\) is NOT the Weil divisor associated to a Cartier divisor for any \(n > 1\). (Hint: Consider a cone, for example \(X : xy - zw = 0\) in \(\mathbf{A}^4_k\). Try to show that \(D = [x = 0, z = 0]\) works.)

Exercise

On a separated scheme \(X\) of finite type over a field: Give an example of a Cartier divisor which is not the difference of two effective Cartier divisors. Hint: Find some \(X\) which does not have any nonempty effective Cartier divisors for example the scheme constructed in [H, III Exercise 5.9]. There is even an example with \(X\) a variety – namely the variety of Exercise 02AS.

Exercise

Example of a nonprojective proper variety. Let \(k\) be a field. Let \(L \subset \mathbf{P}^3_k\) be a line and let \(C \subset \mathbf{P}^3_k\) be a nonsingular conic. Assume that \(C \cap L = \emptyset\). Choose an isomorphism \(\varphi : L \to C\). Let \(X\) be the \(k\)-variety obtained by glueing \(C\) to \(L\) via \(\varphi\). In other words there is a surjective proper birational morphism \[\pi : \mathbf{P}^3_k \longrightarrow X\] and an open \(U \subset X\) such that \(\pi : \pi^{-1}(U) \to U\) is an isomorphism, \(\pi^{-1}(U) = \mathbf{P}^3_k \setminus (L \cup C)\) and such that \(\pi|_L = \pi|_C \circ \varphi\). (These conditions do not yet uniquely define \(X\). In order to do this you need to specify the structure sheaf of \(X\) along points of \(Z = X \setminus U\).) Show \(X\) exists, is a proper variety, but is not projective. (Hint: For existence use the result of Exercise 02A5. For non-projectivity use that \(\Pic(\mathbf{P}^3_k) = \mathbf{Z}\) to show that \(X\) cannot have an ample invertible sheaf.)

Differentials

Definitions and results. Kähler differentials.

1. Let \(R \to A\) be a ring map. The module of Kähler differentials of \(A\) over \(R\) is denoted \(\Omega_{A/R}\). It is generated by the elements \(\text{d}a\), \(a \in A\) subject to the relations: \[\text{d}(a_1 + a_2) = \text{d}a_1 + \text{d}a_2,\quad \text{d}(a_1a_2) = a_1\text{d}a_2 + a_2\text{d}a_1,\quad \text{d}r = 0\] The canonical universal \(R\)-derivation \(\text{d} : A \to \Omega_{A/R}\) maps \(a\mapsto \text{d}a\).

2. Consider the short exact sequence \[0 \to I \to A \otimes_R A \to A \to 0\] which defines the ideal \(I\). There is a canonical derivation \(\text{d} : A \to I/I^2\) which maps \(a\) to the class of \(a \otimes 1 - 1 \otimes a\). This is another presentation of the module of derivations of \(A\) over \(R\), in other words \[(I/I^2, \text{d}) \cong (\Omega_{A/R}, \text{d}).\]

3. For multiplicative subsets \(S_R \subset R\) and \(S_A \subset A\) such that \(S_R\) maps into \(S_A\) we have \[\Omega_{S_A^{-1}A / S_R^{-1}R} = S_A^{-1}\Omega_{A/R}.\]

4. If \(A\) is a finitely presented \(R\)-algebra then \(\Omega_{A/R}\) is a finitely presented \(A\)-module. Hence in this case the fitting ideals of \(\Omega_{A/R}\) are defined.

5. Let \(f : X \to S\) be a morphism of schemes. There is a quasi-coherent sheaf of \({\mathcal O}_X\)-modules \(\Omega_{X/S}\) and a \({\mathcal O}_S\)-linear derivation \[\text{d} : {\mathcal O}_X \longrightarrow \Omega_{X/S}\] such that for any affine opens \(\Spec(A) = U \subset X\), \(\Spec(R) = V \subset S\) with \(f(U) \subset V\) we have \[\Gamma(\Spec(A), \Omega_{X/S}) = \Omega_{A/R}\] compatibly with \(\text{d}\).

Exercise

Let \(k[\epsilon]\) be the ring of dual numbers over the field \(k\), i.e., \(\epsilon^2 = 0\).

1. Consider the ring map \[R = k[\epsilon] \to A = k[x, \epsilon]/(\epsilon x)\] Show that the Fitting ideals of \(\Omega_{A/R}\) are (starting with the zeroth Fitting ideal) \[(\epsilon), A, A, \ldots\]

2. Consider the map \(R = k[t] \to A = k[x, y, t]/(x(y-t)(y-1), x(x-t))\). Show that the Fitting ideals of \(\Omega_{A/R}\) in \(A\) are (assume characteristic \(k\) is zero for simplicity) \[x(2x-t)(2y-t-1)A, \ (x, y, t)\cap (x, y-1, t), \ A, \ A, \ldots\] So the \(0\)-the Fitting ideal is cut out by a single element of \(A\), the \(1\)st Fitting ideal defines two closed points of \(\Spec(A)\), and the others are all trivial.

3. Consider the map \(R = k[t] \to A = k[x, y, t]/(xy-t^n)\). Compute the Fitting ideals of \(\Omega_{A/R}\).

Remark

The \(k\)th Fitting ideal of \(\Omega_{X/S}\) is commonly used to define the singular scheme of the morphism \(X \to S\) when \(X\) has relative dimension \(k\) over \(S\). But as part (a) shows, you have to be careful doing this when your family does not have “constant” fibre dimension, e.g., when it is not flat. As part (b) shows, flatness doesn’t guarantee it works either (and yes this is a flat family). In “good cases” – such as in (c) – for families of curves you expect the \(0\)-th Fitting ideal to be zero and the \(1\)st Fitting ideal to define (scheme-theoretically) the singular locus.

Exercise

Suppose that \(R\) is a ring and \[A = R[x_1, \ldots, x_n]/(f_1, \ldots, f_n).\] Note that we are assuming that \(A\) is presented by the same number of equations as variables. Thus the matrix of partial derivatives \[( \partial f_i / \partial x_j )\] is \(n \times n\), i.e., a square matrix. Assume that its determinant is invertible as an element in \(A\). Note that this is exactly the condition that says that \(\Omega_{A/R} = (0)\) in this case of \(n\)-generators and \(n\) relations. Let \(\pi : B' \to B\) be a surjection of \(R\)-algebras whose kernel \(J\) has square zero (as an ideal in \(B'\)). Let \(\varphi : A \to B\) be a homomorphism of \(R\)-algebras. Show there exists a unique homomorphism of \(R\)-algebras \(\varphi' : A \to B'\) such that \(\varphi = \pi \circ \varphi'\).

Exercise

Find a generalization of the result of Exercise 02FE to the case where \(A = R[x, y]/(f)\).

Exercise

Let \(k\) be a field, let \(f_1, \ldots, f_c \in k[x_1, \ldots, x_n]\), and let \(A = k[x_1, \ldots, x_n]/(f_1, \ldots, f_c)\). Assume that \(f_j(0, \ldots, 0) = 0\). This means that \(\mathfrak m = (x_1, \ldots, x_n)A\) is a maximal ideal. Prove that the local ring \(A_\mathfrak m\) is regular if the rank of the matrix \[(\partial f_j/ \partial x_i)|_{(x_1, \ldots, x_n) = (0, \ldots, 0)}\] is \(c\). What is the dimension of \(A_\mathfrak m\) in this case? Show that the converse is false by giving an example where \(A_\mathfrak m\) is regular but the rank is less than \(c\); what is the dimension of \(A_\mathfrak m\) in your example?

Schemes, Final Exam, Fall 2007

These were the questions in the final exam of a course on Schemes, in the Spring of 2007 at Columbia University.

Exercise

Provide definitions of the following concepts.

1. \(X\) is a scheme

2. the morphism of schemes \(f : X \to Y\) is finite

3. the morphisms of schemes \(f : X \to Y\) is of finite type

4. the scheme \(X\) is Noetherian

5. the \({\mathcal O}_X\)-module \({\mathcal L}\) on the scheme \(X\) is invertible

6. the genus of a nonsingular projective curve over an algebraically closed field

Exercise

Let \(X = \Spec({\mathbf Z}[x, y])\), and let \({\mathcal F}\) be a quasi-coherent \({\mathcal O}_X\)-module. Suppose that \({\mathcal F}\) is zero when restricted to the standard affine open \(D(x)\).

1. Show that every global section \(s\) of \({\mathcal F}\) is killed by some power of \(x\), i.e., \(x^ns = 0\) for some \(n\in {\mathbf N}\).

2. Do you think the same is true if we do not assume that \({\mathcal F}\) is quasi-coherent?

Exercise

Suppose that \(X \to \Spec(R)\) is a proper morphism and that \(R\) is a discrete valuation ring with residue field \(k\). Suppose that \(X \times_{\Spec(R)} \Spec(k)\) is the empty scheme. Show that \(X\) is the empty scheme.

Exercise

Consider the projective⁴ variety \[{\mathbf P}^1 \times {\mathbf P}^1 = {\mathbf P}^1_{{\mathbf C}} \times_{\Spec({\mathbf C})} {\mathbf P}^1_{\mathbf C}\] over the field of complex numbers \({\mathbf C}\). It is covered by four affine pieces, corresponding to pairs of standard affine pieces of \({\mathbf P}^1_{\mathbf C}\). For example, suppose we use homogeneous coordinates \(X_0, X_1\) on the first factor and \(Y_0, Y_1\) on the second. Set \(x = X_1/X_0\), and \(y = Y_1/Y_0\). Then the 4 affine open pieces are the spectra of the rings \[{\mathbf C}[x, y], \quad {\mathbf C}[x^{-1}, y], \quad {\mathbf C}[x, y^{-1}], \quad {\mathbf C}[x^{-1}, y^{-1}].\] Let \(X \subset {\mathbf P}^1 \times {\mathbf P}^1\) be the closed subscheme which is the closure of the closed subset of the first affine piece given by the equation \[y^3(x^4 + 1) = x^4 -1.\]

1. Show that \(X\) is contained in the union of the first and the last of the 4 affine open pieces.

2. Show that \(X\) is a nonsingular projective curve.

3. Consider the morphism \(pr_2 : X \to {\mathbf P}^1\) (projection onto the first factor). On the first affine piece it is the map \((x, y) \mapsto x\). Briefly explain why it has degree \(3\).

4. Compute the ramification points and ramification indices for the map \(pr_2 : X \to {\mathbf P}^1\).

5. Compute the genus of \(X\).

Exercise

Let \(X \to \Spec({\mathbf Z})\) be a morphism of finite type. Suppose that there is an infinite number of primes \(p\) such that \(X \times_{\Spec({\mathbf Z})} \Spec({\mathbf F}_p)\) is not empty.

1. Show that \(X \times_{\Spec({\mathbf Z})}\Spec(\mathbf{Q})\) is not empty.

2. Do you think the same is true if we replace the condition “finite type” by the condition “locally of finite type”?

Schemes, Final Exam, Spring 2009

These were the questions in the final exam of a course on Schemes, in the Spring of 2009 at Columbia University.

Exercise

Let \(X\) be a Noetherian scheme. Let \(\mathcal{F}\) be a coherent sheaf on \(X\). Let \(x \in X\) be a point. Assume that \(\text{Supp}(\mathcal{F}) = \{ x \}\).

1. Show that \(x\) is a closed point of \(X\).

2. Show that \(H^0(X, \mathcal{F})\) is not zero.

3. Show that \(\mathcal{F}\) is generated by global sections.

4. Show that \(H^p(X, \mathcal{F}) = 0\) for \(p > 0\).

Remark

Let \(k\) be a field. Let \(\mathbf{P}^2_k = \text{Proj}(k[X_0, X_1, X_2])\). Any invertible sheaf on \(\mathbf{P}^2_k\) is isomorphic to \(\mathcal{O}_{\mathbf{P}^2_k}(n)\) for some \(n \in \mathbf{Z}\). Recall that \[\Gamma(\mathbf{P}^2_k, \mathcal{O}_{\mathbf{P}^2_k}(n)) = k[X_0, X_1, X_2]_n\] is the degree \(n\) part of the polynomial ring. For a quasi-coherent sheaf \(\mathcal{F}\) on \(\mathbf{P}^2_k\) set \(\mathcal{F}(n) = \mathcal{F} \otimes_{\mathcal{O}_{\mathbf{P}^2_k}} \mathcal{O}_{\mathbf{P}^2_k}(n)\) as usual.

Exercise

Let \(k\) be a field. Let \(\mathcal{E}\) be a vector bundle on \(\mathbf{P}^2_k\), i.e., a finite locally free \(\mathcal{O}_{\mathbf{P}^2_k}\)-module. We say \(\mathcal{E}\) is split if \(\mathcal{E}\) is isomorphic to a direct sum invertible \(\mathcal{O}_{\mathbf{P}^2_k}\)-modules.

1. Show that \(\mathcal{E}\) is split if and only if \(\mathcal{E}(n)\) is split.

2. Show that if \(\mathcal{E}\) is split then \(H^1({\mathbf{P}^2_k}, \mathcal{E}(n)) = 0\) for all \(n \in \mathbf{Z}\).

3. Let \[\varphi : \mathcal{O}_{\mathbf{P}^2_k} \longrightarrow \mathcal{O}_{\mathbf{P}^2_k}(1) \oplus \mathcal{O}_{\mathbf{P}^2_k}(1) \oplus \mathcal{O}_{\mathbf{P}^2_k}(1)\] be given by linear forms \(L_0, L_1, L_2 \in \Gamma(\mathbf{P}^2_k, \mathcal{O}_{\mathbf{P}^2_k}(1))\). Assume \(L_i \not = 0\) for some \(i\). What is the condition on \(L_0, L_1, L_2\) such that the cokernel of \(\varphi\) is a vector bundle? Why?

4. Given an example of such a \(\varphi\).

5. Show that \(\Coker(\varphi)\) is not split (if it is a vector bundle).

Remark

Freely use the following facts on dimension theory (and add more if you need more).

1. The dimension of a scheme is the supremum of the length of chains of irreducible closed subsets.

2. The dimension of a finite type scheme over a field is the maximum of the dimensions of its affine opens.

3. The dimension of a Noetherian scheme is the maximum of the dimensions of its irreducible components.

4. The dimension of an affine scheme coincides with the dimension of the corresponding ring.

5. Let \(k\) be a field and let \(A\) be a finite type \(k\)-algebra. If \(A\) is a domain, and \(x \not = 0\), then \(\dim(A) = \dim(A/xA) + 1\).

Exercise

Let \(k\) be a field. Let \(X\) be a projective, reduced scheme over \(k\). Let \(f : X \to \mathbf{P}^1_k\) be a morphism of schemes over \(k\). Assume there exists an integer \(d \geq 0\) such that for every point \(t \in \mathbf{P}^1_k\) the fibre \(X_t = f^{-1}(t)\) is irreducible of dimension \(d\). (Recall that an irreducible space is not empty.)

1. Show that \(\dim(X) = d + 1\).

2. Let \(X_0 \subset X\) be an irreducible component of \(X\) of dimension \(d + 1\). Prove that for every \(t \in \mathbf{P}^1_k\) the fibre \(X_{0, t}\) has dimension \(d\).

3. What can you conclude about \(X_t\) and \(X_{0, t}\) from the above?

4. Show that \(X\) is irreducible.

Remark

Given a projective scheme \(X\) over a field \(k\) and a coherent sheaf \(\mathcal{F}\) on \(X\) we set \[\chi(X, \mathcal{F}) = \sum\nolimits_{i \geq 0} (-1)^i\dim_k H^i(X, \mathcal{F}).\]

Exercise

Let \(k\) be a field. Write \(\mathbf{P}^3_k = \text{Proj}(k[X_0, X_1, X_2, X_3])\). Let \(C \subset \mathbf{P}^3_k\) be a type \((5, 6)\) complete intersection curve. This means that there exist \(F \in k[X_0, X_1, X_2, X_3]_5\) and \(G \in k[X_0, X_1, X_2, X_3]_6\) such that \[C = \text{Proj}(k[X_0, X_1, X_2, X_3]/(F, G))\] is a variety of dimension \(1\). (Variety implies reduced and irreducible, but feel free to assume \(C\) is nonsingular if you like.) Let \(i : C \to \mathbf{P}^3_k\) be the corresponding closed immersion. Being a complete intersection also implies that \[\xymatrix{ 0 \ar[r] & \mathcal{O}_{\mathbf{P}^3_k}(-11) \ar[r]^-{ \left( \begin{matrix} -G \\ F \end{matrix} \right) } & \mathcal{O}_{\mathbf{P}^3_k}(-5) \oplus \mathcal{O}_{\mathbf{P}^3_k}(-6) \ar[r]^-{(F, G)} & \mathcal{O}_{\mathbf{P}^3_k} \ar[r] & i_*\mathcal{O}_C \ar[r] & 0 }\] is an exact sequence of sheaves. Please use these facts to:

1. compute \(\chi(C, i^*\mathcal{O}_{\mathbf{P}^3_k}(n))\) for any \(n \in \mathbf{Z}\), and

2. compute the dimension of \(H^1(C, \mathcal{O}_C)\).

Exercise

Let \(k\) be a field. Consider the rings \[\begin{align*} A & = k[x, y]/(xy) \\ B & = k[u, v]/(uv) \\ C & = k[t, t^{-1}] \times k[s, s^{-1}] \end{align*}\] and the \(k\)-algebra maps \[\begin{matrix} A \longrightarrow C, & x \mapsto (t, 0), & y \mapsto (0, s) \\ B \longrightarrow C, & u \mapsto (t^{-1}, 0), & v \mapsto (0, s^{-1}) \end{matrix}\] It is a true fact that these maps induce isomorphisms \(A_{x + y} \to C\) and \(B_{u + v} \to C\). Hence the maps \(A \to C\) and \(B \to C\) identify \(\Spec(C)\) with open subsets of \(\Spec(A)\) and \(\Spec(B)\). Let \(X\) be the scheme obtained by glueing \(\Spec(A)\) and \(\Spec(B)\) along \(\Spec(C)\): \[X = \Spec(A) \amalg_{\Spec(C)} \Spec(B).\] As we saw in the course such a scheme exists and there are affine opens \(\Spec(A) \subset X\) and \(\Spec(B) \subset X\) whose overlap is exactly \(\Spec(C)\) identified with an open of each of these using the maps above.

1. Why is \(X\) separated?

2. Why is \(X\) of finite type over \(k\)?

3. Compute \(H^1(X, \mathcal{O}_X)\), or what is its dimension?

4. What is a more geometric way to describe \(X\)?

Schemes, Final Exam, Fall 2010

These were the questions in the final exam of a course on Schemes, in the Fall of 2010 at Columbia University.

Exercise

Provide definitions of the following concepts.

1. a separated scheme,

2. a quasi-compact morphism of schemes,

3. an affine morphism of schemes,

4. a multiplicative subset of a ring,

5. a Noetherian scheme,

6. a variety.

Exercise

Prime avoidance.

1. Let \(A\) be a ring. Let \(I \subset A\) be an ideal and let \(\mathfrak q_1\), \(\mathfrak q_2\) be prime ideals such that \(I \not \subset \mathfrak q_i\). Show that \(I \not \subset \mathfrak q_1 \cup \mathfrak q_2\).

2. What is a geometric interpretation of (1)?

3. Let \(X = \text{Proj}(S)\) for some graded ring \(S\). Let \(x_1, x_2 \in X\). Show that there exists a standard open \(D_{+}(F)\) which contains both \(x_1\) and \(x_2\).

Exercise

Why is a composition of affine morphisms affine?

Exercise

Give examples of the following:

1. A reducible projective scheme over a field \(k\).

2. A scheme with 100 points.

3. A non-affine morphism of schemes.

Exercise

Chevalley’s theorem and the Hilbert Nullstellensatz.

1. Let \(\mathfrak p \subset \mathbf{Z}[x_1, \ldots, x_n]\) be a maximal ideal. What does Chevalley’s theorem imply about \(\mathfrak p \cap \mathbf{Z}\)?

2. In turn, what does the Hilbert Nullstellensatz imply about \(\kappa(\mathfrak p)\)?

Exercise

Let \(A\) be a ring. Let \(S = A[X]\) as a graded \(A\)-algebra where \(X\) has degree \(1\). Show that \(\text{Proj}(S) \cong \Spec(A)\) as schemes over \(A\).

Exercise

Let \(A \to B\) be a finite ring map. Show that \(\Spec(B)\) is a H-projective scheme over \(\Spec(A)\).

Exercise

Give an example of a scheme \(X\) over a field \(k\) such that \(X\) is irreducible and such that for some finite extension \(k'/k\) the base change \(X_{k'} = X \times_{\Spec(k)} \Spec(k')\) is connected but reducible.

Schemes, Final Exam, Spring 2011

These were the questions in the final exam of a course on Schemes, in the Spring of 2011 at Columbia University.

Exercise

Provide definitions of the italicized concepts.

1. a separated scheme,

2. a universally closed morphism of schemes,

3. \(A\) dominates \(B\) for local rings \(A, B\) contained in a common field,

4. the dimension of a scheme \(X\),

5. the codimension of an irreducible closed subscheme \(Y\) of a scheme \(X\),

Exercise

State something formally equivalent to the fact discussed in the course.

1. The valuative criterion of properness for a morphism \(X \to Y\) of varieties for example.

2. The relationship between \(\dim(X)\) and the function field \(k(X)\) of \(X\) for a variety \(X\) over a field \(k\).

3. Fill in the blank: The category of nonsingular projective curves over \(k\) and nonconstant morphisms is anti-equivalent to \(\ldots\ldots\ldots\).

4. Noether normalization.

5. Jacobian criterion.

Exercise

Let \(k\) be a field. Let \(F \in k[X_0, X_1, X_2]\) be a homogeneous form of degree \(d\). Assume that \(C = V_{+}(F) \subset \mathbf{P}^2_k\) is a smooth curve over \(k\). Denote \(i : C \to \mathbf{P}^2_k\) the corresponding closed immersion.

1. Show that there is a short exact sequence \[0 \to \mathcal{O}_{\mathbf{P}^2_k}(-d) \to \mathcal{O}_{\mathbf{P}^2_k} \to i_*\mathcal{O}_C \to 0\] of coherent sheaves on \(\mathbf{P}^2_k\): tell me what the maps are and briefly why it is exact.

2. Conclude that \(H^0(C, \mathcal{O}_C) = k\).

3. Compute the genus of \(C\).

4. Assume now that \(P = (0 : 0 : 1)\) is not on \(C\). Prove that \(\pi : C \to \mathbf{P}^1_k\) given by \((a_0 : a_1 : a_2) \mapsto (a_0 : a_1)\) has degree \(d\).

5. Assume \(k\) is algebraically closed, assume all ramification indices (the “\(e_i\)”) are \(1\) or \(2\), and assume the characteristic of \(k\) is not equal to \(2\). How many ramification points does \(\pi : C \to \mathbf{P}^1_k\) have?

6. In terms of \(F\), what do you think is a set of equations of the set of ramification points of \(\pi\)?

7. Can you guess \(K_C\)?

Exercise

Let \(k\) be a field. Let \(X\) be a “triangle” over \(k\), i.e., you get \(X\) by glueing three copies of \(\mathbf{A}^1_k\) to each other by identifying \(0\) on the first copy to \(1\) on the second copy, \(0\) on the second copy to \(1\) on the third copy, and \(0\) on the third copy to \(1\) on the first copy. It turns out that \(X\) is isomorphic to \(\Spec(k[x, y]/(xy(x + y + 1)))\); feel free to use this. Compute the Picard group of \(X\).

Exercise

Let \(k\) be a field. Let \(\pi : X \to Y\) be a finite birational morphism of curves with \(X\) a projective nonsingular curve over \(k\). It follows from the material in the course that \(Y\) is a proper curve and that \(\pi\) is the normalization morphism of \(Y\). We have also seen in the course that there exists a dense open \(V \subset Y\) such that \(U = \pi^{-1}(V)\) is a dense open in \(X\) and \(\pi : U \to V\) is an isomorphism.

1. Show that there exists an effective Cartier divisor \(D \subset X\) such that \(D \subset U\) and such that \(\mathcal{O}_X(D)\) is ample on \(X\).

2. Let \(D\) be as in (1). Show that \(E = \pi(D)\) is an effective Cartier divisor on \(Y\).

3. Briefly indicate why

   1. the map \(\mathcal{O}_Y \to \pi_*\mathcal{O}_X\) has a coherent cokernel \(Q\) which is supported in \(Y \setminus V\), and

   2. for every \(n\) there is a corresponding map \(\mathcal{O}_Y(nE) \to \pi_*\mathcal{O}_X(nD)\) whose cokernel is isomorphic to \(Q\).

4. Show that \(\dim_k H^0(X, \mathcal{O}_X(nD)) - \dim_k H^0(Y, \mathcal{O}_Y(nE))\) is bounded (by what?) and conclude that the invertible sheaf \(\mathcal{O}_Y(nE)\) has lots of sections for large \(n\) (why?).

Schemes, Final Exam, Fall 2011

These were the questions in the final exam of a course on Commutative Algebra, in the Fall of 2011 at Columbia University.

Exercise

Provide definitions of the italicized concepts.

1. a Noetherian ring,

2. a Noetherian scheme,

3. a finite ring homomorphism,

4. a finite morphism of schemes,

5. the dimension of a ring.

Exercise

State something formally equivalent to the fact discussed in the course.

1. Zariski’s Main Theorem.

2. Noether normalization.

3. Chinese remainder theorem.

4. Going up for finite ring maps.

Exercise

Let \((A, \mathfrak m, \kappa)\) be a Noetherian local ring whose residue field has characteristic not \(2\). Suppose that \(\mathfrak m\) is generated by three elements \(x, y, z\) and that \(x^2 + y^2 + z^2 = 0\) in \(A\).

1. What are the possible values of \(\dim(A)\)?

2. Give an example to show that each value is possible.

3. Show that \(A\) is a domain if \(\dim(A) = 2\). (Hint: look at \(\bigoplus_{n \geq 0} \mathfrak m^n/\mathfrak m^{n + 1}\).)

Exercise

Let \(A\) be a ring. Let \(S \subset T \subset A\) be multiplicative subsets. Assume that \[\{\mathfrak q \mid \mathfrak q \cap S = \emptyset\} = \{\mathfrak q \mid \mathfrak q \cap T = \emptyset\}.\] Show that \(S^{-1}A \to T^{-1}A\) is an isomorphism.

Exercise

Let \(k\) be an algebraically closed field. Let \[V_0 = \{ A \in \text{Mat}(3 \times 3, k) \mid \text{rank}(A) = 1\} \subset \text{Mat}(3 \times 3, k) = k^9.\]

1. Show that \(V_0\) is the set of closed points of a (Zariski) locally closed subset \(V \subset \mathbf{A}^9_k\).

2. Is \(V\) irreducible?

3. What is \(\dim(V)\)?

Exercise

Prove that the ideal \((x^2, xy, y^2)\) in \(\mathbf{C}[x, y]\) cannot be generated by \(2\) elements.

Exercise

Let \(f \in \mathbf{C}[x, y]\) be a nonconstant polynomial. Show that for some \(\alpha, \beta \in \mathbf{C}\) the \(\mathbf{C}\)-algebra map \[\mathbf{C}[t] \longrightarrow \mathbf{C}[x, y]/(f),\quad t \longmapsto \alpha x + \beta y\] is finite.

Exercise

Show that given finitely many points \(p_1, \ldots, p_n \in \mathbf{C}^2\) the scheme \(\mathbf{A}^2_\mathbf{C} \setminus \{p_1, \ldots, p_n\}\) is a union of two affine opens.

Exercise

Show that there exists a surjective morphism of schemes \(\mathbf{A}^1_\mathbf{C} \to \mathbf{P}^1_\mathbf{C}\). (Surjective just means surjective on underlying sets of points.)

Exercise

Let \(k\) be an algebraically closed field. Let \(A \subset B\) be an extension of domains which are both finite type \(k\)-algebras. Prove that the image of \(\Spec(B) \to \Spec(A)\) contains a nonempty open subset of \(\Spec(A)\) using the following steps:

1. Prove it if \(A \to B\) is also finite.

2. Prove it in case the fraction field of \(B\) is a finite extension of the fraction field of \(A\).

3. Reduce the statement to the previous case.

Schemes, Final Exam, Fall 2013

These were the questions in the final exam of a course on Commutative Algebra, in the Fall of 2013 at Columbia University.

Exercise

Provide definitions of the italicized concepts.

1. a radical ideal of a ring,

2. a finite type ring homomorphism,

3. a differential a la Weil,

4. a scheme.

Exercise

State something formally equivalent to the fact discussed in the course.

1. result on hilbert polynomials of graded modules.

2. dimension of a Noetherian local ring \((R, \mathfrak m)\) and \(\bigoplus_{n \geq 0} \mathfrak m^n/\mathfrak m^{n + 1}\).

3. Riemann-Roch.

4. Clifford’s theorem.

5. Chevalley’s theorem.

Exercise

Let \(A \to B\) be a ring map. Let \(S \subset A\) be a multiplicative subset. Assume that \(A \to B\) is of finite type and \(S^{-1}A \to S^{-1}B\) is surjective. Show that there exists an \(f \in S\) such that \(A_f \to B_f\) is surjective.

Exercise

Give an example of an injective local homomorphism \(A \to B\) of local rings, such that \(\Spec(B) \to \Spec(A)\) is not surjective.

Situation

Let \(k\) be an algebraically closed field. Let \(F(X_0, X_1, X_2) \in k[X_0, X_1, X_2]\) be an irreducible polynomial homogeneous of degree \(d\). We let \[D = V(F) \subset \mathbf{P}^2\] be the projective plane curve given by the vanishing of \(F\). Set \(x = X_1/X_0\) and \(y = X_2/X_0\) and \(f(x, y) = X_0^{-d}F(X_0, X_1, X_2) = F(1, x, y)\). We denote \(K\) the fraction field of the domain \(k[x, y]/(f)\). We let \(C\) be the abstract curve corresponding to \(K\). Recall (from the lectures) that there is a surjective map \(C \to D\) which is bijective over the nonsingular locus of \(D\) and an isomorphism if \(D\) is nonsingular. Set \(f_x = \partial f/\partial x\) and \(f_y = \partial f/\partial y\). Finally, we denote \(\omega = \text{d}x/f_y = - \text{d}y/f_x\) the element of \(\Omega_{K/k}\) discussed in the lectures. Denote \(K_C\) the divisor of zeros and poles of \(\omega\).

Exercise

In Situation 09U0 assume \(d \geq 3\) and that the curve \(D\) has exactly one singular point, namely \(P = (1 : 0 : 0)\). Assume further that we have the expansion \[f(x, y) = xy + h.o.t\] around \(P = (0, 0)\). Then \(C\) has two points \(v\) and \(w\) lying over \(P\) characterized by \[v(x) = 1, v(y) > 1 \quad\text{and}\quad w(x) > 1, w(y) = 1\]

1. Show that the element \(\omega = \text{d}x/f_y = - \text{d}y/f_x\) of \(\Omega_{K/k}\) has a first order pole at both \(v\) and \(w\). (The behaviour of \(\omega\) at nonsingular points is as discussed in the lectures.)

2. In the lectures we have shown that \(\omega\) vanishes to order \(d - 3\) at the divisor \(X_0 = 0\) pulled back to \(C\) under the map \(C \to D\). Combined with the information of (1) what is the degree of the divisor of zeros and poles of \(\omega\) on \(C\)?

3. What is the genus of the curve \(C\)?

Exercise

In Situation 09U0 assume \(d = 5\) and that the curve \(C = D\) is nonsingular. In the lectures we have shown that the genus of \(C\) is \(6\) and that the linear system \(K_C\) is given by \[L(K_C) = \{h\omega \mid h \in k[x, y],\ \deg(h) \leq 2\}\] where \(\deg\) indicates total degree⁵. Let \(P_1, P_2, P_3, P_4, P_5 \in D\) be pairwise distinct points lying in the affine open \(X_0 \not = 0\). We denote \(\sum P_i = P_1 + P_2 + P_3 + P_4 + P_5\) the corresponding divisor of \(C\).

1. Describe \(L(K_C - \sum P_i)\) in terms of polynomials.

2. What are the possibilities for \(l(\sum P_i)\)?

Exercise

Write down an \(F\) as in Situation 09U0 with \(d = 100\) such that the genus of \(C\) is \(0\).

Exercise

Let \(k\) be an algebraically closed field. Let \(K/k\) be finitely generated field extension of transcendence degree \(1\). Let \(C\) be the abstract curve corresponding to \(K\). Let \(V \subset K\) be a \(g^r_d\) and let \(\Phi : C \to \mathbf{P}^r\) be the corresponding morphism. Show that the image of \(C\) is contained in a quadric⁶ if \(V\) is a complete linear system and \(d\) is large enough relative to the genus of \(C\). (Extra credit: good bound on the degree needed.)

Exercise

Notation as in Situation 09U0. Let \(U \subset \mathbf{P}^2_k\) be the open subscheme whose complement is \(D\). Describe the \(k\)-algebra \(A = \mathcal{O}_{\mathbf{P}^2_k}(U)\). Give an upper bound for the number of generators of \(A\) as a \(k\)-algebra.

Schemes, Final Exam, Spring 2014

These were the questions in the final exam of a course on Schemes, in the Spring of 2014 at Columbia University.

Exercise

Let \((X, \mathcal{O}_X)\) be a scheme. Provide definitions of the italicized concepts.

1. the local ring of \(X\) at a point \(x\),

2. a quasi-coherent sheaf of \(\mathcal{O}_X\)-modules,

3. a coherent sheaf of \(\mathcal{O}_X\)-modules (please assume \(X\) is locally Noetherian,

4. an affine open of \(X\),

5. a finite morphism of schemes \(X \to Y\).

Exercise

Precisely state a nontrivial fact discussed in the lectures related to each item.

1. on birational invariance of pluri-genera of varieties,

2. being an affine morphism is a local property,

3. the topology of a scheme theoretic fibre of a morphism, and

4. valuative criterion of properness.

Exercise

Let \(X = \mathbf{A}^2_\mathbf{C}\) where \(\mathbf{C}\) is the field of complex numbers. A line will mean a closed subscheme of \(X\) defined by one linear equation \(ax + by + c = 0\) for some \(a, b, c \in \mathbf{C}\) with \((a, b) \not = (0, 0)\). A curve will mean an irreducible (so nonempty) closed subscheme \(C \subset X\) of dimension \(1\). A quadric will mean a curve defined by one quadratic equation \(ax^2 + bxy + cy^2 + dx + ey + f = 0\) for some \(a, b, c, d, e, f \in \mathbf{C}\) and \((a, b, c) \not = (0, 0, 0)\).

1. Find a curve \(C\) such that every line has nonempty intersection with \(C\).

2. Find a curve \(C\) such that every line and every quadric has nonempty intersection with \(C\).

3. Show that for every curve \(C\) there exists another curve such that \(C \cap C' = \emptyset\).

Exercise

Let \(k\) be a field. Let \(b : X \to \mathbf{A}^2_k\) be the blow up of the affine plane in the origin. In other words, if \(\mathbf{A}^2_k = \Spec(k[x, y])\), then \(X = \text{Proj}(\bigoplus_{n \geq 0} \mathfrak m^n)\) where \(\mathfrak m = (x, y) \subset k[x, y]\). Prove the following statements

1. the scheme theoretic fibre \(E\) of \(b\) over the origin is isomorphic to \(\mathbf{P}^1_k\),

2. \(E\) is an effective Cartier divisor on \(X\),

3. the restriction of \(\mathcal{O}_X(-E)\) to \(E\) is a line bundle of degree \(1\).

(Recall that \(\mathcal{O}_X(-E)\) is the ideal sheaf of \(E\) in \(X\).)

Exercise

Let \(k\) be a field. Let \(X\) be a projective variety over \(k\). Show there exists an affine variety \(U\) over \(k\) and a surjective morphism of varieties \(U \to X\).

Exercise

Let \(k\) be a field of characteristic \(p > 0\) different from \(2,3\). Consider the closed subscheme \(X\) of \(\mathbf{P}^n_k\) defined by \[\sum\nolimits_{i = 0, \ldots, n} X_i = 0,\quad \sum\nolimits_{i = 0, \ldots, n} X_i^2 = 0,\quad \sum\nolimits_{i = 0, \ldots, n} X_i^3 = 0\] For which pairs \((n, p)\) is this variety singular?

Commutative Algebra, Final Exam, Fall 2016

These were the questions in the final exam of a course on Commutative Algebra, in the Fall of 2016 at Columbia University.

Exercise

Let \(R\) be a ring. Provide definitions of the italicized concepts.

1. the local ring of \(R\) at a prime \(\mathfrak p\),

2. a finite \(R\)-module,

3. a finitely presented \(R\)-module,

4. \(R\) is regular,

5. \(R\) is catenary,

6. \(R\) is Cohen-Macaulay.

Exercise

Precisely state a nontrivial fact discussed in the lectures related to each item.

1. regular rings,

2. associated primes of Cohen-Macaulay modules,

3. dimension of a finite type domain over a field, and

4. Chevalley’s theorem.

Exercise

Let \(A \to B\) be a ring map such that

1. \(A\) is local with maximal ideal \(\mathfrak m\),

2. \(A \to B\) is a finite⁷ ring map,

3. \(A \to B\) is injective (we think of \(A\) as a subring of \(B\)).

Show that there is a prime ideal \(\mathfrak q \subset B\) with \(\mathfrak m = A \cap \mathfrak q\).

Exercise

Let \(k\) be a field. Let \(R = k[x, y, z, w]\). Consider the ideal \(I = (xy, xz, xw)\). What are the irreducible components of \(V(I) \subset \Spec(R)\) and what are their dimensions?

Exercise

Let \(k\) be a field. Let \(A = k[x, x^{-1}]\) and \(B = k[y]\). Show that any \(k\)-algebra map \(\varphi : A \to B\) maps \(x\) to a constant.

Exercise

Consider the ring \(R = \mathbf{Z}[x, y]/(xy - 7)\). Prove that \(R\) is regular.

Given a Noetherian local ring \((R, \mathfrak m, \kappa)\) for \(n \geq 0\) we let \(\varphi_R(n) = \dim_\kappa(\mathfrak m^n/\mathfrak m^{n + 1})\).

Exercise

Does there exist a Noetherian local ring \(R\) with \(\varphi_R(n) = n + 1\) for all \(n \geq 0\)?

Exercise

Let \(R\) be a Noetherian local ring. Suppose that \(\varphi_R(0) = 1\), \(\varphi_R(1) = 3\), \(\varphi_R(2) = 5\). Show that \(\varphi_R(3) \leq 7\).

Schemes, Final Exam, Spring 2017

These were the questions in the final exam of a course on schemes, in the Spring of 2017 at Columbia University.

Exercise

Let \(f : X \to Y\) be a morphism of schemes. Provide brief definitions of the italicized concepts.

1. the scheme theoretic fibre of \(f\) at \(y \in Y\),

2. \(f\) is a finite morphism,

3. a quasi-coherent \(\mathcal{O}_X\)-module,

4. \(X\) is variety,

5. \(f\) is a smooth morphism,

6. \(f\) is a proper morphism.

Exercise

Precisely but briefly state a nontrivial fact discussed in the lectures related to each item.

1. pushforward of quasi-coherent sheaves,

2. cohomology of coherent sheaves on projective varieties,

3. Serre duality for a projective scheme over a field, and

4. Riemann-Hurwitz.

Exercise

Let \(k\) be an algebraically closed field. Let \(\ell > 100\) be a prime number different from the characteristic of \(k\). Let \(X\) be the nonsingular projective model of the affine curve given by the equation \[y^\ell = x(x - 1)^3\] in \(\mathbf{A}^2_k\). Answer the following questions:

1. What is the genus of \(X\)?

2. Give an upper bound for the gonality⁸ of \(X\).

Exercise

Let \(k\) be an algebraically closed field. Let \(X\) be a reduced, projective scheme over \(k\) all of whose irreducible components have the same dimension \(1\). Let \(\omega_{X/k}\) be the relative dualizing module. Show that if \(\dim_k H^1(X, \omega_{X/k}) > 1\), then \(X\) is disconnected.

Exercise

Give an example of a scheme \(X\) and a nontrivial invertible \(\mathcal{O}_X\)-module \(\mathcal{L}\) such that both \(H^0(X, \mathcal{L})\) and \(H^0(X, \mathcal{L}^{\otimes -1})\) are nonzero.

Exercise

Let \(k\) be an algebraically closed field. Let \(g \geq 3\). Let \(X\) and \(X'\) be smooth projective curves over \(k\) of genus \(g\) and \(g + 1\). Let \(Y \subset X \times X'\) be a curve such that the projections \(Y \to X\) and \(Y \to X'\) are nonconstant. Prove that the nonsingular projective model of \(Y\) has genus \(\geq 2g + 1\).

Exercise

Let \(k\) be a finite field. Let \(g > 1\). Sketch a proof of the following: there are only a finite number of isomorphism classes of smooth projective curves over \(k\) of genus \(g\). (You will get credit for even just trying to answer this.)

Commutative Algebra, Final Exam, Fall 2017

These were the questions in the final exam of a course on commutative algebra, in the Fall of 2017 at Columbia University.

Exercise

Provide brief definitions of the italicized concepts.

1. the left adjoint of a functor \(F : \mathcal{A} \to \mathcal{B}\),

2. the transcendence degree of an extension \(L/K\) of fields,

3. a regular function on a classical affine variety \(X \subset k^n\),

4. a sheaf on a topological space,

5. a local ring, and

6. a morphism of schemes \(f : X \to Y\) being affine.

Exercise

Precisely but briefly state a nontrivial fact discussed in the lectures related to each item (if there is more than one then just pick one of them).

1. Yoneda lemma,

2. Mayer-Vietoris,

3. dimension and cohomology,

4. Hilbert polynomial, and

5. duality for projective space.

Exercise

Let \(k\) be an algebraically closed field. Consider the closed subset \(X\) of \(k^5\) with Zariski topology and coordinates \(x_1, x_2, x_3, x_4, x_5\) given by the equations \[x_1^2 - x_4 = 0,\quad x_2^5 - x_5 = 0,\quad x_3^2 + x_3 + x_4 + x_5 = 0\] What is the dimension of \(X\) and why?

Exercise

Let \(k\) be a field. Let \(X = \mathbf{P}^1_k\) be the projective space of dimension \(1\) over \(k\). Let \(\mathcal{E}\) be a finite locally free \(\mathcal{O}_X\)-module. For \(d \in \mathbf{Z}\) denote \(\mathcal{E}(d) = \mathcal{E} \otimes_{\mathcal{O}_X} \mathcal{O}_X(d)\) the \(d\)th Serre twist of \(\mathcal{E}\) and \(h^i(X, \mathcal{E}(d)) = \dim_k H^i(X, \mathcal{E}(d))\).

1. Why is there no \(\mathcal{E}\) with \(h^0(X, \mathcal{E}) = 5\) and \(h^0(X, \mathcal{E}(1)) = 4\)?

2. Why is there no \(\mathcal{E}\) with \(h^1(X, \mathcal{E}(1)) = 5\) and \(h^1(X, \mathcal{E}) = 4\)?

3. For which \(a \in \mathbf{Z}\) can there exist a vector bundle \(\mathcal{E}\) on \(X\) with \[\begin{matrix} h^0(X, \mathcal{E})\phantom{(1)} = 1 & h^1(X, \mathcal{E})\phantom{(1)} = 1 \\ h^0(X, \mathcal{E}(1)) = 2 & h^1(X, \mathcal{E}(1)) = 0 \\ h^0(X, \mathcal{E}(2)) = 4 & h^1(X, \mathcal{E}(2)) = a \end{matrix}\]

Partial answers are welcomed and encouraged.

Exercise

Let \(X\) be a topological space which is the union \(X = Y \cup Z\) of two closed subsets \(Y\) and \(Z\) whose intersection is denoted \(W = Y \cap Z\). Denote \(i : Y \to X\), \(j : Z \to X\), and \(k : W \to X\) the inclusion maps.

1. Show that there is a short exact sequence of sheaves \[0 \to \underline{\mathbf{Z}}_X \to i_*(\underline{\mathbf{Z}}_Y) \oplus j_*(\underline{\mathbf{Z}}_Z) \to k_*(\underline{\mathbf{Z}}_W) \to 0\] where \(\underline{\mathbf{Z}}_X\) denotes the constant sheaf with value \(\mathbf{Z}\) on \(X\), etc.

2. What can you conclude about the relationship between the cohomology groups of \(X\), \(Y\), \(Z\), \(W\) with \(\mathbf{Z}\)-coefficients?

Exercise

Let \(k\) be a field. Let \(A = k[x_1, x_2, x_3, \ldots]\) be the polynomial ring in infinitely many variables. Denote \(\mathfrak m\) the maximal ideal of \(A\) generated by all the variables. Let \(X = \Spec(A)\) and \(U = X \setminus \{\mathfrak m\}\).

1. Show \(H^1(U, \mathcal{O}_U) = 0\). Hint: Čech cohomology computation.

2. What is your guess for \(H^i(U, \mathcal{O}_U)\) for \(i \geq 1\)?

Exercise

Let \(A\) be a local ring. Let \(a \in A\) be a nonzerodivisor. Let \(I, J \subset A\) be ideals such that \(IJ = (a)\). Show that the ideal \(I\) is principal, i.e., generated by one element (which will turn out to be a nonzerodivisor).

Schemes, Final Exam, Spring 2018

These were the questions in the final exam of a course on schemes, in the Spring of 2018 at Columbia University.

Exercise

Provide brief definitions of the italicized concepts. Let \(k\) be an algebraically closed field. Let \(X\) be a projective curve over \(k\).

1. a smooth algebra over \(k\),

2. the degree of an invertible \(\mathcal{O}_X\)-module on \(X\),

3. the genus of \(X\),

4. the Weil divisor class group of \(X\),

5. \(X\) is hyperelliptic, and

6. the intersection number of two curves on a smooth projective surface over \(k\).

Exercise

Precisely but briefly state a nontrivial fact discussed in the lectures related to each item (if there is more than one then just pick one of them).

1. Riemann-Hurwitz theorem,

2. Clifford’s theorem,

3. factorization of maps between smooth projective surfaces,

4. Hodge index theorem, and

5. Riemann hypothesis for curves over finite fields.

Exercise

Let \(k\) be an algebraically closed field. Let \(X \subset \mathbf{P}^3_k\) be a smooth curve of degree \(d\) and genus \(\geq 2\). Assume \(X\) is not contained in a plane and that there is a line \(\ell\) in \(\mathbf{P}^3_k\) meeting \(X\) in \(d - 2\) points. Show that \(X\) is hyperelliptic.

Exercise

Let \(k\) be an algebraically closed field. Let \(X\) be a projective curve with pairwise distinct singular points \(p_1, \ldots, p_n\). Explain why the genus of the normalization of \(X\) is at most \(-n + \dim_k H^1(X, \mathcal{O}_X)\).

Exercise

Let \(k\) be a field. Let \(X = \Spec(k[x, y])\) be affine \(2\) space. Let \[I = (x^3, x^2y, xy^2, y^3) \subset k[x, y].\] Let \(Y \subset X\) be the closed subscheme corresponding to \(I\). Let \(b : X' \to X\) be the blowing up of the ideal \((x, y)\), i.e., the blow up of affine space at the origin.

1. Show that the scheme theoretic inverse image \(b^{-1}Y \subset X'\) is an effective Cartier divisor.

2. Given an example of an ideal \(J \subset k[x, y]\) with \(I \subset J \subset (x, y)\) such that if \(Z \subset X\) is the closed subscheme corresponding to \(J\), then the scheme theoretic inverse image \(b^{-1}Z\) is not an effective Cartier divisor.

Exercise

Let \(k\) be an algebraically closed field. Consider the following types of surfaces

1. \(S = C_1 \times C_2\) where \(C_1\) and \(C_2\) are smooth projective curves,

2. \(S = C_1 \times C_2\) where \(C_1\) and \(C_2\) are smooth projective curves and the genus of \(C_1\) is \(> 0\),

3. \(S \subset \mathbf{P}^3_k\) is a hypersurface of degree \(4\), and

4. \(S \subset \mathbf{P}^3_k\) is a smooth hypersurface of degree \(4\).

For each type briefly indicate why or why not the class of surfaces of this type contains rational surfaces.

Exercise

Let \(k\) be an algebraically closed field. Let \(S \subset \mathbf{P}^3_k\) be a smooth hypersurface of degree \(d\). Assume that \(S\) contains a line \(\ell\). What is the self square of \(\ell\) viewed as a divisor on \(S\)?

Commutative Algebra, Final Exam, Fall 2019

These were the questions in the final exam of a course on commutative algebra, in the Fall of 2019 at Columbia University.

Exercise

Provide brief definitions of the italicized concepts.

1. a constructible subset of a Noetherian topological space,

2. the localization of an \(R\)-module \(M\) at a prime \(\mathfrak p\),

3. the length of a module over a Noetherian local ring \((A, \mathfrak m, \kappa)\),

4. a projective module over a ring \(R\), and

5. a Cohen-Macaulay module over a Noetherian local ring \((A, \mathfrak m, \kappa)\).

Exercise

Precisely but briefly state a nontrivial fact discussed in the lectures related to each item (if there is more than one then just pick one of them).

1. images of constructible sets,

2. Hilbert Nullstellensatz,

3. dimension of finite type algebras over fields,

4. Noether normalization, and

5. regular local rings.

For a ring \(R\) and an ideal \(I \subset R\) recall that \(V(I)\) denotes the set of \(\mathfrak p \in \Spec(R)\) with \(I \subset \mathfrak p\).

Exercise

Construct infinitely many distinct prime ideals \(\mathfrak p \subset \mathbf{C}[x, y]\) such that \(V(\mathfrak p)\) contains \((x, y)\) and \((x - 1, y - 1)\).

Exercise

Let \(R = \mathbf{C}[x, y, z]/(xy)\). Argue briefly there does not exist a prime ideal \(\mathfrak p \subset R\) such that \(V(\mathfrak p)\) contains \((x, y - 1, z - 5)\) and \((x - 1, y, z - 7)\).

Exercise

Let \(p\) be a prime number (you may assume \(p = 2\) to simplify the formulas). Let \(R\) be a ring such that \(p = 0\) in \(R\).

1. Show that the map \(F : R \to R\), \(x \mapsto x^p\) is a ring homomorphism.

2. Show that \(\Spec(F) : \Spec(R) \to \Spec(R)\) is the identity map.

Recall that a specialization \(x \leadsto y\) of points of a topological space simply means \(y\) is in the closure of \(x\). We say \(x \leadsto y\) is an immediate specialization if there does not exist a \(z\) different from \(x\) and \(y\) such that \(x \leadsto z\) and \(z \leadsto y\).

Exercise

Suppose we have a sober topological space \(X\) containing \(5\) distinct points \(x, y, z, u, v\) having the following specializations \[\xymatrix{ x \ar[d] \ar[r] & u & v \ar[l] \ar[dl] \\ y \ar[r] & z }\] What is the minimal dimension such an \(X\) can have? If \(X\) is the spectrum of a finite type algebra over a field and \(x \leadsto u\) is an immediate specialization, what can you say about the specialization \(v \leadsto z\)?

Exercise

Let \(R = \mathbf{C}[x, y, z]\). Let \(M = R/(x, z)\) and \(N = R/(y, z)\). For which \(i \in \mathbf{Z}\) is \(\text{Tor}_i^R(M, N)\) nonzero?

Exercise

Let \(A \to B\) be a flat local homomorphism of local Noetherian rings. Show that if \(A\) has depth \(k\), then \(B\) has depth at least \(k\).

Algebraic Geometry, Final Exam, Spring 2020

These were the questions in the final exam of a course on Algebraic Geometry, in the Spring of 2020 at Columbia University.

Exercise

Provide brief definitions of the italicized concepts.

1. a scheme,

2. a morphism of schemes,

3. a quasi-coherent module on a scheme,

4. a variety over a field \(k\),

5. a curve over a field \(k\),

6. a finite morphism of schemes,

7. the cohomology of a sheaf of abelian groups \(\mathcal{F}\) over a topological space \(X\),

8. a dualizing sheaf on a scheme \(X\) of dimension \(d\) proper over a field \(k\), and

9. a rational map from a variety \(X\) to a variety \(Y\).

Exercise

Precisely but briefly state a nontrivial fact discussed in the lectures related to each item (if there is more than one then just pick one of them).

1. cohomology of abelian sheaves on a Noetherian topological space \(X\) of dimension \(d\),

2. sheaf of differentials \(\Omega^1_{X/k}\) of a smooth variety over a field \(k\),

3. dualizing sheaf \(\omega_X\) of a smooth projective variety \(X\) over the field \(k\),

4. a smooth proper genus \(0\) curve over an algebraically closed field \(k\), and

5. the genus of a plane curve of degree \(d\).

Exercise

Let \(k\) be a field. Let \(X\) be a scheme over \(k\). Assume \(X = X_1 \cup X_2\) is an open covering with \(X_1\), \(X_2\) both isomorphic to \(\mathbf{P}^1_k\) and \(X_1 \cap X_2\) isomorphic to \(\mathbf{A}^1_k\). (Such a scheme exists, for example you can take \(\mathbf{P}^1_k\) with \(\infty\) doubled.) Show that \(\dim_k H^1(X, \mathcal{O}_X)\) is infinite.

Exercise

Let \(k\) be an algebraically closed field. Let \(Y\) be a smooth projective curve of genus \(10\). Find a good lower bound for the genus of a smooth projective curve \(X\) such that there exists a nonconstant morphism \(f : X \to Y\) which is not an isomorphism.

Exercise

Let \(k\) be an algebraically closed field of characteristic \(0\). Let \[X : T_0^d + T_1^d - T_2^d = 0 \subset \mathbf{P}^2_k\] be the Fermat curve of degree \(d \geq 3\). Consider the closed points \(p = [1 : 0 : 1]\) and \(q = [0 : 1 : 1]\) on \(X\). Set \(D = [p] - [q]\).

1. Show that \(D\) is nontrivial in the Weil divisor class group.

2. Show that \(d D\) is trivial in the Weil divisor class group. (Hint: try to show that both \(d[p]\) and \(d[q]\) are the intersection of \(X\) with a line in the plane.)

Exercise

Let \(k\) be an algebraically closed field. Consider the \(2\)-uple embedding \[\varphi : \mathbf{P}^2 \longrightarrow \mathbf{P}^5\] In terms of the material/notation in the lectures this is the morphism \[\varphi = \varphi_{\mathcal{O}_{\mathbf{P}^2}(2)} : \mathbf{P}^2 \longrightarrow \mathbf{P}(\Gamma(\mathbf{P}^2, \mathcal{O}_{\mathbf{P}^2}(2)))\] In terms of homogeneous coordinates it is given by \[[a_0 : a_1 : a_2] \longmapsto [a_0^2 : a_0a_1 : a_0a_2 : a_1^2 : a_1a_2 : a_2^2]\] It is a closed immersion (please just use this). Let \(I \subset k[T_0, \ldots, T_5]\) be the homogeneous ideal of \(\varphi(\mathbf{P}^2)\), i.e., the elements of the homogeneous part \(I_d\) are the homogeneous polynomials \(F(T_0, \ldots, T_5)\) of degree \(d\) which restrict to zero on the closed subscheme \(\varphi(\mathbf{P}^2)\). Compute \(\dim_k I_d\) as a function of \(d\).

Exercise

Let \(k\) be an algebraically closed field. Let \(X\) be a proper scheme of dimension \(d\) over \(k\) with dualizing module \(\omega_X\). You are given the following information:

1. \(\text{Ext}^i_X(\mathcal{F}, \omega_X) \times H^{d - i}(X, \mathcal{F}) \to H^d(X, \omega_X) \xrightarrow{t} k\) is nondegenerate for all \(i\) and for all coherent \(\mathcal{O}_X\)-modules \(\mathcal{F}\), and

2. \(\omega_X\) is finite locally free of some rank \(r\).

Show that \(r = 1\). (Hint: see what happens if you take \(\mathcal{F}\) a suitable module supported at a closed point.)

Commutative Algebra, Final Exam, Fall 2021

These were the questions in the final exam of a course on commutative algebra, in the Fall of 2021 at Columbia University.

Exercise

Provide brief definitions of the italicized concepts.

1. a multiplicative subset of a ring \(A\),

2. an Artinian ring \(A\),

3. the spectrum of a ring \(A\) as a topological space,

4. a flat ring map \(A \to B\),

5. the height of a prime ideal \(\mathfrak p\) in \(A\), and

6. the functors \(\text{Tor}^A_i(-, -)\) over a ring \(A\).

Exercise

Precisely but briefly state a nontrivial fact discussed in the lectures related to each item (if there is more than one then just pick one of them).

1. Artinian rings,

2. flatness and prime ideals,

3. lengths of \(A/\mathfrak m^n\) for \((A, \mathfrak m)\) Noetherian local,

4. the dimension formula for universally catenary Noetherian rings,

5. completion of a Noetherian local ring, and

6. Matlis duality for Artinian local rings.

Exercise

What is the structure of the group of units of \(\mathbf{Z}[x, 1/x]\) as an abelian group? No explanation necessary.