Non-commutative Algebraic Geometry

S. P. Smith

August 26, 1997

0 Introduction

This is a reasonably faithful account of the five lectures I delivered at the summer course “Geometria Algebraica no Commutativa y Espacios Cuanticos” for graduate students, in Spain, July 25–29, 1994. The material covered was, for the most part, an abridged version of Artin and Zhang’s paper [2].

Fix a field k. Given a ℤ-graded k-algebra, A say, which for simplicity is assumed to be left noetherian and locally finite dimensional, its noncommutative projective scheme is defined to be the pair

proj(A) := (tails(A), A),

where tails(A) is the quotient category of grmod(A), the category of finitely generated graded left A-modules, modulo its full subcategory of finite dimensional modules, and A is the image of the distinguished module _{A}A in tails(A). If A is a quotient of a commutative polynomial ring generated in degree 1, Serre [4] proved that proj(A) is isomorphic (in an obvious sense) to the pair (Coh(O_{X}), O_{X}), where X denotes the projective scheme determined by A, O_{X} is the sheaf of regular functions on X, and Coh(O_{X}) is the category of coherent O_{X}-modules. Thus tails(A) is the non-commutative analogue of Coh(O_{X}), and the objects in tails(A) are the non-commutative geometric objects analogous to sheaves of O_{X}-modules.

For each F ∈ tails(A) there are cohomology groups H^{q}(F), q ≥ 0, which generalize the Čech cohomology groups—if A is commutative as above, then H^{q}(F) coincides with H^{q}(X, F) for F ∈ Coh(O_{X}). The functors H^{q}(−) have the properties one would want/expect for a satisfactory generalization of H^{q}(X, −). In particular, there is a version of Serre’s Finiteness Theorem (4.16) provided a certain technical condition χ holds (see Definition 4.11). Every commutative algebra satisfies χ, but there exist rather nice non-commutative algebras which do not (Example 4.15). We compute the cohomology groups H^{q}(A[d]), d ∈ ℤ, when A is an Artin-Schelter regular algebra. This family of algebras includes the commutative polynomial ring, and in that case H^{q}(A[d]) ≅ H^{q}(ℙ^{n}, O(d)). The Artin-Schelter regular algebras are non-commutative algebras which enjoy many of the properties of polynomial rings; amongst the non-commutative Artin-Schelter regular algebras are most graded iterated Ore extensions, homogenizations of enveloping algebras, and Sklyanin algebras. Artin-Schelter regular algebras always satisfy the condition χ.

The functorial behavior of tails(A), and maps between proj(A) and proj(B) are discussed in Section 5.

The polarized projective scheme associated to A is the triple (tails(A), A, [1]), where [1] is the degree shift functor on grmod(M), namely M[1]_{i} = M_{i+1}. Including [1] with proj(A) is analogous to specifying a line bundle on a scheme X; it is natural to ask whether that line bundle is very ample, i.e., whether it determines an embedding of X in some projective space (or, equivalently, whether it arises from an embedding of X in some ℙ^{n}). This leads to the notion of ampleness for [1] on proj(A) (see Definition 5.18). Whether or not [1] is ample in proj(A) is closely related to the condition χ.

Polarized projective schemes are objects in a category of triples (C, O, s) where C is a k-linear category, O a distinguished object in C, and s is an auto-equivalence of C. The notion of ampleness is defined in this larger context and plays a key role in whether a given triple (C, O, s) arises from a graded algebra A. Indeed, if s is ample, and (C, O, s) satisfies some finiteness conditions, then (C, O, s) ≅ (tails(A), A, [1]) for some left noetherian, locally finite, ℕ-graded algebra A which satisfies χ_{1}. This result gives some idea of the scope of non-commutative algebraic geometry because it says (roughly) which C can be non-commutative schemes. The result may be used to show that some non-commutative algebras behave as if they are commutative from the point of view of tails( ); for example, if A is a twisted homogeneous coordinate ring (see Example 5.16), usually written A = B(X, σ, L), where X is a projective scheme, σ ∈ Aut(X) and L is a σ-ample line bundle on X, then (tails(A), A, [1]) ≅ (Coh(O_{X}), O_{X}, s) for a suitable s (the hypothesis that L is σ-ample guarantees that s is ample). In particular, tails(A) is equivalent to Coh(O_{X}), which allows A to be studied via the methods of algebraic geometry. The utility of this result arises because twisted homogeneous coordinate rings turn up rather often in the theory of non-commutative graded algebras.

1 Graded Algebras and Modules

In all that follows,

• k is a field, and

• A is a ℤ-graded k-algebra; that is A = ⨁_{n∈ℤ} A_{n} and A_{m}A_{n} ⊂ A_{m+n}.

An A-module, M say, is graded if it has a vector space decomposition M = ⨁_{n∈ℤ} M_{n} such that

A_{i}M_{j} ⊂ M_{i+j}

for all i and j. Elements in M_{j} are homogeneous of degree j, and M_{j} is the degree j homogeneous component of M. The graded A-modules are the objects in the category GrMod(A). The full subcategory of GrMod(A) consisting of finitely generated modules is denoted by grmod(A). The morphisms in these categories, denoted Hom_{Gr}(N, M), are the A-module maps f : N → M such that f(N_{i}) ⊂ M_{i} for all i. More generally, if f : N → M satisfies f(N_{i}) ⊂ M_{i+d} for all i ∈ ℤ, we say that the degree of f is d.

We need to consider several other Hom spaces:

• Hom_{A}(N, M) := {all A-module homomorphisms f : N → M};

• Hom_{A}(N, M)_{d} := {f ∈ Hom_{A}(N, M) | deg(f) = d}

• \underline{Hom}_{A}(N, M) := ⨁_{d∈ℤ} Hom_{A}(N, M)_{d}.

Lemma 1.1 If N is finitely generated, then

\underline{Hom}_{A}(N, M) = Hom_{A}(N, M).

Example 1.2 Let V be a graded vector space such that dim_{k} V_{n} ≥ 1 for all n ∈ ℤ. If f : V → k is such that f(V_{n}) ≠ 0 for infinitely many n, then f ∉ \underline{Hom}_{k}(V, k). Thus \underline{Hom}_{k}(V, k) ≠ Hom_{k}(V, k).

The field k itself is a graded algebra concentrated in degree zero. If V is a graded vector space, the graded dual of V is

V^{∗} := \underline{Hom}_{k}(V, k).

Thus V_{−d}^{∗} = Hom_{k}(V_{d}, k). A graded vector space, V say, is locally finite if dim_{k} V_{n} < ∞ for all n. A graded k-algebra generated by a finite number of elements of positive degree is locally finite. Finitely generated modules over a locally finite algebra are locally finite.

We use the notation

M_{≥n} = ⨁_{d≥n} M_{d} and M_{≤n} = ⨁_{d≤n} M_{d}.

We say that M is left (respectively, right) bounded if M_{≤n} = 0 (respectively, M_{≥n} = 0) for some n. From Section 2 onwards our attention is restricted to ℕ-graded algebras. Such an algebra, A say, is left bounded, and so are its finitely generated modules. Further, if M ∈ GrMod(A), so is M_{≥n}.

GrMod(A) is an abelian category and one’s intuition from the category of ungraded modules carries over. A small difference is that A is rarely a generator in GrMod(A); for example Hom_{Gr}(A, M_{≥1}) = 0 for all M. This minor irritation is alleviated by introducing the shift functor [1] : GrMod(A) → GrMod(A) defined as follows: as an A-module M[1] equals M, but the grading is now M[1]_{n} = M_{n+1}. The pair (A, [1]) now acts somewhat like a generator; more precisely P = ⨁_{n∈ℤ} A[n] is a generator. It is an easy but worthwhile exercise to check that \underline{Hom}_{A}(N[i], M[j]) ≅ \underline{Hom}_{A}(N, M)[j − i] as graded vector spaces.

The algebra A is connected if it is ℕ-graded and A_{0} = k. In this case there is a distinguished A-module, namely A/A_{≥0}; it is the only irreducible object in GrMod(A), and is called the trivial module. For connected algebras there is a useful analogue of Nakayama’s Lemma.

Lemma 1.3 Let A be connected. If M ∈ GrMod(A) is left bounded, then M = 0 if and only if k ⊗_{A} M = 0.

Proof. Suppose that M ≠ 0. Since M is bounded below, we can choose 0 ≠ m ∈ M, homogeneous of minimal degree. Such m cannot belong to A_{≥1}M. This is absurd, since k ⊗_{A} M = 0 implies that A_{≥1}M = M, so we conclude that M = 0. ■

Since Hom_{A}(A, −) is exact, so is Hom_{Gr}(A[n], −) for all n ∈ ℤ. Thus A[n] is projective in GrMod(A), whence GrMod(A) has enough projectives. A module M is free if it is a direct sum of shifts of A.

Lemma 1.4 Let A be connected, and M ∈ GrMod(A). If M is bounded below, then

1. M is free if and only if Tor_{1}^{A}(k, M) = 0

2. M is projective if and only if M is free.

Proof. (1) (⇐) Choose a graded vector space V such that V ⊕ A_{≥1}M = M. Then k ⊗_{A} (M/AV) = 0 so, by Nakayama’s Lemma M = AV. Let ψ : A ⊗_{k} V → M be the multiplication map. Since Tor_{1}^{A}(k, M) = 0, there is an exact sequence

0 → k ⊗_{A} ker(ψ) → k ⊗_{A} A ⊗_{k} V ⟶^{1⊗ψ} k ⊗_{A} M → 0.

Since 1 ⊗ ψ is an isomorphism, k ⊗_{A} ker(ψ) = 0. But ker(ψ) is bounded below so, by Nakayama’s Lemma, ψ is an isomorphism.

(2) This follows immediately from (1). ■

The existence of injectives in GrMod(A) is more complicated than the existence of projectives, but we have the following positive result.

Proposition 1.5 GrMod(A) has enough injectives.

As in the category of ungraded modules, E ∈ GrMod(A) is injective if and only if it has no essential extensions. There is an obvious notion of the injective envelope of a module, and it may be characterized as the largest essential extension. Hence we have injective resolutions. If 0 → M → E^{0} →^{d} E^{1} → ⋯ is an injective resolution, we say it is minimal if E^{j} is the injective envelope of dE^{j−1} for all j ≥ 0.

For each q ≥ 0 we may define Ext_{Gr}^{q}(N, −) as the right derived functors of Hom_{Gr}(N, −), and compute these by taking injective resolutions in the usual way; Ext_{Gr}^{q}(N, M) can also be computed by taking projective resolutions of N. We will use the following notation:

Ext_{A}^{q}(N, M) = the usual Ext groups in Mod(A),
Ext_{A}^{q}(N, M)_{d} = the derived functors of Hom_{A}(N, −)_{d},
\underline{Ext}_{A}^{q}(N, M) = ⨁_{n∈ℤ} Ext_{A}^{q}(N, M)_{n}.

Example 1.6 If A is connected, the injective envelope of _{A}k = A/A_{≥1} is A^{∗} = \underline{Hom}_{k}(A, k) with left action of x ∈ A given by (x.λ)(a) = λ(ax) for λ ∈ A^{∗}. The copy of k inside A^{∗} is kϵ, where ϵ : A → k is the projection with kernel A_{≥1}. It is easy to see that A^{∗} is an essential extension of kϵ.

The injectivity of A^{∗} follows from the projectivity of A_{A} as follows. If f : N → M is injective and α : N → A^{∗} are maps in GrMod(A), then β : M → A^{∗} is defined by

β(m)(a) = θ(a)(m),

where θ : A → M^{∗} is a right A-module map such that f^{∗} ∘ θ = α^{∗}, and α^{∗} : A → N^{∗}, f^{∗} : M^{∗} → N^{∗} are the maps dual to α and f. It is easy to check that β is a left A-module map satisfying β ∘ f = α, showing that A^{∗} is injective.

If A, B, C are graded algebras, and _{A}M_{B} and _{A}N_{C} are graded bimodules, then \underline{Ext}_{A}^{q}(N, M) is a graded C-B-bimodule.

The Ext-groups inherit good properties from their second argument.

Proposition 1.7 Let A be left noetherian, and ℕ-graded. If N ∈ grmod(A) and M ∈ GrMod(A), then

1. if M is left (or right) bounded, so is \underline{Ext}_{A}^{q}(N, M);

2. if M is locally finite, so is \underline{Ext}_{A}^{q}(N, M);

3. if M is a graded A-B bimodule, where B is a right noetherian graded algebra, and M ∈ grmod(B), then \underline{Ext}_{A}^{q}(N, M) ∈ grmod(B) too.

Proof. Take a projective resolution for N, each term of which is a finite direct sum of shifts of A. Apply \underline{Hom}_{A}(−, M) to get a complex in which each term is a finite direct sum of shifts of M. Each \underline{Ext}_{A}^{q}(N, M) is a subquotient of these terms, so inherits the relevant property from M. ■

2 Torsion

From now on A is a locally finite, left noetherian, ℕ-graded algebra over a field k. Hence each M ∈ grmod(A) is left bounded and locally finite.

Definition 2.1 The torsion submodule of M ∈ GrMod(A) is

τM := the sum of all finite dimensional submodules of M.

We say that M is torsion (respectively, torsion-free) if τM = M (respectively, τM = 0). We define Tors(A) (respectively, tors(A)) to be the full subcategory of GrMod(A) (respectively, grmod(A)) consisting of the torsion modules.

It follows from the definition that M/τM is torsion-free.

A module M ∈ grmod(A) is torsion if and only if dim_{k} M < ∞ (since M is noetherian, an ascending sum of finite-dimensional submodules stabilizes after finitely many terms). Thus

tors(A) = {finite dimensional modules}.

A useful reformulation of this is that τM = lim_{⟶} \underline{Hom}_{A}(A/A_{≥n}, M).

Proposition 2.2 Tors(A) and tors(A) are dense subcategories of GrMod(A); that is, if 0 → L → M → N → 0 is exact in GrMod(A), then M is torsion if and only if L and N are.

Proof. (⇒) Suppose M is a sum of finite dimensional modules. Then N is the sum of their images, so is torsion. Also, each m ∈ M belongs to a finite sum of finite dimensional modules, so dim_{k}(Am) < ∞, whence every submodule of M is a sum of finite dimensional modules, so is torsion.

(⇐) Suppose L and N are torsion. For m ∈ M, we have an exact sequence

0 → Am ∩ L → Am → Am/Am ∩ L → 0.

By the first part of the proof, Am∩L is torsion since L is, and so is Am/Am∩L since it is isomorphic to Am + L/L, which is a submodule of N. But Am ∩ L and Am/Am ∩ L are noetherian, since A is, whence they are finite dimensional. Thus dim_{k}(Am) < ∞ also. Hence M is a sum of finite dimensional modules, as required. ■

The relation between injective envelopes and torsion is described by the next result.

Lemma 2.3 An essential extension of a torsion (respectively, torsion-free) module is torsion (respectively, torsion-free).

Proof. Let M ⊂ E be an essential extension. If τE ≠ 0, then τE ∩ M = 0, so τM ≠ 0. Thus M torsion-free implies E is too. Conversely, suppose that M = τM. Let e ∈ E. Then Ae ∩ M is torsion, hence finite dimensional since A is noetherian. Thus A_{≥n}e ∩ M = 0 for n ≫ 0, whence dim_{k}(Ae) < ∞, since A is locally finite. Thus E is a sum of finite dimensional modules, hence torsion. ■

3 Tails

Since Tors(A) and tors(A) are dense, there are quotient categories

Tails(A) := GrMod(A)/ Tors(A)
tails(A) := grmod(A)/ tors(A).

We write π : GrMod(A) → Tails(A)

for the quotient functor, and

A = πA.

The objects in a quotient category are the same as those in the original category so, to avoid confusion we will write πM for the image of M ∈ GrMod(A) in Proj(A).

The basic properties of quotient categories may be found in [3].

Theorem 3.1 (Serre) If A is a quotient of the polynomial ring k[X_{0}, …, X_{n}] with deg(X_{i}) = 1 for all i, there is an equivalence of categories

tails(A) ≃ Coh(O_{X}),

the category of coherent O_{X}-modules, where X ⊆ ℙ^{n} is the closed subscheme cut out by the ideal defining A, and O_{X} is the sheaf of regular functions on X.

Thus the objects in tails(A) are the non-commutative analogues of sheaves of O_{X}-modules—they are the objects of non-commutative geometry, and the category tails(A) is the main object of study in non-commutative geometry. To reinforce the analogy with sheaves of O_{X}-modules, we will use script letters to denote objects in Tails(A).

Since [1] sends Tors(A) to Tors(A), the functor [1] passes to Tails(A). Under Serre’s equivalence of categories we have the correspondence

\begin{matrix}A & ↔ & O_{X} \\ A[d] & ↔ & O_{X}(d), \end{matrix}

where O_{X}(d) is the line bundle on X induced from the degree d line bundle on ℙ^{n} (by definition O_{X}(d)(X_{f}) is the degree d component of k[X_{0}, …, X_{n}][f^{−1}], where X_{f} = {p ∈ ℙ^{n} | f(p) ≠ 0}).

A scheme is a pair (X, O_{X}) consisting of a topological space X, and a sheaf of rings O_{X} on X, this data being subject to certain axioms. The space X can be recovered from the pair (Coh(O_{X}), O_{X}), so in a sense the objects of algebraic geometry are pairs (C, O) consisting of a category together with a distinguished object. Hence we make the following definition.

Definition 3.2 The (noetherian) projective scheme associated to a graded algebra A is the pair

proj(A) := (tails(A), A).

The general projective scheme associated to A is

Proj(A) = (Tails(A), A).

The morphisms in Tails(A) can be a little tricky to understand; by definition

Hom_{Tails}(πN, πM) = lim_{⟶} Hom_{Gr}(N′, M/M′)

where the direct limit is taken over all pairs (N′, M′) of submodules of N and M such that N/N′ and M′ ∈ Tors(A), and (N′, M′) ≤ (N′′, M′′) if N′′ ⊆ N′ and M′ ⊆ M′′. The hypotheses on A allow us to simplify this description.

Proposition 3.3 If N ∈ grmod(A) and M ∈ GrMod(A), then

Hom_{Tails}(πN, πM) = lim_{⟶} Hom_{Gr}(N_{≥n}, M).

This direct limit is similar to a union, with the proviso that it is not really a union since the restriction of f : N_{≥n} → M to N_{≥n+1} may be zero, even if f ≠ 0.

The main properties of Tails, and the functor π, which we need are contained in the next few results.

Proposition 3.4 If f : N → M is a morphism in GrMod(A), then

1. ker(πf) = π(ker f) and coker(πf) = π(coker f);

2. πf = 0 if and only if Im(f) is torsion;

3. πf is a monomorphism if and only if ker(f) is torsion;

4. πf is an epimorphism if and only if coker(f) is torsion;

5. πf is an isomorphism if and only if ker(f) and coker(f) are torsion.

Proposition 3.5 1. Tails(A) is an abelian category and π is exact.

2. If D is another abelian category, and F : GrMod(A) → D is an exact functor such that F N = 0 for all N ∈ Tors(A), then there is a unique functor G : Tails(A) → D such that F = G ∘ π.

3. A functor G : Tails(A) → D is exact if and only if G ∘ π is.

We mention two applications of Proposition 3.5. First, π(M/τM) ≅ πM, so given F ∈ Tails(A), F ≅ πN for some torsion free N. Second, since A is ℕ-graded each M_{≥n} is a submodule of M, and πM ≅ πM_{≥n}; we call M_{≥n} a tail of M, and this explains the name of the quotient category—its objects are determined by the tails of A-modules. More precisely, we have the following result.

Proposition 3.6 If M, N ∈ grmod(A), then πM ≅ πN if and only if M_{≥n} ≅ N_{≥n} for some n.

Proof. Suppose that πM ≅ πN. By Proposition 3.3, the isomorphism is given by πf for some f : N_{≥n} → M. Thus ker(f) and coker(f) are torsion, and hence finite dimensional by the noetherian hypotheses. It follows that for r ≫ 0, f : N_{≥r} → M_{≥r} is an isomorphism, as required. The converse is trivial. ■

Theorem 3.7 The functor π has a right adjoint ω : Tails(A) → GrMod(A).

We will make frequent use of the adjoint isomorphism

Hom_{Tails}(πN, F) ≅ Hom_{Gr}(N, ωF).

This implies that ωF is torsion-free since, if N is torsion then πN = 0, which ensures that both the above homomorphism groups are zero.

Proposition 3.8 ωπM ≅ lim_{⟶} \underline{Hom}_{A}(A_{≥n}, M)

Proof. The proof is a “finger exercise”:

ωπM = \underline{Hom}_{A}(A, ωπM)  because _{A}A is finitely generated,
= ⨁_{d∈ℤ} Hom_{Gr}(A, ωπM[d])
= ⨁_{d∈ℤ} Hom_{Tails}(πA, πM[d]) by the adjoint isomorphism,
= ⨁_{d∈ℤ} lim_{⟶} Hom_{Gr}(A_{≥n}, M[d]) by Proposition 3.3,
= lim_{⟶} ⨁_{d∈ℤ} Hom_{Gr}(A_{≥n}, M[d])
= lim_{⟶} \underline{Hom}_{A}(A_{≥n}, M).  ■

Notation. It is convenient to write

\underline{Hom}_{Tails}(F, G) := ⨁_{d∈ℤ} Hom_{Tails}(F, G[d]).

With this notation, the proof of Proposition 3.8 says that ωF ≅ \underline{Hom}(A, F); in fact, there is a natural equivalence of functors

ω ≃ \underline{Hom}(A, −).

We also note that there is a natural map

ρ : A → \underline{Hom}(A, A) = ⨁_{d∈ℤ} Hom(A, A[d])

sending a ∈ A_{d} to πρ_{a}, where ρ_{a} : A → A is right multiplication by a. It is easy to check that ρ is an anti-homomorphism of graded algebras, so each \underline{Hom}(A, F) has a natural left A-module structure. Of course,

\underline{Hom}(A, F) ≅ lim_{⟶} \underline{Hom}_{A}(A_{≥n}, ωF)

already has a natural left A-module structure coming from the right action of A on A_{≥n}. These two actions of A on \underline{Hom}(A, F) coincide.

Although Proposition 3.8 gives an explicit description of ω, its existence and basic properties are usually established by defining ω as follows. Given M ∈ GrMod(A), let E denote the injective envelope of \overline{M} = M/τM. Then ωπM is defined to be the largest graded submodule, H say, of E such that \overline{M} ⊂ H and H/\overline{M} is torsion. Thus H/\overline{M} = τ(E/\overline{M}), and there is an exact sequence

0 → τM → M → ωπM → torsion → 0;

the last term in this sequence will be described in Proposition 4.6.

Example 3.9 Let A = k[x]. One can check directly that E = k[x, x^{−1}] is an injective A-module, and hence is the injective envelope of A in GrMod(A). (Notice this shows that, in contrast to projectives, injectives in GrMod(A) need not be injective in Mod(A).) Since E/A is torsion it follows that ωπA ≅ E. (We will see later that for the polynomial ring in ≥ 2 variables, ωπA ≅ A.) In particular, ωπA is not a finitely generated A-module.

The following result is crucial.

Proposition 3.10 π ∘ ω ≃ Id.

Proof. We must show that the natural map πωF → F is an isomorphism for all F ∈ Tails(A). By Yoneda’s lemma, it is enough to show that the map

Hom_{Tails}(πN, πωF) → Hom_{Tails}(πN, F)

is an isomorphism for all N ∈ GrMod(A); in fact, it suffices to do this for finitely generated N, by writing an arbitrary module as a direct limit of finitely generated ones. The map in question is the horizontal map in the following diagram

[figure: lim_{⟶} Hom_{Gr}(N_{≥n}, ωF) ≅ Hom_{Tails}(πN, πωF); Hom_{Tails}(πN, F); Hom_{Gr}(N, ωF); adjoint isomorphism; π]

where the isomorphism on the left is a consequence of Proposition 3.3 and the torsion-freeness of ωF. It suffices to show that the vertical map, which is π on morphisms, is an isomorphism. The functoriality of the adjoint isomorphism yields a commutative diagram

Hom_{Gr}(N, ωF) ⟶^{∼} Hom_{Tails}(πN, F)
↓  ↓
Hom_{Gr}(N_{≥n}, ωF) ⟶^{∼} Hom_{Tails}(πN_{≥n}, F)

where the vertical maps are restriction, and the horizontal maps are the adjoint isomorphisms. Since the inclusion map N_{≥n} → N induces an isomorphism πN_{≥n} → πN the right hand vertical map is an isomorphism, hence so is the left hand one. Hence all maps in lim_{⟶} Hom_{Gr}(N_{≥n}, ωF) are isomorphisms, so

lim_{⟶} Hom_{Gr}(N_{≥n}, ωF) ≅ Hom_{Gr}(N, ωF),

which is precisely what we required. ■

4 Cohomology

We will define the cohomology groups H^{q}(F) for F ∈ Tails(A), and prove a version of Serre’s Finiteness Theorem. An essential preliminary step is to understand injectives in Tails(A).

Proposition 4.1 1. Tails(A) has enough injectives.

2. If Q ∈ Tails(A) is injective, then ωQ is a torsion-free injective.

3. If Q ∈ GrMod(A) is torsion-free injective, then πQ is injective and Q ≅ ωπQ.

Proof. (2) The adjoint isomorphism gives the natural equivalence

Hom_{Tails}(−, Q) ∘ π ≃ Hom_{Gr}(−, ωQ).

The left hand side is a composition of exact functors, so we conclude that ωQ is injective. It is torsion-free since ωF is always torsion-free.

(3) By definition of ω, Q ≅ ωπQ, so

Hom_{Tails}(πN, πQ) ≅ Hom_{Gr}(N, ωπQ) ≅ Hom_{Gr}(N, Q)

for all N ∈ GrMod(A). That is

Hom_{Tails}(−, πQ) ∘ π ≃ Hom_{Gr}(−, Q).

By hypothesis the right hand side is exact, hence so is Hom_{Tails}(−, πQ) by Proposition 3.5. Thus πQ is injective.

(1) Let F ∈ Tails(A), and let f : ωF → Q be the inclusion of ωF in its injective envelope. Since ωF is torsion-free, so is Q. Thus πQ is injective by (3). Also ker(πf) = π(ker f), so πf : πωF ≃ F → πQ is injective, which shows F embeds in an injective, as required. ■

Lemma 4.2 1. Each injective in GrMod(A) decomposes as a direct sum of a torsion injective and a torsion-free injective.

2. If A is connected, then every torsion injective is a direct sum of shifts of A^{∗} = \underline{Hom}_{k}(A, k).

Proof. (1) Let E an injective. Being injective it contains a copy of the injective envelope of τE, say I. Since I is injective, E = I ⊕ Q for some other submodule Q; being a summand of an injective, Q is also injective, and torsion-free since τE ⊂ I. Finally, by Lemma 2.3, I is torsion.

(2) Let I be a torsion injective in GrMod(A). If 0 ≠ M ∈ Tors(A), then \underline{Hom}_{A}(k, M) ≠ 0. We may consider S = \underline{Hom}_{A}(k, I) as a submodule of I; it is a (possibly infinite) direct sum of shifts of_{A}k. If M is a non-zero submodule of I then, since M is torsion, Hom_{A}(k, M) ≠ 0, whence M ∩ S ≠ 0, so S is essential in I; thus I = E(S). Since A is left noetherian, a direct sum of injective modules is injective, whence E(S) is a (possibly infinite) direct sum of shifts of E(_{A}k) ≅ A^{∗}. ■

If F ∈ Tails(A), then Hom_{Tails}(F, −) is left exact, so we may define its right derived functors, and compute them via injective resolutions. That is, if G → E^{•} is an injective resolution in Tails(A), then

Ext^{q}(F, G) := h^{q}(Hom_{Tails}(F, E^{•})),

the q^{th} homology group of the complex. We also define

\underline{Ext}^{q}(F, G) := ⨁_{d∈ℤ} Ext^{q}(F, G[d]).

These Ext groups are k-vector spaces. We will show that they can be computed in terms of Ext groups in GrMod(A) by using ω.

Proposition 4.3 Let N ∈ grmod(A) and M ∈ GrMod(A). Let E^{•}M be a minimal injective resolution of M, and write E^{•}M = I^{•}M ⊕ Q^{•}M, where I^{•}M is the torsion part of E^{•}M (it is a subcomplex) and Q^{•}M is a torsionfree complement. Write N = πN and M = πM. Then

1. Ext^{q}(N , M) = h^{q}(Hom_{Gr}(N, Q^{•}M))

2. \underline{Ext}^{q}(N , M) ≅ lim_{⟶} \underline{Ext}_{A}^{q}(N_{≥n}, M)

Proof. (1) Although Q^{•}M is not usually a subcomplex of E^{•}M, we may identify it with the complex E^{•}M/I^{•}M. The exactness of π implies that M → πE^{•} ≃ πQ^{•} is an injective resolution of M in Tails(A). But

Hom(N , πQ^{•}) ≅ Hom_{Gr}(N, ωπQ^{•} ≅ Q^{•}),

so the result follows.

(2) First observe that lim_{⟶} \underline{Hom}_{A}(N_{≥n}, I^{•}) = 0: if f : N_{≥n} → I^{•}, then N_{≥n}/ ker f is finite dimensional because I^{•} is torsion and N is noetherian, whence N_{≥r} ⊆ ker f for r ≫ 0, which implies that in the direct limit f becomes zero. Therefore

lim_{⟶} \underline{Ext}_{A}^{q}(N_{≥n}, M) = lim_{⟶} h^{q}(\underline{Hom}_{A}(N_{≥n}, I^{•} ⊕ Q^{•}))
= h^{q}(lim_{⟶} \underline{Hom}_{A}(N_{≥n}, Q^{•}))
≅ h^{q}(\underline{Hom}(N , πQ^{•}))
= \underline{Ext}^{q}(N , M),

as required. ■

Definition 4.4 For F ∈ Tails(A) we define the cohomology groups

H^{q}(F) := Ext^{q}(A, F)

and the cohomology modules

\underline{H}^{q}(F) := \underline{Ext}^{q}(A, F),

which are graded by

\underline{H}^{q}(F)_{d} := \underline{Ext}^{q}(A, F[d]).

We have already observed that \underline{Hom}(A, −) ≃ ω, so the \underline{H}^{q}(−) are the right derived functors of ω. In particular, if 0 → F′ → F → F′′ → 0 is exact, there is a long exact cohomology sequence

0 → H^{0}(F′) → H^{0}(F) → H^{0}(F′′) → H′(F′) → H′(F) → · · · .

The Čech cohomology groups H^{q}(X, −), defined for O_{X}-modules, are the derived functors of the global section functor Γ(X, −). But Γ(X, F) ≅ Hom_{O_{X}}(O_{X}, F), so H^{q}(X, −) are the derived functors of Hom(O_{X}, −). Hence by Serre’s equivalence of categories (Theorem 3.1), this definition of cohomology reduces to the classical one for projective schemes.

The following result is mostly a specialization of earlier results.

Proposition 4.5 Let M ∈ GrMod(A), and write M = πM. Then

1. \underline{H}^{0}(M) ≅ ωπM;

2. \underline{H}^{q}(M) ≅ lim_{⟶} \underline{Ext}_{A}^{q}(A_{≥n}, M);

3. \underline{H}^{q}(M) ≅ lim_{⟶} \underline{Ext}_{A}^{q+1}(A/A_{≥n}, M) for q ≥ 1;

4. \underline{H}^{q}(M) ≅ h^{q+1}(I^{•}M) for q ≥ 1, where I^{•}M is the torsion part of the injective resolution of M.

Proof. (1) and (2) follow from the previous result and Proposition 3.8. (3) For q ≥ 1, the long exact sequence for \underline{Ext}_{A}(−, M) gives

\underline{Ext}_{A}^{q}(A_{≥n}, M) ≅ \underline{Ext}_{A}^{q+1}(A/A_{≥n}, M)

since A is projective, so (3) follows from (2).

(4) Consider the exact sequence of complexes

0 → I^{•}M → E^{•}M → Q^{•}M → 0.

Since Q^{•}M is torsion free, and A/A_{≥n} is torsion, there is an isomorphism of complexes

\underline{Hom}_{A}(A/A_{≥n}, I^{•}M) ≅ \underline{Hom}_{A}(A/A_{≥n}, E^{•}M).

Taking direct limits and homology yields

h^{q+1}(lim_{⟶} \underline{Hom}_{A}(A/A_{≥n}, I^{•}M)) ≅ \underline{H}^{q}(M).

But I^{•}M is torsion, so the sum of its finite dimensional submodules, whence

lim_{⟶} \underline{Hom}_{A}(A/A_{≥n}, I^{•}M) ≅ I^{•}M. ■

Each \underline{H}^{q}(M) has a natural left A-module structure arising from the right action of A on A_{≥n} (in Proposition 4.5(2) say). In fact, \underline{H}^{q}(M) becomes a graded left A-module with degree d component being H^{q}(M[d]).

Although lim_{⟶} \underline{Ext}_{A}^{1}(A/A_{≥n}, M) does not appear in the statement of the previous Proposition, it is an important object as the next, and later, results show.

Proposition 4.6 For each M ∈ GrMod(A), there is an exact sequence

0 → τM → M → ωπM → lim_{⟶} \underline{Ext}_{A}^{1}(A/A_{≥n}, M) → 0.

Proof. Over directed sets lim_{⟶} is an exact functor, so taking direct limits of the exact sequences

0 → \underline{Hom}_{A}(A/A_{≥n}, M) → \underline{Hom}_{A}(A, M) → \underline{Hom}_{A}(A_{≥n}, M) → \underline{Ext}_{A}^{1}(A/A_{≥n}, M) → 0

yields the result, because lim_{⟶} \underline{Hom}_{A}(A/A_{≥n}, M) = τM. ■

After defining sheaf cohomology, one of the first exercises is to compute the cohomology groups of line bundles on ℙ^{n}, i.e. H^{q}(ℙ^{n}, O(d)). We will now carry out a slight generalization of this. The non-commutative algebras in the next definition are good analogues of polynomial rings.

Definition 4.7 A locally finite connected k-algebra, A say, is Artin-Schelter regular of dimension n + 1 if

• gl . dim(A) = n + 1 < ∞,

• GK dim(A) < ∞, and

• Ext_{A}^{i}(_{A}k, A) = \{ \begin{matrix}0 & if i ≠ n + 1 \\ k & if i = n + 1 \end{matrix} , i.e. A is Gorenstein.

Polynomial rings, and more generally iterated Ore extensions

k[X_{0}][X_{1}; σ_{1}, δ_{1}] ⋯ [X_{n}; σ_{n}, δ_{n}]

where each σ_{i} is an automorphism and deg(X_{i}) = 1 for all i, are Artin-Schelter regular; so too are the Sklyanin algebras.

Example 4.8 Let A be Artin-Schelter regular of dimension n + 1. We compute H^{q}(A[d]) for d ∈ ℤ. For simplicity suppose that n + 1 ≥ 2.

First we show that \underline{H}^{0}(A) = ωπA. The Gorenstein property ensures that \underline{Hom}_{A}(A/A_{≥1}, M) = 0, whence τA = 0. Also, \underline{Ext}_{A}^{1}(A/A_{≥1}, M) = 0 by the Gorenstein property, whence \underline{Ext}_{A}^{1}(A/A_{≥n}, M) = 0 for all n (by induction). Hence by Proposition 4.6, A ≅ ωπA. That is,

\underline{H}^{0}(A) = A  and  H^{0}(A[d]) = A_{d}.

Now suppose that q ≥ 1. Since \underline{Ext}_{A}^{n+1}(k, A) ≅ k[l], the trivial right Amodule shifted by some integer l, it follows that for any finite dimensional A-module T, \underline{Ext}_{A}^{n+1}(T, A) ≅ T^{∗}[l]; one argues by induction on the length of T, the case of a shift of k being obviously true. Hence

\underline{H}^{q}(A) = lim_{⟶} \underline{Ext}_{A}^{q+1}(A/A_{≥n}, A)
= lim_{⟶} \{ \begin{matrix}0 & q ≠ n, \\ (A/A_{≥n})^{∗}[l] & q = n. \end{matrix}
= \{ \begin{matrix}0 & q ≠ n \\ A^{∗}[l] & q = n. \end{matrix}

Thus H^{n}(A[d]) = (A^{∗})_{l+d} = (A_{−l−d})^{∗}.

When A is a polynomial ring on n+1 generators the Koszul complex gives a linear resolution of the trivial module _{A}k, so l = n + 1, whence we recover the usual result for H^{q}(ℙ^{n}, O(d)).

Before proving Serre’s Finiteness Theorem, we need some technical results.

Lemma 4.9 Write [l, r] = {T ∈ GrMod(A) | T_{<l} = T_{>r} = 0}. If \underline{Ext}_{A}^{j}(A/A_{≥1}, M) ∈ [l′, r′] for all j ≤ i, and T ∈ [l, r], then

\underline{Ext}_{A}^{j}(T, M) ∈ [l′ − r, r′ − l]

for all j < i.

Proof. By induction on r − l, we reduce to r − l = 1, in which case T is a direct sum of shifts of A/A_{≥1}; the lemma is easy for such T. ■

Proposition 4.10 Let M ∈ grmod(A) and fix i ≥ 0. The following are equivalent:

1. for all j ≤ i, \underline{Ext}_{A}^{j}(A/A_{≥1}, M) is finite dimensional;

2. for all j ≤ i, \underline{Ext}_{A}^{j}(A/A_{≥n}, M) is finite dimensional for all n;

3. for all j ≤ i and all N ∈ grmod(A), \underline{Ext}_{A}^{j}(N/N_{≥n}, M) has a right bound independent of n;

4. for all j ≤ i and all N ∈ grmod(A), lim_{⟶} \underline{Ext}_{A}^{j}(N/N_{≥n}, M) is right bounded.

Proof. First, by Proposition 1.7, if T ∈ grmod(A), \underline{Ext}_{A}^{q}(T, M) is a subquotient of a finite direct sum of shifts of M, so is left bounded and locally finite.

We will prove the result by induction on i. For i = 0, (1)–(4) all hold because dim_{k} T < ∞ implies that \underline{Hom}_{A}(T, M) ⊆ \underline{Hom}_{A}(T, τM) which is finite dimensional since dim_{k}(τM) < ∞; notice that (4) holds because lim_{⟶} \underline{Hom}_{A}(A/A_{≥n}, M) = τM. So suppose the Proposition is true for i − 1; i.e., the four conditions are equivalent.

(1) ⇔ (2) If (1) holds, the previous lemma implies that \underline{Ext}_{A}^{j}(A/A_{≥n}, M) is bounded, and hence finite dimensional by the first paragraph; thus (2) holds. The converse is a tautology.

(1) ⇒ (3) The exact sequence 0 → T → N/N_{≥n+1} → N/N_{≥n} → 0 yields an exact sequence.

\underline{Ext}_{A}^{j−1}(T, M) → \underline{Ext}_{A}^{j}(N/N_{≥n}, M) → \underline{Ext}_{A}^{j}(N/N_{≥n+1}, M) → \underline{Ext}_{A}^{j}(T, M).

But T ∈ [n, n], so by Lemma 4.9, the first and last terms are bounded, and their right bounds approaches −∞ as n → ∞. Hence, given d ∈ ℤ, there is a natural isomorphism

\underline{Ext}_{A}^{j}(N/N_{≥n}, M)_{≥d} →^{∼} \underline{Ext}_{A}^{j}(N/N_{≥n+1}, M)_{≥d}  (1)

for n ≫ 0. By Lemma 4.9, these are right bounded, so have a right bound which is independent of n.

(3) ⇒ (4) This is immediate.

(4) ⇒ (1) Consider the exact sequence

\underline{Ext}_{A}^{i−1}(A_{≥1}/A_{≥n}, M) → \underline{Ext}_{A}^{i}(A/A_{≥1}, M) → \underline{Ext}_{A}^{i}(A/A_{≥n}, M).

By hypothesis the direct limit of the last term is right bounded. Since (4) holds for i, and hence for i − 1, the direct limit of the first term is right bounded. Hence so is the direct limit of the middle term. But that is simply \underline{Ext}_{A}^{i}(A/A_{≥1}, M), which we already know is left bounded and locally finite, whence it is finite dimensional. Thus (1) is true. ■

Definition 4.11 Let M ∈ grmod(A). We say that

• χ_{i}(M) holds if the equivalent conditions of Proposition 4.10 hold;

• χ(M) holds if χ_{i}(M) holds for all i;

• A satisfies χ if χ(M) holds for all M ∈ grmod(A).

Proposition 4.12 If M ∈ grmod(A), the following are equivalent:

1. χ_{1}(M) holds;

2. coker(M → ωπM) is right bounded;

3. (ωπM)_{≥d} is finitely generated for all d ∈ ℤ.

Proof. We will use the exact sequence

0 → τM → M → ωπM → lim_{⟶} \underline{Ext}_{A}^{1}(A/A_{≥n}, M) → 0.

The equivalence of (1) and (2) is a restatement of the equivalence of (1) and (4) in Proposition 4.10, noting that the proof of (4) implies (1) only used the truth of (4) for N = A.

(1) ⇒ (3). Fix d ∈ ℤ, and consider

M_{≥d} → (ωπM)_{≥d} → lim_{⟶} \underline{Ext}_{A}^{1}(A/A_{≥n}, M)_{≥d} → 0.  (2)

By hypothesis the first term is finitely generated. Since χ_{1}(M) holds, part (3) of Proposition 4.10 ensures that the last term of (2) is right bounded and hence finite dimensional. It follows that (ωπM)_{≥d} is finitely generated too.

(3) ⇒ (1). The hypothesis ensures that the last term of (2) is finitely generated, but it is also torsion, hence finite dimensional. Therefore part (4) of Proposition 4.10 holds for i = 1 (with N = A) and, as noted, this ensures that part (1) of Proposition 4.10 holds too; i.e., χ_{1}(M) holds. ■

Rephrasing part (2) of Proposition 4.12, if A satisfies χ_{1}, then ωπM is finitely generated up to torsion whenever M ∈ grmod(A) (Example 3.9 showed that ωπM is not generally finitely generated). Part (3) of Proposition 4.12 says that if ωπM is considered as a rather nice module with respect to torsion, then M is not too far from being nice—at least M_{≥d} ≅ (ωπM)_{≥d} for d ≫ 0.

The condition χ is a non-commutative phenomenon. The next two results show that quotients of polynomial rings satisfy it, and the example which follows these positive results shows a rather nice non-commutative algebra which does not satisfy χ_{1}.

Theorem 4.13 Noetherian Artin-Schelter regular algebras satisfy χ.

Proof. Let A be such an algebra, and M ∈ grmod(A). We proceed by induction on pdim(M). If pdim(M) = 0, then A is a finite direct sum of shifts of A; but \underline{Ext}_{A}^{j}(A/A_{≥1}, A) is finite dimensional by the Gorenstein hypothesis, so χ_{1}(M) holds. If pdim(M) > 0, write 0 → K → P → M → 0 with P projective, and pdim(K) = pdim(M) − 1. By the induction hypothesis, the last term of the exact sequence

\underline{Ext}_{A}^{j}(k, P) → \underline{Ext}_{A}^{j}(k, M) → \underline{Ext}_{A}^{j+1}(k, K)

is finite dimensional, as is the first term, whence so is the middle term. ■

Proposition 4.14 If A is noetherian and satisfies χ_{i}, so does A/I for all ideals I.

Proof. Write B = A/I and let M ∈ grmod(B). We will proceed by induction on i; since B satisfies χ_{0}, we will assume the result is true for i−1. Thus B satisfies χ_{i−1}, and we must show B satisfies χ_{i}.

Consider the spectral sequence

E_{2}^{pq} = \underline{Ext}_{B}^{P}(Tor_{q}^{A}(B, A/A_{≥n}), M) ⇒ \underline{Ext}_{A}^{p+q}(A/A_{≥n}, M).

Since A is noetherian, each term in the minimal projective resolution of B_{A} is a finite direct sum of shifts of A, whence each Tor_{q}^{A}(B, A/A_{≥n}) is finite dimensional. In particular, it is right bounded. Since A is projective,

Tor_{q}^{A}(B, A/A_{≥n}) ≅ Tor_{q−1}^{A}(B, A_{≥n})

for q ≥ 2, and

Tor_{1}^{A}(B, A/A_{≥n}) ⊆ B ⊗_{A} A_{≥n}.

By taking a minimal resolution of A_{≥n}, it is easy to see that

Tor_{q−1}^{A}(B, A_{≥n}) ∈ [n, ∞)

for all q ≥ 1, whence

Tor_{q}^{A}(B, A/A_{≥n}) ∈ [n, ∞)

for all q ≥ 1. Since B satisfies χ_{i−1}, Lemma 4.9 with T = Tor_{q}^{A}(B, A/A_{≥n}) implies that, for all p ≤ i−1, the right bound of E_{2}^{pq} tends to −∞ as n → ∞. Thus, given d ∈ ℤ, p ≤ i − 1, and q ≥ 1,

(E_{2}^{pq})_{≥d} = 0

for n ≫ 0. Hence, for all p ≤ i,

(E_{2}^{p0})_{≥d} ≅ \underline{Ext}_{A}^{p}(A/A_{≥n}, M)_{≥d}

for all n ≫ 0. That is, for all p ≤ i, and all n ≫ 0,

\underline{Ext}_{B}^{p}(B/B_{≥n}, M)_{≥d} ≅ \underline{Ext}_{A}^{p}(A/A_{≥n}, M)_{≥d}.

But A satisfies χ_{i}, so the condition in part (3) of Proposition 4.10 implies that B satisfies χ_{i} too. ■

Example 4.15 Fix 0 ≠ q ∈ k, and suppose that q is not a root of unity. Let B = k[x, y], with defining relation xy − qyx = y^{2}. (It is easy to show that B ≅ k[u, v] with relation vu = quv.) Define A = k + xB.

It is standard that B is (right and left) noetherian, and not too difficult to deduce from that that A is also noetherian. As a right A-module, B is finitely generated, namely B = A + yA. In contrast, as a left A-module, B is not finitely generated: indeed, as a left A-module,

B/A ≅ k[−1] ⊕ k[−2] ⊕ ⋯

is an infinite direct sum of shifts of the trivial A-module _{A}k = A/A_{≥1}. To see this, simply observe that B/A has a basis given by the images of {y^{i}|i ≥ 1}, and that A_{≥1}y = xBy ⊆ A. Since A is a domain, τA = 0, whence A ⊆ ωπA. Since Fract(A) = Fract(B), B is an essential extension of A; since _{A}(B/A) is torsion, it follows from the definition of ω that A ⊂ B ⊂ ωπA. Thus coker(A → ωπA) is not right bounded, so χ_{1}(A) does not hold. Alternatively, one can see from the description of B/A that \underline{Ext}_{A}^{1}(A/A_{≥1}, A) is not finite dimensional.

Theorem 4.16 (Serre’s Finiteness Theorem). Let F ∈ tails(A). If A satisfies χ, then

1. dim_{k} H^{q}(F) < ∞ for all q, and

2. if q ≥ 1, then H^{q}(F[n]) = 0 for n ≫ 0.

Conversely, if A satisfies χ_{1}, and (2) holds for all F ∈ tails(A), then A satisfies χ.

Proof. Write F = πM where M ∈ grmod(A).

Suppose that q = 0. Since χ_{1}(M) holds, (ωπM)_{≥0} is finitely generated, hence locally finite. In particular, (ωπM)_{0} = H^{0}(F) is finite dimensional.

Suppose that q ≥ 1. Since A satisfies χ_{q+1},

lim_{⟶} \underline{Ext}_{A}^{q+1}(A/A_{≥n}, M)

is right bounded; but this equals \underline{H}^{q}(F), so (2) follows because \underline{H}^{q}(F)_{n} = H^{q}(F[n]). The proof of Proposition 4.10 showed that, given d ∈ ℤ,

lim_{⟶} \underline{Ext}_{A}^{q+1}(A/A_{≥n}, M)_{≥d} ≅ \underline{Ext}_{A}^{q+1}(A/A_{≥r}, M)_{≥d}

for r ≫ 0; in particular, this is locally finite, which proves (1) for q ≥ 1.

Conversely, (2) implies that \underline{H}^{i−1}(F) is right bounded for i ≥ 2, but this is isomorphic to lim_{⟶} \underline{Ext}_{A}^{i}(A/A_{≥n}, M); thus, since χ_{1}(M) holds, condition (4) in Proposition 4.10 is satisfied for all i. Thus A satisfies χ. ■

5 Non-commutative Schemes

Let A be a left noetherian, locally finite, ℕ-graded k-algebra. We already defined the projective scheme associated to A as the pair proj(A) = (tails(A), A). Such pairs are the objects in the category Pairs: the objects are pairs (C, O) consisting of a k-linear abelian category C, together with a distinguished object O; the morphisms are pairs

(f, θ) : (C_{1}, O_{1}) → (C_{2}, O_{2})

consisting of a covariant k-linear functor f : C_{1} → C_{2} and a morphism θ : fO_{1} → O_{2}.

Definition 5.1 A map F : proj(B) → proj(A) of schemes is a natural equivalence class of morphisms

(f, θ) : (tails(A), A) → (tails(B), B)

such that f is right exact, and θ : fA → B is an isomorphism.

There is a similar notion of map between general projective schemes Proj(A) = (Tails(A), A).

Warning: The map F goes in the opposite direction to the functor f. We have deliberately not defined a category of schemes—possibly the notion of map is too restrictive, and other morphisms should be permitted; in any case, whatever the appropriate definition should be, the maps above should be allowed.

Given a homomorphism f : A → B of graded algebras, we have the induction and restriction functors

f^{∗} : GrMod(A) → GrMod(B),
f_{∗} : GrMod(B) → GrMod(A),

defined by

\begin{aligned}f^{∗}M & = & B ⨂_{A} M, and \\ f_{∗}N & = & _{A}N. \end{aligned}

These are an adjoint pair:

Hom(f^{∗}M, N) ≅ Hom(M, f_{∗}N).

Proposition 5.2 If f : A → B is a homomorphism of graded algebras, there are induced functors

1. f^{∗} : tails(A) → tails(B) which is exact;

2. f_{∗} : tails(B) → tails(A) if _{A}B is finitely generated up to torsion (i.e., πB ∈ tails(A));

3. f^{∗} : Tails(A) → Tails(B) and f^{∗} : tails(A) → tails(B) if either B_{A} is finitely generated, or coker(f) is right bounded; in this case, we obtain a map proj(B) → proj(A).

Proof. The existence of f^{∗} or f_{∗} at the level of Tails or tails is proved by checking that induction or restriction of a torsion module is again torsion. The details are straightforward. ■

If f : A → A/I is the natural map to a quotient of A, then f_{∗} : Tails(A/I) → Tails(A) is fully faithful, and we think of the induced map proj(A/I) → proj(A) as being a closed embedding. We usually identify proj(A/I) with its image in proj(A).

If u is a homogeneous regular normalizing element of A and g : A → A[u^{−1}] is the natural map, then g_{∗} : Tails(A[u^{−1}]) → Tails(A) is fully faithful, and we should think of proj(A[u^{−1}]_{0}) as being the (open) complement to proj(A/(u)) in proj(A). Suppose that u is not in A_{0}. Then A[u^{−1}] cannot have any torsion modules (because there is a unit of positive degree), so tails(A[u^{−1}]) ≃ grmod(A[u^{−1}]). If u is of degree one, or if A is generated by A_{0} and A_{1}, then A[u^{−1}] is a strongly graded algebra, meaning that the product of the degree i and j components equals the degree i + j component, and therefore has the property that grmod(A[u^{−1}]) ≃ mod(A[u^{−1}]_{0}), the equivalence being given by M ↦ M_{0}. Thus the open complement to proj(A/(u)) is the ‘affine scheme’ mod(A[u^{−1}]_{0}).

Although we have restricted our attention to ℕ-graded algebras in these talks, the main ideas extend to ℤ-graded algebras. In particular, there is a ℤgraded version of Proposition 5.2. In particular, we have the next Proposition which establishes an equivalence of categories Tails(A_{≥0}) ≃ Tails(A). Thus, as far as projective schemes are concerned, we can replace A by the ℕ-graded algebra A_{≥0}; it is for this reason that our restriction to ℕ-graded algebras is reasonable.

Proposition 5.3 If f : A → B is a homomorphism of graded algebras such that ker(f) is torsion and coker(f) is right bounded, then f^{∗} and f_{∗} induce equivalences Tails(A) ≃ Tails(B) and tails(A) ≃ tails(B). In particular, proj(A) ≃ proj(B).

We omit the proof of the next two results, which may be found in [2] and [5] respectively.

Proposition 5.4 If A is left noetherian, and generated over A_{0} by A_{1}, then proj(A) ≅ proj(A^{(d)}), where A^{(d)} = ⊕_{n∈ℤ}A_{nd} is the d^{th} Veronese subalgebra of A, with grading defined by A_{n}^{(d)} = A_{nd}.

Proposition 5.5 Let A and B be ℕ-graded k-algebras, generated over A_{0} by A_{1}. Define their Segre product

A ∘ B = ⨁_{n∈ℤ} A_{n} ⊗_{k} B_{n}

with grading (A ∘ B)_{n} = A_{n} ⊗ B_{n}, and multiplication inherited from that on A ⊗_{k} B. Then there are maps

proj(A ∘ B) → proj(A)  and  proj(A ∘ B) → proj(B).

The maps in the previous proposition are not induced by algebra homomorphisms since the natural map A → A⊗_{k} B does not have image in A∘B.

Twisting. We now describe an important construction which gives rise to a map of schemes which does not arise from an algebra homomorphism. In particular, it shows that there may be a wide range of algebras having the same scheme associated to them—for example, a non-commutative algebra may determine the same scheme as a commutative algebra.

If σ is a graded algebra automorphism, σ ∈ Aut_{k}(A), then the twisted algebra ^{σ}A is ^{σ}A = A as a graded vector space, but with multiplication

a⊙b = a^{σ^{n}}b

if a ∈ A_{m}, b ∈ A_{n}.

Proposition 5.6 There is an isomorphism

proj(A) ≅ proj(^{σ}A).

Proof. In fact there is an equivalence of categories

Φ : GrMod(A) → GrMod(^{σ}A)

sending A to ^{σ}A which is defined as follows. if M ∈ GrMod(A), then ΦM is the ^{σ}A-module defined by ΦM = M as a graded vector space, and

a⊙m = a^{σ^{n}}m

if a ∈ ^{σ}A_{i}, m ∈ M_{n}. The details are easy to check (see [7]). ■

Notation We usually write ^{σ}M for the ^{σ}A-module ΦM defined in the proof of Proposition 5.6.

Example 5.7 Let A = k[x, y] be the commutative polynomial ring , and σ the automorphism defined by x^{σ} = x and y^{σ} = qy where q is some fixed non-zero scalar. Then ^{σ}A = k[u, v] with defining relation vu = quv. Here proj(^{σ}A) ≅ Coh(ℙ^{1}).

Continuing with this idea, if A = k[x, y, z] with defining relations zy = αyz, xz = βzx, yx = γxy then proj(A) contains three copies of ℙ^{1}, namely the three coordinate axes proj(A/(x)), proj(A/(y)) and proj(A/(z)).

In the proof of Proposition 5.6, we usually have

(^{σ}M)[1] ≅ ^{σ̸}(M[1]);

the [1] on the left is the shift functor for ^{σ}A, whereas the [1] on the right is the shift functor for A. To see this let a ∈ ^{σ}A_{i} and m ∈ M_{j+1}. If we consider m ∈ (^{σ}M)[1]_{j}, then a⊙m = a^{σ^{j+1}}m; on the other hand, if we consider m ∈ ^{σ}(M[1])_{j}, then a⊙m = a^{σ^{j}}m. The relation between the shift functors for ^{σ}A and A is described by the next lemma.

Lemma 5.8 Let Φ : GrMod(A) → GrMod(^{σ}A) be the functor in Proposition 5.6. Let σ_{∗} : GrMod(A) → GrMod(A) be the restriction functor arising from the algebra homomorphism σ : A → A. There is a natural equivalence

Φ ∘ σ_{∗} ∘ [1] ≃ [1] ∘ Φ.

Proof. (Probably this should be done behind closed doors.) Let M ∈ GrMod(A); then σ_{∗}M = M as a graded vector space, and the A-action on σ_{∗}M is given by

a · m = a^{σ}m.

for a ∈ A and m ∈ σ_{∗}M. As vector spaces, both (Φ∘σ_{∗}∘[1])M and ([1]∘Φ)M equal M; let

ρ : (Φ ∘ σ_{∗} ∘ [1])M → ([1] ∘ Φ)M

be the identity map. We will show that ρ is an isomorphism of ^{σ}A-modules. First the degree j component of (Φ ∘ σ_{∗} ∘ [1])M equals σ_{∗}(M[1])_{j} = M_{j+1}, as does the degree j component of ([1] ∘ Φ)M. Thus ρ is a graded vector space map. If m ∈ ^{σ}(σ_{∗}M[1])_{j}, then

a⊙m = a^{σ^{j}} · m = a^{σ^{j+1}}m;

on the other hand, ρ(m) ∈ (^{σ}M)[1]_{j} = (^{σ}M)_{j+1}, so

a⊙ρ(m) = a^{σ^{j+1}}m.

That is, ρ(a⊙m) = a⊙ρ(m), as required. ■

Remark 5.9 The twist (^{σ}A, ⊙) defined prior to Proposition 5.6 should really be called the left twist of A. We may also define the right twist (A^{σ}, ∗); as a graded vector space A^{σ} = A, and it is endowed with multiplication

a ∗ b = ab^{σ^{m}}

for a ∈ A_{m}, b ∈ A_{n}. It is an easy exercise to show that the map θ : ^{σ}A → A^{σ^{−1}} defined by θ(a) = σ^{−i}(a) for a ∈ A_{i} is a graded algebra isomorphism.

Definition 5.10 Given a graded algebra A, we call

(tails(A), A, [1]) or (Tails(A), A, [1])

the polarized projective scheme associated to A.

These are objects of the following category. Definition 5.11 The category of triples, Trip, has as its objects triples (C, O, s) where

• C is a k-linear abelian category,

• O is a distinguished object of C, and

• s : C → C is an auto-equivalence, and morphisms

(f, θ, μ) : (C_{1}, O_{1}, s_{1}) → (C_{2}, O_{2}, s_{2}),

where • f : C_{1} → C_{2} is a k-linear covariant functor,

• θ : fO_{1} → O_{2} is a morphism, and

• μ : f ∘ s_{1} → s_{2} ∘ f is a natural transformation. Definition 5.12 A map F : (Proj(B), s_{2}) → (Proj(A), s_{1}) of polarized schemes is a natural equivalence class of morphisms

(f, θ, μ) : (Tails(A), A, s_{1}) → (Tails(B), B, s_{2})

such that f is right exact, θ is an isomorphism, and μ is a natural equivalence.

For example, there are isomorphisms of polarized schemes

(Tails(^{σ}A),^{σ}A, [1]) →^{∼} (Tails(A), A, σ_{∗} ∘ [1])

and

(Proj(A), [d]) →^{∼} (Proj(A^{(d)}), [1])

by Lemma 5.8 and Proposition 5.4 respectively.

A graded algebra associated to (C, O, s). We may associate to a triple (C, O, s) a ℤ-graded algebra

B(C, O, s) = ⨁_{n∈ℤ} B_{n}

where

B_{n} = Hom_{C}(O, s^{n}(O)),

and the product rule B_{m} × B_{n} → B_{m+n} is given by composition:

(f, g) ↦ (s^{n}f) ∘ g.

It is easy to see this is associative.

Proposition 5.13 The rule above gives a covariant functor

B : Trip → GrAlg

to the category of graded k-algebras.

Proof. This is straightforward, although the notation can get a little unwieldy. ■

Example 5.14 If R is a k-algebra and σ ∈ Aut_{k}(R), then

B(Mod(R^{op}), R_{R}, σ^{∗}) ≅ R[x, x^{−1}; σ],

the skew Laurent polynomial ring, in which deg(R) = 0 and deg(x) = 1.

Example 5.15 Let X be a projective scheme, and L a coherent O_{X}-module. Let s = L ⊗_{O_{X}} −. Since Hom(O_{X}, F) ≅ H^{0}(X, F) for any O_{X}-module F, we have

B(Coh(O_{X}), O_{X}, s) ≅ ⨁_{n∈ℤ} H^{0}(X, L^{⊗m})

with its natural commutative multiplication.

Example 5.16 We now combine the ideas in the previous two examples. Let X be a projective scheme, L a coherent O_{X}-module, and σ ∈ Aut_{k} X. For an O_{X}-module F, we write F^{σ} = σ^{∗}F; we have σ^{∗}F ≅ σ_{∗}^{−1}F, and also O_{X}^{σ} ≅ O_{X}. Now define s = (L ⊗_{O} −) ∘ σ^{∗}, and consider B(Coh(O_{X}), O_{X}, s). Then

s^{n}O = L ⊗ L^{σ} ⊗ ⋯ ⊗ L^{σ^{n−1}},

so

B_{n} = H^{0}(X, L ⊗ L^{σ} ⊗ ⋯ ⊗ L^{σ^{n−1}}),

and the product rule B_{m} ⊗ B_{n} → B_{m+n} is given by

(f, g) ↦ s^{n}(f) ⊗ g.

Thus B is a twisted homogeneous coordinate ring in the sense of Artin-van den Bergh [1]; to be consistent with their notation, we have

B(Coh(O_{X}), O_{X}, s) ≅ B(X, σ^{−1}, L)

(this isomorphism is analogous to the isomorphism ^{σ}A ≅ A^{σ^{−1}} remarked on earlier). Equivalently,

B(Coh(O_{X}), O_{X}, (L ⊗_{O} −) ∘ σ_{∗}) ≅ B(X, σ, L).

The next result is an important special case of the functoriality of B in Proposition 5.13.

Proposition 5.17 Let A be left noetherian, locally finite and ℕ-graded.

1. B(GrMod(A^{op}), A_{A}, [1]) ≅ A.

2. B(Tails(A^{op}), A, [1]) ≅ ωπA as a graded vector space; thus ωπA has a graded algebra structure.

3. Let f = B(π), where π : GrMod(A^{op}) → Tails(A^{op}) is the quotient functor. Then f : A → ωπA is a graded algebra homomorphism.

4. If A satisfies χ_{1}, then f^{∗} induces isomorphisms of polarized schemes

(Proj(A), [1]) ≅ (Proj(ωπA), [1])

and

(proj(A), [1]) ≅ (proj(ωπA), [1]).

Proof. (1) Just use the definition of B.

(2) We have

B(Tails(A^{op}), A, [1]) = ⨁_{n∈ℤ} Hom_{Tails}(A, A[n])
= \underline{Hom}(πA, πA)
≅ \underline{Hom}_{Gr}(A, ωπA)
≅ ωπA.

(3) The map, f say, arising from the functoriality of B is

A_{n} = Hom_{Gr}(A, A[n]) → Hom_{Tails}(πA, πA[n])
= Hom_{Gr}(A, ωπA[n])
= (ωπA)_{n}

which coincides with the map A → ωπA in the definition of ω.

(4) If A satisfies χ_{1}, then coker(f) is right bounded by Proposition 4.12. On the other hand ker(f) = τA is torsion, so the result follows from Proposition 5.3. ■

Part (4) of this proposition says that as far as the projective scheme proj(A) is concerned, one may replace A by ωπA (at least if A satisfies χ_{1}), which is generally a better algebra than A (as in (as in (as in (as in (as in (as in (as in (as in (as in Example 4.15).

Notation Given a triple (C, O, s), we write

• F(n) = s^{n}F for objects F in C, and

• H^{0}(F) = Hom_{C}(O, F).

Definition 5.18 Let (C, O, s) be a triple. We say that s is ample if

1. for each F in C, there exist positive integers n_{1}, …, n_{p} and an epimorphism

⨁_{i=1}^{p} O(−n_{i}) → F,

and

2. if f : F → G is an epimorphism in C, then the induced map

H^{0}(F(n)) → H^{0}(G(n))

is surjective for n ≫ 0.

Condition (1) says that O (with the help of the shift s) is something like a generator in C, and that the objects are in a sort of finitely generated, and condition (2) says that O (with the help of s) is something like a projective object.

Proposition 5.19 If A satisfies χ_{1}, then [1] is ample for (tails(A), A, [1]).

Proof. Let M ∈ grmod(A), and define F = πM. Then F ≅ π(M_{≥1}). Since A is left noetherian, M_{≥1} ∈ grmod(A) too, whence there is a surjection

⨁_{i=1}^{p} A[−n_{i}] → M_{≥1}

for some positive integers n_{1}, …, n_{p}. Applying π, this gives an epimorphism

⨁_{i=1}^{p} A[−n_{i}] → F

so condition (1) of Definition 5.18 is satisfied.

Let f : F → G be an epimorphism in tails(A), and write G = πN where N ∈ grmod(A). Now

⨁_{n∈ℤ} H^{0}(F[n]) = ⨁_{n∈ℤ} Hom_{Tails}(A, F[n])
= ⨁_{n∈ℤ} Hom_{Tails}(πA, πM[n])
= ⨁_{n∈ℤ} Hom_{Gr}(A, ωπM[n])
= ωπM.

Hence we must show that the map ωπM → ωπN induced by f is surjective in high degree. By Proposition 3.3, f is of the form πg for some A-module map g : M_{≥n} → N_{≥n}, for n ≫ 0. Hence we must show that g induces a surjection ωπ(M_{≥n}) → ωπ(N_{≥n}). Since f = πg is an epimorphism, ker(g) and coker(g) are torsion, hence finite dimensional as M_{≥n}, N_{≥n} ∈ grmod(A). Thus, for n ≫ 0, g : M_{≥n} → N_{≥n} is surjective. But A satisfies χ_{1}, so for n ≫ 0 (ωπM)_{≥n} = M_{≥n} and (ωπN)_{≥n} = N_{≥n}, whence the result. ■

Example 5.20 Let A and B be the algebras in Example 4.15, and define R = A[t], the polynomial extension with deg(t) = 1. Then [1] is not ample for (tails(R), R, [1]). Let N = R/(z). The point of Example 4.15 is that N has an essential extension by a torsion module which is not right bounded; thus coker(N → ωπN) is not right bounded. Let M = R, let g : M → N be the natural map, and let f : F = πM → G = πN denote πg. Certainly f is an epimorphism because g is surjective, so if ωπM → ωπN is not surjective in high degree, then [1] is not ample. However, \underline{Ext}_{R}^{1}(k, R) ≅ \underline{Hom}_{A}(k, A) = 0, so χ_{1}(M) holds. Thus ωπM/M is right bounded; since ωπN/N is not right bounded, the map ωπM → ωπN cannot be surjective in high degree.

The next Theorem is one of the main results in [2]; it gives some idea of which categories can arise as non-commutative schemes. Theorem 5.21 Let (C, O, s) be a triple such that

1. s is ample,

2. O is a noetherian object in C, and

3. Hom_{C}(O, F) is a finite dimensional for all F in C. Then A := B(C, O, s)_{≥0} is

• right noetherian,

• locally finite,

• satisfies χ_{1}, and

• (C, O, s) ≅ (tails(A^{op}), A, [1]).

The only comment we will make concerning the proof of Theorem 5.21 is to describe the functor implementing the equivalence of categories between C and tails(A^{op}). Let B = B(C, O, s); a graded B-module is of course an A-module. The equivalence is given by

F ↦ πΓF

where π : GrMod(A^{op}) → Tails(A^{op}) is the quotient functor and

Γ : C → GrMod(B^{op})

is the functor defined by

ΓF = ⨁_{n∈ℤ}(ΓF)_{n} = ⨁_{n∈ℤ} Hom_{C}(O, s^{n}F)

with right B-module structure

f.b = s^{n}(f) ∘ b

for f ∈ (ΓF)_{m}, b ∈ B_{n}.

Perhaps the simplest illustration of Theorem 5.21 is the following: if (C, O, s) = (Mod(R^{op}), R_{R}, Id), then B = R[x, x^{−1}] the Laurent polynomial extension, and ΓM = M[x, x^{−1}]; since B is strongly graded Mod(R^{op}) is equivalent to GrMod(B) which is equivalent to Tails(B). Since R[x] → B has right bounded cokernel Tails(R[x]) is equivalent to Tails(B). Thus Mod(R^{op}) is equivalent to Tails(R[x]), and R[x] is the ring A in the statement of the Theorem. Thus the theorem confirms what we already know.

Although Proposition 5.17, which says that tails(A) is equivalent to tails(ωπA)_{≥0} when χ_{1} holds, has a simple proof, it can also be deduced by applying Theorem 5.21 to (tails(A), A, [1]).

Another important consequence of Theorem 5.21 is Theorem 5.24 below on twisted homogeneous coordinate rings.

Definition 5.22 Let X be a noetherian scheme and σ ∈ Aut X. An invertible O_{X}-module L is σ-ample if, for every F ∈ Coh(O_{X})

H^{q}(X, L ⊗ L^{σ} ⊗ ⋯ ⊗ L^{σ^{n}} ⊗ F) = 0

for all q > 0 and all n ≫ 0.

The point is that σ-ampleness ensures that a certain shift functor is ample—the key to proving this is the next Lemma.

Lemma 5.23 [1, Lemma 3.2] Let L be a σ-ample line bundle on a scheme X. If F ∈ Coh(O_{X}), then s^{n}F is generated by its global sections for n ≫ 0.

Theorem 5.24 Let L be a σ-ample line bundle on a scheme X. Then B = B(X, σ, L)_{≥0} is noetherian and

tails(B) ≃ Coh(O_{X}).

Proof. Apply the Theorem to the triple

(Coh(O_{X}), O_{X}, s)

where s = (L ⊗_{O_{X}} −) ∘ σ^{∗}, as in Example 5.16. By standard commutative theory, O_{X} is a noetherian object in Coh(O_{X}) and H^{0}(F) = Hom(O_{X}, F) = H^{0}(X, F) is finite dimensional for all F ∈ Coh(O_{X}).

Next we show that s is ample. By applying s^{−n} to the result in Lemma 5.23, it follows that there is an epimorphism (s^{−n}O)^{p} → F for some large p. Hence condition (1) in Definition 5.18 holds. Now let f : F → G be an epimorphism in Coh(O_{X}). Write K = ker(f); since s is an exact functor, there is a long exact sequence in cohomology

0 → H^{0}(s^{n}K) → H^{0}(s^{n}F) → H^{0}(s^{n}G) → H^{1}(s^{n}K).

Since L is σ-ample, H^{1}(s^{n}K) = 0 for n ≫ 0, whence

H^{0}(s^{n}F) → H^{0}(s^{n}G)

is surjective; thus condition (2) in Definition 5.18 holds.

Hence the hypotheses of Theorem 5.21 hold. By its conclusion B is right noetherian and tails(B^{op}) ≅ Coh(O_{X}). ■

References

[1] M. Artin and M. van den Bergh, Twisted Homogeneous Coordinate Rings, J. Algebra 133 (1990) 249–271.

[2] M. Artin and J. Zhang, Non-commutative Projective Schemes, Adv. in Math., to appear.

[3] P. Gabriel, Des Catégories Abéliennes, Bull. Soc. Math. France, 90 (1962) 323–448.

[4] J. P. Serre, Faisceaux Algébriques Cohérents, Ann. of Math., 61 (1955) 197–278.

[5] K. van Rompay, Segre product of Artin-Schelter regular algebras of dimension 2 and embeddings in quantum spaces of dimension 4, preprint (1994).

[6] J. T. Stafford and J. Zhang, Examples in Non-commutative Projective Geometry, Proc. Camb. Phil. Soc., to appear.

[7] J. Zhang, Twisted Graded Algebras and Equivalences of Graded Categories, submitted.