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Finite-type Noetherian spaces and affine-line product dimension

PDF p. 13: Proof of Proposition 6.1: Noetherianity and ambient affine-line dimensions for the classical comparison

Statement

Assume the Axiom of Choice. Let $k$ be a field and $T$ a finite-type $k$ -scheme, without a separatedness assumption. The underlying space $∣ T ∣$ is Noetherian. If $T$ is integral and $d = \dim T$ , then $T \times_{k} A_{k}^{1}$ is an integral finite-type $k$ -scheme of dimension $d + 1$ . Here $A_{k}^{1} = Spec k [t]$ , products are scheme fibre products over $k$ , and dimension is chain dimension of the underlying Noetherian space.

Statement references

The Axiom of Choice

Field

Schemes and morphisms over a base

Locally finite type and finite type morphisms

Integral schemes

Noetherian topological spaces via ACC on opens or DCC on closed subsets

Chain dimension and the empty-space convention

Existence of all scheme fibre products

The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution

Schemes

Affine schemes and their coordinate rings

Morphisms of schemes

Fibre product of schemes

The underlying space of an affine spectrum

Irreducible topological spaces and irreducible subsets in the subspace topology

Affine open subschemes

Subalgebra generated by a subset, algebras of finite type, and module-finite algebras

Algebras over a commutative ring, central structure maps, and algebra homomorphisms

Facts & Assumptions

Given: The Axiom of Choice, the field $k$ and the finite-type $k$ -scheme $T$ of the statement. Integrality is assumed only for the product assertion.

Goal: Establish the global Noetherian-space conclusion and the integrality, finite type and dimension of the affine-line product.

[A1]

Choice meets the assumptions of affine quasi-compactness, the Noetherian-spectrum theorem, the prime/closed-set correspondence and the polynomial-dimension theorem below. The Axiom of Choice

[F1]

A $k$ -scheme has a structure map to $Spec k$ . Finite type means locally of finite type and quasi-compact; locally of finite type supplies affine neighbourhoods $Spec A$ with $A$ finitely generated over $k$ . A quasi-compact morphism has quasi-compact inverse images of quasi-compact opens, and a quasi-compact scheme has a finite subcover of every open cover. Under Choice every affine scheme is quasi-compact. Field, Schemes and morphisms over a base, Locally finite type and finite type morphisms, Quasi-compact and quasi-separated morphisms, Quasi-compact and quasi-separated schemes, Every affine scheme is quasi-compact, Schemes, Affine schemes and their coordinate rings, Affine open subschemes, Morphisms of schemes, Subalgebra generated by a subset, algebras of finite type, and module-finite algebras

[F2]

A field is Noetherian. A commutative ring is Noetherian when every ideal is finitely generated; a finite-type algebra over a Noetherian ring is Noetherian, and under Choice its spectrum is a Noetherian space. A space is Noetherian exactly when every descending chain of closed subsets stabilizes. A field has only the zero ideal and itself, hence is Noetherian, Noetherian commutative rings and modules, Every algebra of finite type over a Noetherian ring is a Noetherian ring, The spectrum of a Noetherian ring is a Noetherian topological space, Noetherian topological spaces via ACC on opens or DCC on closed subsets

[F3]

An integral scheme is nonempty, reduced and irreducible; its nonempty affine opens have domain coordinate rings. Here a scheme is reduced when it has an affine open cover by affine schemes whose coordinate rings have no nonzero nilpotents. Scheme fibre products have the open affine tensor-product charts. Polynomial rings over domains are domains. Integral schemes, Existence of all scheme fibre products, A polynomial ring over an integral domain is an integral domain, The underlying space of an affine spectrum, Prime ideals and maximal ideals in a commutative ring, Zero divisor, and integral domain: a commutative ring with 1 ≠ 0 and no zero divisors, Reduced affine schemes, The nilradical and reduced rings, Fibre product of schemes

[F4]

Chain dimension counts strict chains of nonempty irreducible closed subsets. Krull dimension counts strict prime chains. For a nonzero Noetherian ring $A$ , its spectrum is Noetherian under Choice. The prime/irreducible-closed-set correspondence reverses inclusion, so $\dim Spec A = \dim A$ for such $A$ . Noether normalization makes a finite-type $k$ -domain $A$ module-finite over a polynomial subring in some finite number $n$ of variables; the normalization dimension bound then gives $\dim A \leq n < \infty$ . Under Choice a Noetherian ring $A$ of finite Krull dimension satisfies $\dim A [t] = \dim A + 1$ . Integral finite-type schemes and all their nonempty opens have the same dimension, as supplied by the finite-type geometry interface. Chain dimension and the empty-space convention, Krull dimension of a nonzero ring, The spectrum of a Noetherian ring is a Noetherian topological space, A Zariski-closed subset is irreducible exactly when its radical defining ideal is prime, and then it has a unique generic point, Noether normalisation yields module finiteness over a polynomial subring, A finite affine extension of a polynomial ring has dimension at most the number of variables, A Noetherian polynomial ring has dimension one larger, Regularity and dimensions for finite-type cycle arguments

[F5]

For a commutative ring $R$ , $R [t]$ consists of finitely supported coefficient sequences, with coefficientwise addition and convolution multiplication. For a commutative unital $k$ -algebra $A$ , the tensor product $A \otimes_{k} k [t]$ is generated additively by elementary tensors, with additivity in both entries and balancing $(a c) \otimes p = a \otimes (c p)$ for $c \in k$ . Since $A$ and $k [t]$ are commutative unital $k$ -algebras, its multiplication is $(a \otimes p) (b \otimes q) = a b \otimes p q$ and its unit is $1 \otimes 1$ . The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution, The tensor product M⊗_R N from the additive group underlying the free ℤ-module on M× N, elementary tensors, and finite tensor sums, The tensor product of R-algebras has multiplication (a⊗ b)(a'⊗ b')=aa'⊗ bb', Algebras over a commutative ring, central structure maps, and algebra homomorphisms

[F6]

A space is irreducible if it is nonempty and is not the union of two proper closed subsets; irreducibility of a subset uses its subspace topology, whose opens are intersections with ambient opens. For a subset $S$ of a space $Z$ , its closure is the intersection of all closed subsets of $Z$ containing $S$ . A point $η$ is generic for a closed subset $C$ when the closure of ${η}$ is $C$ ; a subset is dense when its closure is the whole space. Irreducible topological spaces and irreducible subsets in the subspace topology, Generic points of irreducible closed subsets

Proof

1.1givenA1F1F2

The affine scheme $Spec k$ is quasi-compact by [F1], so finite type makes $T$ quasi-compact. Extract a finite affine open cover $T = ⋃_{i = 1}^{m} U_{i}$ with $U_{i} = Spec A_{i}$ and each $A_{i}$ finitely generated over $k$ . For empty $T$ use the empty cover. By [F2], $k$ and then every $A_{i}$ are Noetherian, so every $∣ U_{i} ∣$ is Noetherian under [A1]. Given a descending chain $C_{n}$ of closed subsets of $∣ T ∣$ , the chain $C_{n} \cap U_{i}$ stabilizes for each $i$ . The maximum of the finitely many stabilization indices makes every restriction constant and therefore makes $C_{n} = ⋃_{i} (C_{n} \cap U_{i})$ constant. The empty case is immediate. This proves that $∣ T ∣$ is Noetherian.

2.1A1F1F2F3F4F5F6step 1.1∎

Suppose $T$ is integral, and discard empty charts of the cover in step 1.1. By [F3], $P = T \times_{k} A_{k}^{1}$ has open affine cover $V_{i} = Spec (A_{i} \otimes_{k} k [t]) = Spec A_{i} [t]$ . Using the constructions in [F5], the ring identification sends $a \otimes p (t)$ to $a p (t)$ , with inverse $\sum_{n} a_{n} t^{n} \mapsto \sum_{n} a_{n} \otimes t^{n}$ ; the tensor relations verify multiplicativity and both inverse identities. Each $A_{i} [t]$ is a domain, hence has no nonzero nilpotents, and is finitely generated over $k$ : adjoining $t$ to a finite algebra-generating list for $A_{i}$ gives such a list for $A_{i} [t]$ . Thus $P$ is reduced and locally of finite type. The finite cover by quasi-compact affine charts makes $P$ quasi-compact. The only opens of $Spec k$ are itself and the empty set, since a field has only its zero prime, so this also makes the structure morphism quasi-compact; hence $P$ is finite type. Any two nonempty $U_{i}$ meet by irreducibility of $T$ : otherwise their complements would be two proper closed subsets covering $T$ , contrary to [F6]. An affine neighbourhood $Spec B \subset U_{i} \cap U_{j}$ has domain coordinate ring, and its product chart $Spec B [t]$ is a nonempty open in $V_{i} \cap V_{j}$ , since the zero ideal is prime in $B [t]$ . Thus the irreducible charts $V_{i}, V_{j}$ have nonempty open intersection. A generic point lies in every nonempty open, since a closed complement containing it would contain its closure. It remains generic in a nonempty open, since every nonempty relatively open subset there is open in the ambient space and contains that point. Thus each chart's generic point lies in the nonempty open $Spec B [t]$ and is generic there, so uniqueness of its affine generic point identifies all chart generic points. Their common point lies in every nonempty open of $P$ , because such an open meets some $V_{i}$ . It is therefore dense in $P$ : a proper closed subset containing it would have nonempty open complement. A union of two proper closed subsets cannot contain this point, so [F6] proves irreducibility and hence integrality. Apply step 1.1 also to this finite-type $P$ to obtain its Noetherian underlying space. For one nonempty chart $U_{i}$ , [F4] gives $\dim A_{i} = \dim U_{i} = \dim T = d < \infty$ . Together with the Noetherianity of $A_{i}$ from step 1.1, the polynomial-dimension theorem gives $\dim A_{i} [t] = d + 1$ . The nonzero finite-type $k$ -algebra $A_{i} [t]$ is Noetherian by [F2], so the spectrum-dimension identity of [F4] applies to it. Thus $\dim V_{i} = d + 1$ , and integrality and finite type of $P$ make this also $\dim P$ . No step requires separatedness over $k$ .

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Classical Chow relations and Bloch boundaries

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