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Finite morphism

In algebraic geometry, a finite morphism is a morphism of schemes f: X \to S that is affine and such that, for every affine open subscheme \operatorname{Spec}(R) \subset S, the preimage f^{-1}(\operatorname{Spec}(R)) = \operatorname{Spec}(A) corresponds to a ring homomorphism R \to A making A a finite R-module.^[1] Finite morphisms exhibit several key properties that make them fundamental in the study of scheme theory. They are necessarily of finite type, separated, quasi-compact, proper, and integral, meaning the ring extensions satisfy monic polynomial equations over the base ring.^[1] Moreover, finite morphisms are stable under composition and base change, allowing them to behave well in families and diagrams of schemes.^[1]^[2] These morphisms generalize closed immersions. For instance, the morphism induced by a finite étale extension of fields, such as \operatorname{Spec}(\mathbb{Q}(\sqrt{2})) \to \operatorname{Spec}(\mathbb{Q}), is finite, illustrating their appearance in number theory and arithmetic geometry.^[2]

Definitions

Ring homomorphisms

In commutative algebra, a ring homomorphism \phi: A \to B between commutative rings with identity is called a finite homomorphism if B is a finitely generated A-module via the A-module structure induced by \phi, meaning there exist finitely many elements b_1, \dots, b_n \in B such that every element of B can be written as \sum_{i=1}^n a_i b_i with a_i \in A.^[4] Equivalent characterizations of finite homomorphisms include the existence of a surjective A-module homomorphism from a finite free A-module A^n onto B, and the property that B admits a presentation as a quotient A[x_1, \dots, x_n]/I where I is an ideal of the polynomial ring A[x_1, \dots, x_n] generated by monic polynomials (one in each variable x_i).^[4] These formulations highlight the close connection between finite homomorphisms and integral extensions, as finite homomorphisms are integral and of finite type as algebras.^[4] The notion of finite homomorphisms originated in the development of commutative algebra, with foundational roots in the study of Noetherian rings and David Hilbert's basis theorem from the 1890s, which established that polynomial rings over Noetherian rings are Noetherian.^[5] Integral extensions were introduced by Richard Dedekind around 1882, with key properties such as the lying-over theorem formalized by Wolfgang Krull in the 1930s, building on his work in dimension theory and prime ideal behavior under extensions.^[5] A key example arises in finite field extensions, such as the extension \mathbb{Q}(\sqrt{2})/\mathbb{Q} induced by the homomorphism \phi: \mathbb{Q} \to \mathbb{Q}[\sqrt{2}], where \mathbb{Q}[\sqrt{2}] is finitely generated as a \mathbb{Q}-module by the basis \{1, \sqrt{2}\}, with the action determined by the minimal polynomial x^2 - 2 = 0 satisfied by \sqrt{2}.^[4] To sketch why finite generation as an A-module implies a finite presentation of B as an A-algebra, select module generators b_1, \dots, b_n \in B. The universal property of the polynomial ring yields a surjective A-algebra homomorphism \psi: A[x_1, \dots, x_n] \to B with \psi(x_i) = b_i, so B \cong A[x_1, \dots, x_n]/\ker(\psi). Since B is finitely generated as an A-module, the elements b_i satisfy integral relations over A, allowing \ker(\psi) to be generated by a finite set of monic polynomials expressing these dependencies, thus ensuring finite presentation.^[4]

Morphisms of schemes

A morphism f: X \to Y of schemes is finite if it is affine and, for every affine open subscheme V = \Spec B \subset Y, the preimage f^{-1}(V) = \Spec A where the corresponding ring homomorphism B \to A makes A a finite B-module.^[1] This condition ensures that the morphism is locally modeled on finite ring extensions, as defined for ring homomorphisms.^[1] When Y is affine, say Y = \Spec B, the morphism f: X \to Y corresponds precisely to a finite ring homomorphism B \to A with X = \Spec A.^[1] Assuming familiarity with affine schemes, this equivalence highlights how finite morphisms geometrize the algebraic concept in the affine case. An equivalent local condition is that f is affine and, for every point x \in X with y = f(x), the induced stalk map f^\#: \mathcal{O}_{Y,y} \to \mathcal{O}_{X,x} makes \mathcal{O}_{X,x} a finite module over \mathcal{O}_{Y,y}.^[1] Finite morphisms are stable under base change: if f: X \to Y is finite and g: Z \to Y is any morphism of schemes, then the base-changed morphism X \times_Y Z \to Z is finite.^[1]

Properties

Affine and finite-type aspects

Finite morphisms of schemes are affine, meaning that the preimage under f: X \to Y of any affine open subscheme of Y is affine. This property arises directly from the definition: when restricted to an affine open U = \Spec A \subset Y, the preimage f^{-1}(U) = \Spec B, where the ring homomorphism A \to B endows B with the structure of a finite A-module. Consequently, finite morphisms preserve the affine nature of open subschemes, allowing reductions to local ring-theoretic computations.^[6] The affine structure of finite morphisms is explicitly realized through the relative Spec functor. For a finite morphism f, the scheme X can be reconstructed as X = \Spec_Y(f_* \mathcal{O}_X), where f_* \mathcal{O}_X is a sheaf of \mathcal{O}_Y-algebras that is locally finite as an \mathcal{O}_Y-module. This construction underscores how finite morphisms "tame" potentially infinite-dimensional geometry by embedding the source into a finite extension of the target's structure sheaf.^[7] Every finite morphism is of finite type. This holds because, on affine opens, the corresponding ring homomorphism A \to B renders B finitely generated as an A-algebra.^[7] Finite morphisms are also quasi-compact, since the finite module structure ensures that the preimage of any quasi-compact open in Y is a finite union of affine opens, each covered by finitely many standard opens.^[8] For a finite morphism f: X \to Y with Y Noetherian, the pushforward f_* \mathcal{O}_X is a coherent sheaf of \mathcal{O}_Y-algebras, locally free as a finite \mathcal{O}_Y-module on affine opens. This coherence reflects the finite generation and relations inherent to the module structure, enabling effective control over cohomology and other invariants. However, not all affine morphisms of finite type are finite; for example, the morphism \Spec k[t,u] \to \Spec k induced by the inclusion k \hookrightarrow k[t,u] is affine and of finite type, but k[t,u] is not finite as a k-module.^[7]

Integral and flat conditions

A finite morphism of schemes is always integral, meaning that locally on affine opens, it corresponds to a ring homomorphism making the target algebraically integral over the source via monic polynomials.^[8] Conversely, an integral morphism that is also locally of finite type is finite, as the finite generation ensures the integral extension is finitely generated as a module.^[8] Regarding flatness, a finite morphism is flat if and only if it is faithfully flat.^[9] This equivalence holds because, locally on the source and target, the corresponding ring homomorphism between local rings is flat precisely when it is faithfully flat, and the quasi-compactness of finite morphisms ensures the global condition aligns.^[9] For example, a finite unramified morphism, such as one induced by a separable field extension without ramification, is flat, as it is étale and hence faithfully flat of constant rank.^[10] More strongly, if the morphism is additionally birational, it is an isomorphism, reflecting that the integral closure in the function field yields no new points or structure.^[11] The Cohen-Seidenberg theorems apply directly to finite extensions, as finiteness implies integrality, guaranteeing properties like the going-up theorem: for primes \mathfrak{p} \subset \mathfrak{p}' in the base ring and a prime \mathfrak{q} in the extension mapping to \mathfrak{p}, there exists \mathfrak{q}' \supset \mathfrak{q} mapping to \mathfrak{p}'.^[12] This ensures lying-over and chain conditions that preserve dimension and prime ideals under finite morphisms.^[12] Every finite morphism is proper, as it is affine, separated, of finite type, and universally closed.^[13] However, the converse does not hold; for instance, the structure morphism of projective space over an affine scheme is proper but not finite, as it is not affine.^[13]

Examples and applications

Classical examples from algebra

A fundamental class of finite morphisms arises in the context of finite field extensions. Consider a field k and an irreducible polynomial f(x) \in k of degree n. The quotient ring A = k/(f(x)) is a field extension K = k(\alpha), where \alpha is the image of x in A, and the natural ring homomorphism \phi: k \to K given by evaluation makes K a finite k-module with basis \{1, \alpha, \dots, \alpha^{n-1}\}, so the rank is n = [K:k].^[14] This illustrates how the degree of the extension corresponds to the module rank under the homomorphism.^[15] Finite morphisms also appear among Artinian rings, particularly finite-dimensional algebras over a field. For instance, let k be a field and consider the ring homomorphism \psi: k \to M_n(k), where M_n(k) is the ring of n \times n matrices over k. Here, M_n(k) is a finite-dimensional k-algebra of dimension n^2, hence Artinian, and acts as a free k-module of rank n^2 via the scalar matrix embedding, making \psi finite.^[16] More generally, any ring homomorphism from k to a finite-dimensional k-algebra B (which is automatically Artinian) renders B a finite k-module.^[17] In the setting of Dedekind domains, finite extensions of rings of integers provide concrete examples. The ring \mathbb{Z}[\sqrt{-5}] is the ring of integers of the number field \mathbb{Q}(\sqrt{-5}), and the inclusion \iota: \mathbb{Z} \to \mathbb{Z}[\sqrt{-5}] is a finite morphism with \mathbb{Z}[\sqrt{-5}] free of rank 2 as a \mathbb{Z}-module, generated by $1 and \sqrt{-5}. For the prime ideal (2) \subset \mathbb{Z}, it factors as (2) \mathcal{O}_K = \mathfrak{p}^2 where \mathfrak{p} = (2, 1 + \sqrt{-5}), so the ramification index is 2; similarly, (5) = \mathfrak{q}^2 with \mathfrak{q} = (5, \sqrt{-5}) and ramification index 2.^[18] A contrasting non-example is the inclusion \mathbb{[Q](/page/Q)} \to \overline{\mathbb{[Q](/page/Q_Sharp)}}, the algebraic closure, which induces an infinite field extension of infinite degree, so \overline{\mathbb{[Q](/page/Q)}} is not finitely generated as a \mathbb{[Q](/page/Q_Sharp)}-module, hence not finite. In Galois theory, finite separable morphisms play a key role: a finite field extension L/K is Galois if and only if it is normal and separable, with the Galois group \mathrm{Gal}(L/K) controlling the intermediate extensions via the fundamental theorem.^[19]

Geometric examples in schemes

One prominent geometric example of a finite morphism arises in the study of covers of curves. Consider an elliptic curve E over an algebraically closed field k of characteristic not 2 or 3; it admits a degree 2 morphism \pi: E \to \mathbb{P}^1_k that is a branched double cover, ramified at four distinct points corresponding to the Weierstrass points.^[20] This morphism is finite because it is proper (as a morphism between projective curves) and quasi-finite (with finite fibers of cardinality at most 2), and more precisely, it satisfies the integral condition locally via the corresponding ring extension.^[1] The degree of such a cover can be related to the ramification via the Hurwitz formula, which states that for a finite morphism f: C \to D of degree d between smooth projective curves of genera g_C and g_D, the formula $2g_C - 2 = d(2g_D - 2) + \deg R holds, where R is the ramification divisor; for the elliptic case with g_C = 1 and g_D = 0, d=2, this gives \deg R = 4.^[21] Another illustrative example is the normalization of singular curves, which provides a finite birational morphism. For the cuspidal curve defined by the ideal (y^2 - x^3) in \mathbb{A}^2_k, the normalization map \nu: \mathbb{A}^1_k \to C, given parametrically by x \mapsto t^2, y \mapsto t^3, is a finite morphism because the coordinate ring k[x,y]/(y^2 - x^3) is a finite module over k via the integral extension generated by these relations.^[22] This morphism resolves the singularity at the origin while remaining birational, highlighting how finite morphisms preserve integral closure in the geometric setting.^[23] In the projective setting, finite morphisms from projective schemes often manifest as branched covers. For instance, a double cover of \mathbb{P}^2_k branched along a smooth quartic curve yields a K3 surface as the source, with the projection map being a finite morphism of degree 2, proper and integral by virtue of the associated line bundle \mathcal{O}(2) and the ramification along the branch locus.^[22] The Veronese embedding v_d: \mathbb{P}^n_k \to \mathbb{P}^N_k (with N = \binom{n+d}{d} - 1) is a closed immersion and thus a finite morphism; it is proper and quasi-finite.^[24]^[1] Open immersions serve as a key non-example: while they are morphisms of finite type (locally isomorphic to polynomial ring inclusions), they are not finite because they fail to be proper, with infinite fibers over points in the complement.^[25] A scheme-specific example over finite fields is the absolute Frobenius morphism F: X \to X on a scheme X of finite type over \mathbb{F}_p, defined by raising coordinates to the p-th power on affine opens. This is a finite morphism because, locally on affines \operatorname{Spec} A \to \operatorname{Spec} A, the ring map A \to A, a \mapsto a^p, renders A integral over its image (each element satisfies a monic polynomial via the freshman’s dream in characteristic p), and the pushforward F_* \mathcal{O}_X is coherent.^[26]

Role in descent and cohomology

Finite morphisms play a crucial role in descent theory, particularly for quasi-coherent sheaves on schemes. When a finite morphism is faithfully flat, it qualifies as an fpqc covering, enabling effective descent for quasi-coherent sheaves. Specifically, the theorem on fpqc descent asserts that for any fpqc covering \{U_i \to S\} of a scheme S, descent data on quasi-coherent \mathcal{O}_S-modules are effective, meaning the associated sheaf on S exists and the comparison functor is an equivalence of categories.^[27] This extends to sheaves of finite presentation, where finite morphisms facilitate gluing local data globally, as the finite presentation condition ensures compatibility under base change along such coverings.^[27] In sheaf cohomology, finite morphisms ensure the finiteness of cohomology groups for coherent sheaves. For instance, if f: X \to \Spec k is a finite morphism over a field k and \mathcal{F} is a coherent sheaf on X, the cohomology groups H^i(X, \mathcal{F}) are finite-dimensional vector spaces over k. This follows from the fact that X is affine (as the spectrum of a finite-dimensional k-algebra), so higher cohomology vanishes, and H^0(X, \mathcal{F}) is finitely generated as a module over the finite-dimensional algebra.^[28] More generally, under a finite morphism, the direct image functor f_* preserves coherence, contributing to bounded cohomology in the target.^[28] A key application in étale cohomology involves finite morphisms preserving cohomological purity and influencing the Brauer group through period-index relations. Finite étale morphisms act as cohomological equivalences in the étale topology, preserving the purity of cohomology sheaves, such as the isomorphism between nearby and vanishing cycles for smooth varieties.^[29] For the Brauer group \text{Br}(X) \cong H^2_{\text{ét}}(X, \mathbb{G}_m)_{\text{tors}}, finite morphisms correspond to splitting extensions, where the index of a class \alpha \in \text{Br}(X) is the minimal degree of a finite étale cover splitting \alpha, and period-index bounds quantify this via results like \text{ind}(\alpha) \mid \text{per}(\alpha)^5 for classes over function fields of transcendence degree 2.^[30] In arithmetic geometry, finite flat morphisms are essential for constructing Néron models of abelian varieties over Dedekind bases. They appear in Weil restrictions and descent procedures, ensuring that Néron models commute with étale base change and that torsors under finite flat group schemes descend effectively.^[31] For example, in the theory of Jacobians, finite flat morphisms over proper flat curves help establish the Néron model structure for the Picard scheme when the special fiber satisfies multiplicity conditions.^[31] In modern derived algebraic geometry, finite morphisms preserve the category of perfect complexes. For a proper morphism f: X \to Y of spectral algebraic spaces that is locally almost of finite presentation, the direct image f_* sends almost perfect objects in \text{QCoh}(X) to almost perfect objects in \text{QCoh}(Y), with finite morphisms providing a prime example due to their properness and Tor-dimension bounds.^[32] This result, developed in the spectral setting, extends classical finiteness properties to derived stacks.^[32]

References

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29.44 Integral and finite morphisms - Stacks Project
It is clear that integral/finite morphisms are separated and quasi-compact. It is also clear that a finite morphism is a morphism of finite type.
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Commutative Algebra and Some of its Applications Historical Notes
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### Summary of Finite Morphisms from Stacks Project (Tag 01WG)
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Lemma 29.44.4 (01WJ)—The Stacks project
A finite morphism is integral. An integral morphism which is locally of finite type is finite. Proof. See Algebra, Lemma 10.36.3 and Lemma 10.36.5.
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Ntron models were invented by A. Ntron in the early 1960's with the intention to study the integral structure of abelian varieties over number fields.
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