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Flat morphism

In algebraic geometry, a flat morphism between schemes f: X \to Y is defined as one where, for every point x \in X, the stalk \mathcal{O}_{X,x} is a flat \mathcal{O}_{Y,f(x)}-module via the induced map on structure sheaves.^[1] This local ring condition captures a form of "flatness" in the module-theoretic sense, ensuring that the morphism preserves exactness of sequences of quasi-coherent sheaves upon pullback.^[1] Flat morphisms play a central role in the study of families of algebraic varieties, as they often ensure that fibers vary in a controlled manner, such as maintaining the same Hilbert polynomial, which helps avoid pathological jumps under additional assumptions like finite presentation and properness.^[2] This property is essential for deformation theory and moduli problems, where flatness ensures stability under base change—meaning that if f is flat, then the pullback of f along any morphism to Y remains flat.^[1] Additionally, flatness is preserved under composition and is a local property on both source and target schemes, making it versatile for gluing constructions in algebraic geometry.^[1] Key examples include all open immersions, which are flat by inducing isomorphisms on local rings, and étale morphisms (smooth of relative dimension zero), which are flat.^[1] Flat morphisms of finite presentation are universally open, meaning they are open and remain open under arbitrary base changes, a property crucial for coherence in intersection theory and cohomology.^[1] In broader contexts, such as formal algebraic spaces or stacks, flatness extends these behaviors, facilitating descent and relative spec constructions.^[3]

Definition and Basics

Definition in ring theory

In commutative algebra, a ring homomorphism \phi: A \to B is called flat if B is a flat A-module via the structure induced by \phi. An A-module M is flat if the functor -\otimes_A M from the category of A-modules to itself is exact, meaning that for every short exact sequence $0 \to N' \to N \to N'' \to 0 of A-modules, the sequence $0 \to N' \otimes_A M \to N \otimes_A M \to N'' \otimes_A M \to 0 is also exact. Equivalently, tensoring with M preserves injections: if f: N' \to N is an injective A-module homomorphism, then f \otimes_A \id_M: N' \otimes_A M \to N \otimes_A M is injective.^[4]^[5] There are several equivalent characterizations of flatness. One is the vanishing of higher Tor groups: M is a flat A-module if and only if \Tor_i^A(N, M) = 0 for all i \geq 1 and all A-modules N. Another is in terms of ideals: M is flat over A if and only if for every (finitely generated) ideal I \subseteq A, the natural multiplication map I \otimes_A M \to M given by i \otimes m \mapsto i \cdot m is injective, which means that I \otimes_A M \cong I M as submodules of M. A related condition for a flat homomorphism A \to B is that every finitely generated ideal of A extends to a finitely generated ideal in B, since flat base change preserves finite generation.^[4]^[5] The notion of flat modules was introduced by Jean-Pierre Serre in his 1956 paper Géométrie algébrique et géométrie analytique, where it arose in the context of comparing algebraic and analytic geometries.^[6] Alexander Grothendieck built upon Serre's work in the Éléments de géométrie algébrique (EGA), particularly developing flatness as a key tool for descent theory and properties of morphisms in algebraic geometry.^[7] A basic example of a flat ring homomorphism is the localization A \to A_f at an element f \in A, or more generally, localization at a multiplicative subset S \subseteq A, which is always flat.^[4]

Extension to schemes

In the context of schemes, the notion of flatness extends the algebraic definition from ring homomorphisms to morphisms of schemes by localizing at points via stalks of structure sheaves. Specifically, a morphism f: X \to Y of schemes is flat if, for every point x \in X with image y = f(x) \in Y, the induced local ring homomorphism \mathcal{O}_{Y,y} \to \mathcal{O}_{X,x} is flat as a map of rings.^[8] This condition is local on both the source and target in the Zariski topology, meaning that flatness holds globally if and only if it holds Zariski-locally on X and Y. Equivalently, f is flat if and only if the stalks \mathcal{O}_{X,x} are flat \mathcal{O}_{Y,y}-modules for all x \in X.^[8] A key tool for verifying flatness in practice is the local criterion of flatness, particularly in the Noetherian setting. For a local homomorphism R \to S of Noetherian local rings with maximal ideal \mathfrak{m} in R and residue field \kappa = R/\mathfrak{m}, and for a finite S-module M, M is flat over R if \Tor_1^R(\kappa, M) = 0.^[9] This vanishing condition can be checked using Nakayama's lemma, as it ensures that the minimal number of generators of M equals the dimension of the vector space M/\mathfrak{m}M over \kappa, reflecting faithful preservation of resolutions. For morphisms that are locally of finite presentation, additional criteria apply: a local ring map R \to S with S essentially of finite presentation over R and a finite presentation S-module M is flat if M/IM is flat over R/I for the relevant ideal I and certain Tor-vanishing or injectivity conditions hold on the special fiber M/\mathfrak{m}M.^[10] These fiber conditions ensure that flatness descends appropriately without introducing torsion or dimension irregularities. While flat morphisms generalize open immersions—since open immersions induce isomorphisms on local rings and thus are flat—they need not be open in general without further assumptions like local finite presentation.^[8] In contrast, faithfully flat morphisms, which are flat and induce surjective maps on the underlying topological spaces (i.e., surjective on spectra), guarantee surjectivity on points.^[11]

Examples

Positive examples

One prominent example of a flat morphism is the projection morphism from the affine line bundle over the affine line, given by \operatorname{Spec}(k) \to \operatorname{Spec}(k), where k is a field. This morphism corresponds to the ring extension k[t,x] over k, which is free as a module with basis \{1, x\}, hence flat. Similarly, the structure morphism \mathbb{P}^n_A \to \operatorname{Spec}(A) for any commutative ring A is flat. This follows because \mathbb{P}^n_A admits an open cover by affine schemes \operatorname{Spec}(A[x_0, \dots, x_n]_{x_i}), each of which is a localization of the polynomial ring A[x_0, \dots, x_n] over A; polynomial rings are free (hence flat) over the base ring, and localizations preserve flatness.^[12] Any smooth morphism of schemes is flat. By definition, a smooth morphism is locally isomorphic to the projection \mathbb{A}^n_S \to S for some scheme S and integer n \geq 0, and such projections are flat since they arise from polynomial rings, which are free modules over the base.^[12] Étale morphisms provide a subclass of smooth morphisms that are also flat. An étale morphism is smooth of relative dimension zero, meaning it is locally isomorphic to \operatorname{Spec}(R) \to \operatorname{Spec}(R) for a local ring R, preserving flatness via the same free module structure.^[12]

Miracle flatness

The miracle flatness theorem gives a powerful criterion for determining when a morphism of schemes is flat, especially in geometric families where the source satisfies the Cohen-Macaulay condition and the target is regular. Let f: X \to Y be a morphism of finite type between Noetherian schemes, with Y regular and X Cohen-Macaulay. Assume that the fibers of f are equidimensional. Then f is flat if and only if all fibers have the same dimension.^[13] A proof of the theorem proceeds locally on stalks. For a local homomorphism R \to S of Noetherian local rings with R regular and S Cohen-Macaulay, the conditions imply \dim S = \dim R + \dim(S/mS), where m is the maximal ideal of R. Flatness then holds precisely when \dim(S/mS) = \dim S - \dim R, ensuring constant fiber dimension. The argument uses the Auslander-Buchsbaum formula, which states that for a finitely generated module M over a regular local ring R of finite projective dimension, \mathrm{pd}_R M + \mathrm{depth} M = \mathrm{depth} R, combined with depth equalities from the Cohen-Macaulay hypothesis to show that higher Tor groups vanish, verifying flatness via the local criterion.^[14]^[15] This theorem finds important applications in the geometry of fibrations. For instance, an elliptic fibration over a smooth curve is flat if and only if there are no multiple fibers, as the total space is Cohen-Macaulay and the fibers are equidimensional of dimension 1.^[16] It is also used in the construction and analysis of moduli spaces of curves, where it helps establish flatness of universal families or projections in k-moduli stacks over quadric surfaces.^[17] The result originates from criteria for flatness developed in Grothendieck's Éléments de géométrie algébrique (EGA IV), with the term "miracle flatness" coined by Robin Hartshorne to highlight its surprising efficacy in verifying flatness under mild hypotheses.^[18]^[19]

Hilbert schemes

The Hilbert scheme \Hilb^P_X of a projective scheme X over a base scheme with respect to a fixed Hilbert polynomial P parametrizes closed subschemes of X with Hilbert polynomial P, specifically representing the functor that sends a locally Noetherian base scheme S to the set of S-flat families of such closed subschemes in the base change X_S.^[20] This flatness condition ensures that the fibers over points of S vary continuously, maintaining constant Hilbert polynomial and providing a geometric parameter space for deformations of subschemes.^[20] In deformation theory, these schemes capture infinitesimal deformations of subschemes as flat morphisms, allowing the study of moduli problems where flat families correspond to points in the Hilbert scheme.^[21] The points of \Hilb^P_X thus correspond to flat morphisms from the universal subscheme \mathcal{Z} \subset X \times \Hilb^P_X to the base scheme \Hilb^P_X itself, where \mathcal{Z} is the universal flat family over the Hilbert scheme.^[20] Grothendieck's representability theorem establishes that the Hilbert functor is representable by a scheme, implying that this universal subscheme \mathcal{Z} is flat over \Hilb^P_X, which guarantees the properness and universality of the parameterization for projective X.^[20] A representative example arises in the study of curves, where the Hilbert scheme \Hilb^{2g-2}_{\mathbb{P}^3} for genus g=4 parametrizes canonical curves of degree $6 in \mathbb{P}^3, and flat families over this scheme yield moduli spaces of such curves, facilitating the construction of the moduli space of curves via GIT quotients.^[22] In general, for higher genus, flat families in analogous Hilbert schemes of canonical curves in \mathbb{P}^{g-1} provide essential tools for understanding the geometry of the moduli stack of curves through their embedding properties.^[23]

Non-Examples

Blowup morphisms

A blowup morphism is defined as the projection \pi: \Bl_I X \to X from the blowup of a scheme X along a closed subscheme defined by an ideal sheaf \mathcal{I}, where \Bl_I X = \Proj_X \bigoplus_{n \geq 0} \mathcal{I}^n.^[24] This morphism is proper and birational, replacing the center Z = V(\mathcal{I}) with the projectivized normal cone.^[25] The blowup introduces an exceptional divisor E = \pi^{-1}(Z), which is an effective Cartier divisor isomorphic to the projectivized normal bundle \mathbb{P}(N_{Z/X}) over Z, where N_{Z/X} is the normal bundle of the embedding Z \hookrightarrow X.^[24] This divisor captures the directions transverse to Z and plays a central role in the geometry of the blowup.^[26] Blowup morphisms are generally not flat because the dimensions of the fibers vary: over points outside the center Z, the fiber is a single point (dimension 0), while over points in Z, the fiber is the exceptional divisor, which has positive dimension equal to \dim X - \dim Z - 1.^[25] This violation of equidimensionality across fibers prevents the structure sheaf of \Bl_I X from being flat over that of X.^[27] Homologically, flatness fails as higher Tor groups, such as \Tor_1(\mathcal{O}_{\Bl_I X}, k(p)), are nonzero for points p in Z, indicating torsion in the fibers.^[28] A concrete example is the blowup of the affine plane \mathbb{A}^2_k = \Spec k[x,y] along the origin, defined by the ideal \mathcal{I} = (x,y). The exceptional divisor is \mathbb{P}^1_k, so the fiber over the origin has dimension 1, whereas fibers over other points are single points of dimension 0.^[24] This dimension jump confirms non-flatness, as the condition \dim Z \geq \dim X - 1 (here, $0 \not\geq 1) is not satisfied.^[25] As a consequence, blowup morphisms are not stable under base change in general; pulling back along a non-flat morphism can exacerbate the fiber dimension irregularities.^[25]

Morphisms with infinite resolutions

A classic example of a ring homomorphism that is not flat is the projection \pi: k[\epsilon] \to k, where k[\epsilon] = k[\epsilon]/(\epsilon^2) is the ring of dual numbers over a field k, and \pi(\epsilon) = 0. To see that \pi is not flat, consider the short exact sequence $0 \to (\epsilon) \to k[\epsilon] \to k \to 0. Tensoring with k over k[\epsilon] yields $0 \to k \to k \to k \to 0, but the middle map is multiplication by \epsilon, which is zero, so the tensored sequence is not exact.^[29] Equivalently, the natural map k/(\epsilon)k \to (\epsilon)k is k \to 0, which is not an isomorphism, violating the criterion for flatness over k[\epsilon].^[29] The failure of flatness here stems from the infinite projective dimension of k as a k[\epsilon]-module. The minimal projective resolution of k is the infinite complex

\cdots \to k[\epsilon] \xrightarrow{\cdot \epsilon} k[\epsilon] \xrightarrow{\cdot \epsilon} k[\epsilon] \to k \to 0,

where the maps alternate multiplication by \epsilon, and each kernel and image is (\epsilon). This infinite resolution implies that higher Tor groups do not vanish; for instance, \Tor_i^{k[\epsilon]}(k, k) \neq 0 for all i \geq 1. Since the global dimension of k[\epsilon] equals the projective dimension of its residue field k, which is infinite, the ring homomorphism \pi exhibits non-vanishing higher Tor groups.^[30] In general, a ring homomorphism A \to B is flat if and only if B has Tor-dimension 0 as an A-module, meaning \Tor_i^A(B, N) = 0 for all i > 0 and all A-modules N. Morphisms like \pi fail this condition because the Tor-dimension of B over A is infinite, often occurring in Artinian rings where modules admit non-finite projective resolutions, such as injective hulls of simple modules in non-semisimple Artinian rings. Such homomorphisms do not preserve exactness of tensor products in the derived category, as the derived tensor product B \otimes^A_L N has non-vanishing homology in negative degrees for some N.^[31]^[28]

Fundamental Properties

Preservation under composition and base change

Flat morphisms exhibit stability under basic algebraic operations, notably composition and base change, which are fundamental to their utility in algebraic geometry. Specifically, if f: X \to Y and g: Y \to Z are morphisms of schemes that are both flat, then the composition g \circ f: X \to Z is flat.^[32] This property holds locally at points: if a quasi-coherent sheaf on X is flat over Y at a point x \in X and Y is flat over Z at the image of x, then the sheaf is flat over Z at x.^[32] A key stability feature is preservation under base change. If f: X \to Y is a flat morphism of schemes and g: Y' \to Y is an arbitrary morphism, then the base change X' = X \times_Y Y' \to Y' is also flat.^[33] In terms of sheaves, if \mathcal{F} is a quasi-coherent \mathcal{O}_X-module flat over Y at a point x \in X, then the pullback (g')^* \mathcal{F} on X' is flat over Y' at the corresponding point x' \in X'.^[33] This ensures that flatness behaves well under fiber products, facilitating constructions in relative geometry. These preservation properties follow from the underlying commutative algebra of flat modules. Flatness of a module M over a ring A means that tensoring with M preserves exact sequences, and this extends to ring maps via the corresponding module structure. For composition, if B is flat over A and C is flat over B, then C is flat over A because the functor -\otimes_B C (exact by flatness of C) composed with -\otimes_A B (exact by flatness of B) yields -\otimes_A C, which is thus exact; transitivity arises similarly for module categories.^[34] Base change preservation stems from the fact that pullback corresponds to tensor product, which commutes with the exactness-preserving tensor with the original flat module.^[35] When combined with finite presentation, flatness implies that the direct image of the structure sheaf is locally free. A morphism of schemes that is flat and locally of finite presentation is such that on affine opens \operatorname{Spec} B \to \operatorname{Spec} A corresponding to the ring map A \to B, B is a finitely presented flat A-module, hence finite projective over A.^[36] More precisely, over a ring R, a finitely presented flat R-module M is finite projective, hence locally free of finite rank; this equivalence holds because localizations at primes show M free locally, ensuring projectivity.^[36] For Noetherian bases, this often simplifies to finite flat implying finite locally free.^[37]

Faithfully flat morphisms

A morphism of schemes f: X \to Y is said to be faithfully flat if it is flat and the induced map on underlying topological spaces f: X \to Y is surjective.^[8] For an affine morphism \operatorname{Spec} B \to \operatorname{Spec} A corresponding to a ring homomorphism A \to B, surjectivity on spectra is equivalent to every prime ideal \mathfrak{p} \subset A contracting from some prime ideal \mathfrak{q} \subset B, meaning every prime of A lifts to a prime of B.^[8] Equivalently, A \to B is faithfully flat if B is a flat A-module and, for every A-module M, the condition M \otimes_A B = 0 implies M = 0. This module-theoretic characterization highlights the "faithful" aspect, as the functor M \mapsto M \otimes_A B not only preserves exactness (due to flatness) but also reflects it, detecting properties like injectivity or zero modules on the original side. For instance, if a homomorphism \phi: M \to N of A-modules becomes injective after base change to B, then \phi was already injective.^[38] Faithfully flat morphisms are preserved under composition, so the composite of two faithfully flat morphisms is again faithfully flat.^[38] A standard example of faithfully flat morphisms arises in étale geometry: an étale cover of a scheme, which is a surjective étale morphism, is faithfully flat because étale morphisms are flat and the surjectivity ensures the covering property.^[39] More generally, base change along a faithfully flat morphism reflects exactness of sequences of quasi-coherent sheaves; that is, a sequence of sheaves on Y is exact if and only if its pullback to X is exact.^[29] This detection property underpins many descent results in algebraic geometry, allowing properties defined over X to be checked after faithfully flat base change.^[29]

Advanced Properties

Topological aspects

A key topological feature of flat morphisms arises in the Zariski topology on schemes. A morphism f: X \to Y of schemes that is flat and locally of finite presentation is open, in the sense that the image of every open subset of X is open in Y.^[40] Such morphisms are in fact universally open: after any base change Y' \to Y, the resulting morphism X \times_Y Y' \to Y' remains open. This stability follows from the facts that both flatness and local finite presentation are preserved under arbitrary base change.^[40] The topology of fibers over points in the base is also preserved under base change for flat morphisms. Specifically, if f: X \to Y is flat and y \in Y, then for any Y' \to Y with y' \in Y' mapping to y, the fiber (X \times_Y Y')_{y'} is isomorphic to the base change of the original fiber X_y along the residue field extension k(y) \to k(y'); since flatness is stable under base change, this preserves the scheme-theoretic and topological structure of the fibers.^[41] The condition of local finite presentation is essential for these openness properties, as flat morphisms need not be open without it. For instance, the morphism \operatorname{Spec}(\mathbb{Q}) \to \operatorname{Spec}(\mathbb{Z}) induced by the inclusion \mathbb{Z} \hookrightarrow \mathbb{Q} is flat, since \mathbb{Q} is a flat \mathbb{Z}-module, but not open: the image of the unique nonempty open set in \operatorname{Spec}(\mathbb{Q}) is the generic point of \operatorname{Spec}(\mathbb{Z}), which is not open in the Zariski topology. This morphism fails to be locally of finite presentation.

Dimension theory

In algebraic geometry, for a flat morphism f: X \to Y that is locally of finite type between Noetherian schemes, the Krull dimension exhibits additivity along the fibers. Specifically, for any point x \in X with y = f(x), the equality \dim_x X = \dim_y Y + \dim_x (X_y) holds, where X_y denotes the fiber over y and \dim_x (X_y) is the Krull dimension of the local ring at x in the fiber scheme.^[42] This formula underscores the "openness" of the relative dimension function y \mapsto \sup_{x \in X_y} \dim_x (X_y), which is locally constant on Y.^[43] Flatness further ensures that the fibers inherit favorable cohomological properties from the total space. In particular, if X is Cohen-Macaulay, then each fiber X_y is also Cohen-Macaulay, implying that the cohomological dimension of the local ring \mathcal{O}_{X_y, x} (defined as the projective dimension of its residue field) equals the Krull dimension of X_y.^[44] This equality, often referred to in terms of arithmetic rank, highlights how flatness aligns homological and geometric dimensions in the fibers, facilitating computations in derived categories and intersection theory. In Noetherian settings, flat morphisms of locally finite type preserve the purity of dimensions in the sense that the dimension additivity ensures consistent relative dimensions locally, complementing the miracle flatness theorem, which conversely guarantees flatness when fiber dimensions match the expected additivity.^[45]

Descent and effective descent

In the context of a faithfully flat morphism f: X \to Y of schemes, descent theory provides a framework for transferring modules from X to Y via a descent datum. Specifically, a quasi-coherent sheaf \mathcal{M} on X is equipped with an isomorphism \rho: p_1^*\mathcal{M} \to p_2^*\mathcal{M} over X \times_Y X, where p_1, p_2 are the projections, such that the cocycle condition holds: the composition p_{13}^*\rho \circ p_{12}^*\rho = p_{23}^*\rho over X \times_Y X \times_Y X.^[46] This datum ensures that \mathcal{M} glues compatibly, allowing reconstruction of a unique quasi-coherent sheaf on Y.^[47] A key result is Grothendieck's theorem on effective descent, which states that if f: X \to Y is flat and surjective, then the category of quasi-coherent sheaves on X with descent data is equivalent to the category of quasi-coherent sheaves on Y.^[46] This equivalence implies that every such sheaf on X descends effectively to Y, establishing flat surjective morphisms as effective descent morphisms in the fpqc topology.^[47] The theorem relies on the faithfully flat case for modules and extends to sheaves via the affine nature of quasi-coherent objects.^[46] Flat base change also enables the descent of geometric properties from X to Y. For instance, if f is flat and X is reduced, then Y is reduced; similarly, normality and regularity of X descend to Y under flat base change, provided the fibers satisfy appropriate conditions like geometric normality or regularity.^[48]^[49]^[50] These results follow from the preservation of nilpotent elements, integrally closed rings, and regular local rings under flat extensions, ensuring that local properties transfer globally.^[51] In modern derived algebraic geometry, flatness plays a crucial role in extending descent to stacks, addressing limitations in classical theory by ensuring compatibility with higher categorical structures. For a flat morphism of spectral schemes, the ∞-category of quasi-coherent sheaves satisfies descent with respect to the flat topology, allowing stacks to be reconstructed from their global sections via hyperdescent.^[52] This framework, developed in the context of ∞-topoi, guarantees that quasi-coherent stacks on derived schemes descend effectively under flat covers, bridging algebraic and derived settings.^[52]

References

[1]
Section 29.25 (01U2): Flat morphisms—The Stacks project
29.25 Flat morphisms. Flatness is one of the most important technical tools in algebraic geometry. In this section we introduce this notion.
[2]
Section 88.13 (0GC8): Flat morphisms—The Stacks project
88.13 Flat morphisms. In this section we define flat morphisms of locally Noetherian formal algebraic spaces. Lemma 88.13.1.
[3]
[PDF] Introduction to ComIllutative Algebra
i) If M and N are flat A-modules, then so is M @A N. ii) If B is a flat A-algebra and N is a flat B-module, then N is flat as an A-module. 9. Let 0 -+ M' -+ ...
[4]
[PDF] Hideyuki Matsumura - Commutative Algebra
Part I is a self-contained exposition of basis concepts such as flatness, dimen- sion, depth, normal rings, and regular local rings. Part II deals with the ...
[5]
Géométrie algébrique et géométrie analytique - Numdam
1-42 · Suivant. Géométrie algébrique et géométrie analytique. Serre, Jean-Pierre. Annales de l'Institut Fourier, Tome 6 (1956), pp. 1-42. Résumé. Toute variété ...
[6]
https://www.numdam.org/item/AIF_1956__6__1_0/
[7]
Section 41.9 (0250): Flat morphisms—The Stacks project
Open immersions are flat. (This is clear because it induces isomorphisms on local rings.) Flat morphisms are stable under base change and composition. Morphisms ...Missing: geometry | Show results with:geometry
[8]
Lemma 10.99.7 (00MK): Local criterion for flatness—The Stacks ...
Lemma 10.99.7 (Local criterion for flatness). Let R \to S be a local homomorphism of local Noetherian rings. Let \mathfrak m be the maximal ideal of R, ...
[9]
Section 10.128 (00R3): More flatness criteria—The Stacks project
Here is the version of the local criterion of flatness for the case of local ring maps which are locally of finite presentation. Then M is flat over R.
[10]
https://stacks.math.columbia.edu/tag/00R3
[11]
Section 27.21 (01OA): Projective bundles—The Stacks project
... projective space bundle over \mathop{\mathrm{Spec}}(k). By the discussion above a k-valued point p of \mathbf{P}(V) corresponds to a surjection of k-vector ...
[12]
41.13 Étale and smooth morphisms - Stacks Project
An étale morphism is smooth of relative dimension zero. The projection \mathbf{A}^ n_ S \to S is a standard example of a smooth morphism of relative dimension ...
[13]
Lemma 10.128.1 (00R4)—The Stacks project
Miracle flatness. Lemma 10.128.1. Let R \to S be a local homomorphism of Noetherian local rings. Assume. R is regular,. S Cohen-Macaulay,. \dim (S) = \dim (R) + ...
[14]
[PDF] arXiv:2501.02639v1 [math.AG] 5 Jan 2025
Jan 5, 2025 · Miracle flatness refers to a specific context in which a family of algebraic varieties is necessarily flat. We shall see that miracle flatness.
[15]
Question about the proof of Theorem 23.1 in Matsumura
Jun 12, 2021 · The question is part of the proof of "Miracle Flatness Theorem". I will give a complete proof of this theorem. Theorem. Let (R,m)→(S,n) be ...
[16]
Section 10.111 (090U): Auslander-Buchsbaum—The Stacks project
Proof. We prove this by induction on \text{depth}(M). The most interesting case is the case \text{depth}(M) = 0. In this case, let. 0 \to R^{n_ e} \to R^{n_{ ...Missing: miracle flatness
[17]
[PDF] MIT Open Access Articles LAGRANGIAN FIBRATIONS OF ...
Mar 11, 2019 · In fact, by miracle flatness, smoothness of B is equivalent to flatness of the morphism. The second theorem implies the first one. Indeed ...
[18]
[PDF] k-moduli of curves on a quadric surface and k3 surfaces - NSF-PAR
Hence, miracle flatness implies that pr1 is flat and hence smooth. The proof is finished. 3. Overview of previous results, Laza–O'Grady, and VGIT. We refer ...
[19]
How to prove that a morphism is flat - Math Stack Exchange
May 9, 2019 · In this case you can use so-called miracle flatness (Hartshorne Exercise III.10.9):. if f:Y→X is a morphism with Y Cohen-Macaulay, X regular ...algebraic geometry - Hartshorne Ex III 9.3(a) - Math Stack ExchangeDoes the fibres being equal dimensional imply flatness?More results from math.stackexchange.com
[20]
https://arxiv.org/pdf/math.AG/0504590
[21]
[PDF] Construction of Hilbert and Quot Schemes - arXiv
Abstract. This is an expository account of Grothendieck's construction of Hilbert and Quot Schemes, following his talk 'Techniques de construction et théor` ...
[22]
[PDF] the hilbert scheme
The Hilbert scheme parameterizes subschemes of projective ... Then there exists a unique closed subscheme Z ∈ Pn. X flat over X, whose restriction to Pn.
[23]
[PDF] arXiv:2504.00789v1 [math.AG] 1 Apr 2025
Apr 1, 2025 · We thus obtain a flat family of bielliptic canonical curves in Pg−1, which induces a rational map P ⇢ Hilb(2g−2)t+1−g,Pg−1 dominating the locus.
[24]
(PDF) The bielliptic locus in the Hilbert scheme of canonical curves ...
In this paper we prove the unirationality of the locus of bielliptic curves in the Hilbert scheme of canonical curves of genus g ≥ 11 g \geq 11 .
[25]
Section 31.32 (01OF): Blowing up—The Stacks project
The blowing up is characterized as the “smallest” scheme over X such that the inverse image of Z is an effective Cartier divisor.
[26]
[PDF] When blowups are flat
May 25, 2017 · A blowup is often given as the standard example of a non-flat morphism. On the other hand, if we have a flat morphism from X to Y and we make.
[27]
[PDF] Flatness, blowups, and valuations - Piotr Achinger
The basic example of a non-flat morphism of schemes is that of a blowup. The deep theorem of. Gruson and Raynaud, dubbed flattening by blowup, implies that in a ...
[28]
[PDF] FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASSES 41 AND 42
We say that a ring homomorphism B → A is flat if A is flat as a B-module. (We don't care about the algebra structure of A.) Here are two key examples of flat ...
[29]
Section 15.69 (0A5M): Projective dimension—The Stacks project
We say K has finite projective dimension if K can be represented by a bounded complex of projective modules.Missing: ε² → flat infinite
[30]
Section 15.67 (0651): Tor dimension—The Stacks project
Jul 24, 2016 · Instead of resolving by projective modules we can look at resolutions by flat modules. This leads to the following concept. Definition 15.67.1.
[31]
Section 10.75 (00LY): Tor groups and flatness—The Stacks project
In this section we use some of the homological algebra developed in the previous section to explain what Tor groups are.Missing: dual | Show results with:dual
[32]
https://stacks.math.columbia.edu/tag/01U6
[33]
https://stacks.math.columbia.edu/tag/01U8
[34]
https://stacks.math.columbia.edu/tag/00HC
[35]
https://stacks.math.columbia.edu/tag/00HI
[36]
https://stacks.math.columbia.edu/tag/00NX
[37]
Lemma 29.48.2 (02KB)—The Stacks project
f is finite locally free,. f is finite, flat, and locally of finite presentation. If S is locally Noetherian these are also equivalent to. f is finite and flat.
[38]
10.39 Flat modules and flat ring maps - Stacks Project
A ring map R \to S is called faithfully flat if S is faithfully flat as an R-module. Here is an example of how you can use the flatness condition. Lemma 10.39.2 ...
[39]
Section 59.27 (03PF): Étale coverings—The Stacks project
... étale covering. Since an étale morphism is flat, and the elements of the covering should cover its target, the property fp (faithfully flat) is satisfied.
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Lemma 29.25.10 (01UA)—The Stacks project
A flat morphism locally of finite presentation is universally open. Proof. This follows from Lemmas 29.25.9 and Lemma 29.23.2 above.<|control11|><|separator|>
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https://stacks.math.columbia.edu/tag/01U9
[42]
https://stacks.math.columbia.edu/tag/02JS
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29.28 Morphisms and dimensions of fibres - Stacks Project
Let f : X \to Y be a morphism of finite type with Y quasi-compact. Then the dimension of the fibres of f is bounded.
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76.26 Cohen-Macaulay morphisms - Stacks Project
Let f : X \to Y be a flat morphism of locally Noetherian algebraic spaces over S. If X is Cohen-Macaulay, then f is Cohen-Macaulay and \mathcal{O}_{Y,
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41.10 Topological properties of flat morphisms - Stacks Project
An important reason to study flat morphisms is that they provide the adequate framework for capturing the notion of a family of schemes parametrized by the ...
[46]
[PDF] Descent for algebraic schemes
With the obvious notion of morphism, the pairs (𝑀,𝜌) consisting of a right 𝐵-module and a descent datum form a category 𝖣𝖾𝗌𝖼(𝐵∕𝐴). Theorem 12.8 (Faithfully flat ...
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Section 59.16 (03O6): Faithfully flat descent—The Stacks project
In this section we discuss faithfully flat descent for quasi-coherent modules. More precisely, we will prove quasi-coherent modules satisfy effective descent ...
[48]
https://stacks.math.columbia.edu/tag/033F
[49]
https://stacks.math.columbia.edu/tag/033G
[50]
https://stacks.math.columbia.edu/tag/07NG
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Section 10.164 (033D): Descending properties—The Stacks project
Descending properties of rings, often easier than ascending, include Noetherian, reduced, normal, and regular rings, and properties (S_k) and (R_k) under ...Missing: reducedness | Show results with:reducedness
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[PDF] Derived Algebraic Geometry XI: Descent Theorems
Sep 28, 2011 · Our main result is that, in many cases, a quasi-coherent stack on. X can be recovered from its ∞-category of global sections (Theorem 8.6). We ...