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Flat module

A flat module over a ring R is an R-module M such that the tensor product functor -\otimes_R M (or equivalently M \otimes_R -) is exact, meaning it preserves exact sequences of R-modules.^[1] This condition implies that for any injection of R-modules N' \hookrightarrow N, the induced map M \otimes_R N' \to M \otimes_R N is also injective.^[1] The concept 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 coherent sheaves on projective varieties over the complex numbers.^[2] Flat modules generalize free modules, as every free R-module is flat, and they are characterized homologically by the vanishing of the first derived functor \Tor_1^R(N, M) = 0 for all R-modules N.^[3] Every projective module is flat, since projective modules are direct summands of free modules and direct summands preserve exactness under tensor product.^[4] Notable properties include the fact that flatness is preserved under arbitrary direct limits, base change, and localization: if M is flat over R, then M \otimes_R R' is flat over any R-algebra R', and for any multiplicative set S \subset R, M is flat over R if and only if the localized module S^{-1}M is flat over S^{-1}R.^[1] Over principal ideal domains, flat modules are precisely the torsion-free modules.^[4] Moreover, a finitely presented flat module over a commutative ring is projective.^[3] In algebraic geometry, flat modules underlie the notion of flat morphisms of schemes, which ensure that the dimensions and arithmetic properties of fibers vary "continuously" across the base.^[5]

Definition and Characterizations

Definition

In ring theory, a right R-module M is called flat if, for every injective R-linear map f: {}_R N \to {}_R P of left R-modules, the induced map M \otimes_R f: M \otimes_R N \to M \otimes_R P is also injective. Equivalently, the functor -\otimes_R M preserves all exact sequences of left R-modules. This notion 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 coherent sheaves.^[2] A basic consequence is that every free right R-module is flat. To verify this, let F = \bigoplus_{i \in I} R e_i be free on basis \{e_i\}_{i \in I}, and consider an injection N \to P. Then F \otimes_R N \to F \otimes_R P decomposes as the direct sum \bigoplus_{i \in I} (N \to P), and since each component N \to P is injective and direct sums preserve injectivity, the overall map is injective.^[1]

Equivalent Characterizations

A flat right R-module M is equivalently characterized by the condition that for every (left) ideal I \subseteq R, the natural map I \otimes_R M \to R \otimes_R M \cong M is injective, meaning IM = I \otimes_R M.^[6] This injectivity ensures that the tensor product preserves the exactness of the sequence $0 \to I \to R$. This condition can be restricted to finitely generated ideals without loss of generality: M is flat if and only if I \otimes_R M \to M is injective for every finitely generated ideal I \subseteq R.^[6] The equivalence arises because arbitrary ideals are directed colimits of their finitely generated subideals, and tensor products commute with these colimits, preserving exactness. Over a commutative ring R, flatness of an R-module M is equivalently given by the vanishing \Tor_1^R(R/I, M) = 0 for all ideals I \subseteq R.^[7] Since R/I is the cokernel of I \to R, this condition tests flatness on cyclic quotients by ideals. The ideal characterization relates to the general preservation of exact sequences because every module admits a presentation as a cokernel of a map between free modules, and free modules are flat; the ideal condition suffices to verify exactness in such presentations via finite approximations and colimit arguments.^[6]

Properties and Relations

Relation to Torsion-Free Modules

Over an integral domain R, every flat R-module is torsion-free.^[8] To see this, consider a nonzero element r \in R. The sequence

$0 \to R \xrightarrow{\cdot r} R

is exact, and tensoring with a flat module M yields the exact sequence

$0 \to M \xrightarrow{\cdot r} M,

so multiplication by r on M is injective. Thus, no nonzero element of M is annihilated by a nonzero element of R, meaning M has no torsion elements.^[9] The converse does not hold in general: there exist torsion-free modules over integral domains that are not flat. A standard counterexample is the ideal M = (x, y)R in the polynomial ring R = k[x, y] over a field k. As a submodule of the torsion-free module R, M is torsion-free, but it fails to be flat because no nonzero proper ideal in a polynomial ring in two or more indeterminates is flat.^[10]^[11] However, the converse holds in special cases. For instance, over a Dedekind domain R, an R-module is flat if and only if it is torsion-free; in the finitely generated case, such modules are even projective.^[8] This equivalence arises because Dedekind domains are Prüfer domains, where torsion-freeness ensures the module preserves exactness under tensor products.^[10] Flatness is a stronger condition than torsion-freeness, as it requires the module to preserve exactness of all short exact sequences upon tensoring, not merely the injectivity of multiplication maps by nonzero domain elements.^[8] Free modules over any ring are flat and hence torsion-free over domains, but the relations highlight that flatness captures a broader homological property.^[9]

Relation to Free and Projective Modules

Projective modules form a subclass of flat modules. To see this, recall that a module P is projective if it is a direct summand of some free module F = R^{(I)} for a set I, so P \oplus Q \cong F for some Q. Free modules are flat because the functor -\otimes_R F is exact: it preserves exact sequences as tensoring with a direct sum of copies of R reduces to the identity functor on modules being exact. Moreover, direct summands of flat modules are flat, since if $0 \to A \to B \to C \to 0 is exact and M is flat with M \cong P \oplus Q, then the sequence remains exact after tensoring with M by additivity of the tensor product. Thus, every projective module is flat.^[12] Free modules are a special case of projective modules. A free module F = R^{(I)} is projective because any surjection R^{(J)} \twoheadrightarrow N admits a section when J is sufficiently large, allowing homomorphisms from F to lift over surjections via basis selection. Consequently, every free module is projective and hence flat.^[13] The inclusion is proper: flat modules need not be projective. A standard example occurs over the ring R = C^\infty(\mathbb{R}) of smooth real-valued functions on \mathbb{R}, where the finitely generated module M = R / I—with I the ideal of functions vanishing to infinite order at $0—is flat but not projective. This module is finitely presented and flat yet fails projectivity due to the non-Noetherian nature of R and the specific embedding properties of I.^[14] Over principal ideal domains (PIDs), the notions align more closely in the finitely generated case. For a PID R, an R-module is flat if and only if it is torsion-free.^[4] Furthermore, a finitely generated torsion-free module over a PID is free by the structure theorem, which decomposes such modules as direct sums of copies of R. Thus, over PIDs, finitely generated flat modules are precisely the free modules (and hence projective). This identifies flatness with the weaker torsion-free condition in the finitely generated setting, though torsion-freeness alone does not imply projectivity in general.^[15]

Non-Examples

A prominent non-example of a flat module is the quotient \mathbb{Q}/\mathbb{Z} over the ring of integers \mathbb{Z}. This module is torsion, meaning every element has finite order, and thus it is not torsion-free; as established in the relation to torsion-free modules, flat \mathbb{Z}-modules coincide precisely with torsion-free ones.^[16] Finite abelian groups provide further illustrations of non-flat modules over \mathbb{Z}. For instance, \mathbb{Z}/n\mathbb{Z} for any integer n > 1 is not flat. These are torsion modules, hence not torsion-free and therefore not flat over \mathbb{Z}. Explicitly, flatness fails the preservation of exact sequences: the short exact sequence $0 \to \mathbb{Z} \xrightarrow{\cdot n} \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} \to 0 tensors to $0 \to \mathbb{Z}/n\mathbb{Z} \xrightarrow{0} \mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} \to 0, which is not exact at the middle term since the zero map is not injective.^[17] Over the polynomial ring k[x, y] in two variables where k is a field, the residue field module k[x, y]/(x, y) \cong k is not flat. This module is supported solely at the maximal ideal (x, y) corresponding to the origin and fails flatness because it does not preserve the exactness of the sequence $0 \to (x, y) \to k[x, y] \to k \to 0; tensoring with k yields $0 \to (x, y)/(x, y)^2 \to k \to k \to 0, where (x, y)/(x, y)^2 \cong k^2 maps to zero (as elements of the maximal ideal act trivially on k), rendering the map non-injective.^[16]

Behavior under Direct Sums, Products, and Limits

Flat modules are preserved under the formation of direct sums. Specifically, the direct sum of any family of flat modules over a ring R is flat, as the tensor product functor with a fixed module preserves colimits, and direct sums are colimits in the category of modules.^[18] This holds for arbitrary index sets, so arbitrary coproducts of flat modules are flat.^[18] In contrast, products of flat modules do not necessarily preserve flatness. For a ring R, arbitrary direct products of flat R-modules are flat if and only if R is coherent, meaning every finitely generated ideal of R is finitely presented.^[19] Over commutative rings, this condition may fail; for example, the infinite direct product of copies of the ring R = \mathbb{Q}[y, x_i \mid i \in \mathbb{N}] / (y x_i)_{i \in \mathbb{N}} is not flat, even though each copy of R is flat as an R-module.^[20] Over non-commutative rings, counterexamples exist even when coherence holds in one direction, but the equivalence to coherence persists for the appropriate side (left or right). A seminal result characterizes this behavior precisely in terms of coherence. Regarding limits, flatness is preserved under inverse limits of flat modules provided the inverse system satisfies the Mittag-Leffler condition. This condition ensures that the images of the transition maps stabilize in a certain way, allowing the tensor product to commute with the limit up to exactness.^[21] For instance, in the context of completions over Noetherian rings, if each module in the system is flat over the corresponding quotient ring and the system is Mittag-Leffler, the inverse limit remains flat.^[22]

Homological Aspects

Characterization via Tor Functors

A right R-module M is flat if and only if \Tor_i^R(N, M) = 0 for all left R-modules N and all i \geq 1.^[23]^[24] This homological characterization arises from viewing flatness through the lens of derived functors in homological algebra, where the Tor groups measure the deviation of the tensor product functor from exactness.^[25] The equivalence to the preservation of exactness under tensoring follows from the right exactness of the tensor functor - \otimes_R M, which always preserves surjections.^[23] Vanishing of the higher Tor groups \Tor_i^R(N, M) for i \geq 2 is automatic once \Tor_1^R(N, M) = 0 for all N, as this condition ensures the functor is left exact, hence fully exact.^[24] Specifically, \Tor_1^R(N, M) = 0 for all N implies that for any injection $0 \to K \to L, the induced map K \otimes_R M \to L \otimes_R Mis injective, completing the exactness preservation.[25] In practice, it suffices to verify\Tor_1^R(R/\mathfrak{a}, M) = 0for all ideals\mathfrak{a} \subseteq R$ to establish flatness, as this captures the necessary injectivity conditions.^[23] The Tor groups are computed using projective resolutions: take a projective resolution \cdots \to P_1 \to P_0 \to N \to 0 of N, tensor with M to form the complex \cdots \to P_1 \otimes_R M \to P_0 \otimes_R M \to 0, and \Tor_i^R(N, M) is the i-th homology of this complex.^[26] For flat M, the tensored complex remains exact (up to homology in degree 0), so all higher homology groups vanish.^[24] This computational approach highlights why flatness aligns with the absence of "torsion" in the homological sense.^[25] The condition \Tor_1^R(N, M) = 0 for all left R-modules N is equivalent to the full vanishing \Tor_i^R(N, M) = 0 for all i \geq 1 and all N. This equivalence holds because \Tor_1 = 0 implies M is flat, and flatness ensures all higher Tor groups vanish.^[23]

Flat Resolutions

In homological algebra, a flat resolution of a module M over a ring R is an exact sequence

\cdots \to F_2 \to F_1 \to F_0 \to M \to 0

in which each F_i is a flat R-module.^[27] Every R-module admits a flat resolution. Since free modules are flat and every module has a free resolution, the existence follows immediately from the construction of free resolutions via the forgetful functor to sets or by iteratively embedding into free modules.^[28]^[29] Flat resolutions generalize projective resolutions, as every projective module is flat, so any projective resolution is a flat resolution. However, the converse does not hold, and the minimal length of a flat resolution (the flat dimension of M) is at most the projective dimension of M. For instance, over the ring of integers \mathbb{Z}, the rationals \mathbb{Q} form a flat \mathbb{Z}-module (as torsion-free modules over principal ideal domains are flat) with flat dimension 0, but its projective dimension is 1, as \mathbb{Q} is not free.^[28]^[29] Flat resolutions play a key role in computing Tor groups: given a flat resolution F_\bullet \to M of M, the groups \Tor_i^R(M, N) are the homology groups of the complex F_\bullet \otimes_R N for any R-module N. This holds because flatness ensures that tensoring the resolution with N preserves exactness in sufficiently high degrees, mirroring the behavior for projective resolutions but potentially allowing shorter or simpler constructions when projectives are unavailable.^[27]

Flatness in Ring Extensions and Geometry

Flat Ring Extensions

A flat ring extension occurs when an R-algebra S is flat as an R-module, meaning that the tensor functor - \otimes_R S preserves exact sequences of R-modules.^[1] This property ensures that exactness in the category of R-modules is preserved under base change to S, which is crucial for transferring homological information between the two rings.^[24] A prominent example of a flat ring extension is the polynomial ring R over R, which is free as an R-module with basis \{1, x, x^2, \dots \}, and hence flat. More generally, polynomial rings in any number of variables R[x_1, \dots, x_n] are faithfully flat over R.^[24] If S is flat over R, then for any S-module M, the restriction of scalars yields an R-module that relates back through the adjunction with induction, preserving flatness: specifically, if M is flat over S, then M is flat over R, as flatness composes under the change of rings. As a non-example, consider the R-algebra S = R/(2x) over R = \mathbb{Z}. Here, S \cong \mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} as a \mathbb{Z}-module, which is not flat due to the torsion submodule \mathbb{Z}/2\mathbb{Z}.^[30] This fails in characteristic not dividing 2; in characteristic 2, the ideal (2x) = (0), so S = R is flat.^[31]

Local Nature of Flatness

In commutative algebra, flatness of a module is a local property with respect to the prime ideals of the base ring. Specifically, let R be a commutative ring and M an R-module. Then M is flat over R if and only if the localized module M_\mathfrak{p} is flat over the localized ring R_\mathfrak{p} for every prime ideal \mathfrak{p} \subset R.^[1]^[32] This equivalence follows from the fact that the tensor product functor commutes with localization. For any R-module N, there is a natural isomorphism (N \otimes_R M)_\mathfrak{p} \cong N_\mathfrak{p} \otimes_{R_\mathfrak{p}} M_\mathfrak{p}.^[33] Thus, if $0 \to N' \to N \to N'' \to 0 is a short exact sequence of R-modules, tensoring with M yields an exact sequence if and only if the localized sequence $0 \to N'_\mathfrak{p} \to N_\mathfrak{p} \to N''_\mathfrak{p} \to 0 remains exact after tensoring with M_\mathfrak{p} over R_\mathfrak{p} for every prime \mathfrak{p}. Conversely, suppose the tensor product with M fails to be exact globally, so the kernel K of N' \otimes_R M \to N \otimes_R M is nonzero. Then K_\mathfrak{p} \neq 0 for some \mathfrak{p}, contradicting local flatness. Since exactness of sequences is preserved under localization (a local property), global flatness holds if and only if it holds locally.^[32]^[34] Equivalently, in the context of the spectrum \operatorname{Spec}(R), flatness of M is equivalent to flatness of the stalk M_\mathfrak{p} over R_\mathfrak{p} at every point \mathfrak{p} \in \operatorname{Spec}(R). This stalkwise characterization underscores the geometric interpretation of flatness in algebraic geometry, where it ensures that tensoring preserves exactness on fibers or stalks.^[1] In the non-commutative setting, however, flatness is not generally a local property in an analogous sense, as the construction of localizations requires additional conditions (such as the Ore condition) on the multiplicative sets, and the commutation with tensor products may fail without commutativity.

Flat Morphisms of Schemes

A morphism f: X \to Y of schemes is called flat if, for every point x \in X, the stalk \mathcal{O}_{X,x} is a flat module over the stalk \mathcal{O}_{Y,f(x)} via the induced map on local rings.^[35] Equivalently, the morphism is flat if the structure sheaf \mathcal{O}_X is a flat sheaf of f^{-1}\mathcal{O}_Y-modules.^[35] This geometric notion extends the algebraic concept of flatness from modules over rings to the setting of schemes, where flatness is checked locally on affine opens: if f restricts to a morphism \operatorname{Spec} B \to \operatorname{Spec} A on affine opens, then B must be flat as an A-module.^[35] Flatness of schemes manifests in several key properties that underpin relative dimension theory in algebraic geometry. For instance, if f is faithfully flat, then it is an open morphism, meaning the image of open sets in X remains open in Y.^[35] Moreover, flat morphisms preserve the dimensions of fibers in a stable way under base change, ensuring that the arithmetic genus and other invariants of fibers remain constant across families.^[35] These properties arise because flatness implies that tensor products with the structure sheaf remain exact, preventing torsion or dimension jumps in the fibers. A prominent example of a flat morphism is a smooth morphism of schemes, which is locally of finite presentation and flat by definition, as the local rings satisfy the necessary regularity conditions for flatness.^[35] For instance, the projection \mathbb{A}^n_k \to \operatorname{Spec} k is smooth and hence flat over any field k.^[35] The connection to module theory is direct through the stalks: flatness of f at a point corresponds precisely to the local ring of the source being flat over the local ring of the target, mirroring the torsion-freeness and exactness criteria for modules.^[35] This local characterization underscores why flat morphisms geometrize the module-theoretic notion in the scheme setting. The concept of flat morphisms was formalized by Alexander Grothendieck in Éléments de géométrie algébrique (EGA IV), where it plays a central role in developing relative dimension and intersection theory for families of varieties.

Faithful Flatness

Definition and Basic Examples

In the context of modules over a commutative ring R, an R-module M is faithfully flat if M is flat and the functor -\otimes_R M from the category of R-modules to itself is faithful in the categorical sense, meaning it reflects exactness: a sequence of R-modules is exact if and only if the tensored sequence -\otimes_R M is exact.^[1] This condition ensures that M not only preserves exactness (due to flatness) but also detects it, providing a stronger tool for studying module structures.^[24] An equivalent characterization is that M is flat and M \otimes_R N = 0 implies N = 0 for every R-module N.^[24] This annihilation condition highlights the "detecting" property, distinguishing faithfully flat modules from merely flat ones. Basic examples include free R-modules of positive finite rank, which are faithfully flat since they preserve and reflect exactness via the properties of bases.^[24] Over a field k, any nonzero vector space V is faithfully flat, as all k-modules are flat (being free) and V \otimes_k W = 0 if and only if W = 0.^[3] In contrast, the countably infinite direct sum \bigoplus_{n=1}^\infty \mathbb{Z} over \mathbb{Z} is flat (as a direct sum of flat modules) but not faithfully flat, since there exists a nonzero \mathbb{Z}-module N (such as the cokernel of the inclusion \bigoplus \mathbb{Z} \to \prod \mathbb{Z}) with \bigoplus \mathbb{Z} \otimes_\mathbb{Z} N = 0.^[3]

Faithfully Flat Local Homomorphisms

A local homomorphism \phi: (R, \mathfrak{m}) \to (S, \mathfrak{n}) between local rings is faithfully flat if S is flat as an R-module.^[36] Flatness ensures the faithfulness property and that the induced morphism \operatorname{Spec}(S) \to \operatorname{Spec}(R) has a non-empty closed fiber.^[37] Such homomorphisms are equivalently characterized as flat maps where the induced morphism \mathrm{Spec}(S) \to \mathrm{Spec}(R) has a non-empty closed fiber.^[37] For local rings, the unique closed point corresponding to \mathfrak{m} must lie in the image, which follows from flatness ensuring the fiber ring S/\mathfrak{m}S is nonzero.^[36] This geometric perspective highlights how faithfully flat local homomorphisms maintain surjectivity on closed points, distinguishing them from general flat maps. A canonical example is the natural map from a Noetherian local ring (R, \mathfrak{m}) to its completion \hat{R} = \varprojlim R/\mathfrak{m}^n, which is faithfully flat.^[38] The flatness arises from the inverse limit construction preserving exact sequences in the Noetherian setting, while \mathfrak{m}\hat{R} = \mathfrak{n} holds for the maximal ideal \mathfrak{n} of \hat{R}.^[38] Faithfully flat local homomorphisms play a key role in descent theory, allowing properties such as Noetherianity, reducedness, normality, and regularity to descend from S to R.^[39] They also underpin variants of Hensel's lemma, where henselization or completion enables lifting ideals or solutions modulo \mathfrak{m} to the target ring while preserving exactness.^[40] These maps preserve applications of Nakayama's lemma, ensuring that local freeness or generation properties of modules over S imply analogous properties over R via base change and contraction.^[41] For instance, if a finitely generated S-module is projective, its pullback to R is projective, as verified using Nakayama after tensoring with residue fields.^[41]

Advanced Topics

Flat Covers

In module theory, a flat cover of an R-module M is a surjective R-module homomorphism \pi: F \to M, where F is a flat R-module, such that for any other flat R-module F' and surjective homomorphism \sigma: F' \to M, there exists a unique R-module homomorphism \phi: F' \to F satisfying \pi \circ \phi = \sigma. Moreover, if h: F \to F is an endomorphism with \pi \circ h = \pi, then h is an automorphism of F, ensuring minimality. The existence of flat covers for every R-module M was established in 2001 using the complete cotorsion pair generated by flat modules and their left orthogonal class.^[42] Earlier, in the 1990s, flat covers were proven to exist for all modules over right coherent rings, leveraging properties of pure-injective modules and Auslander-Buchweitz approximations.^[43] Flat covers are closely related to pure submodules: the kernel of a flat cover \pi: F \to M is a pure submodule of F, and constructing flat covers often involves quotienting flat modules by pure submodules to achieve the universal property. This connection arises because pure submodules of flat modules remain flat, facilitating the approximation process.^[44] In approximation theory, flat covers provide minimal flat approximations for modules over artinian rings, where the structure of such covers reveals information about minimal flat resolutions of artinian modules. For instance, over a noetherian ring, the flat cover of an artinian module decomposes into direct sums of indecomposable flat modules, aiding the study of their homological properties.^[45] Flat covers are unique up to isomorphism in categories admitting pullbacks, as the universal property ensures that any two flat covers of the same module are connected by an isomorphism compatible with the surjections.

Flat Modules in Constructive Mathematics

In constructive mathematics, the classical characterization of flat modules via the vanishing of the first Tor functor relies on the law of excluded middle to establish certain exactness properties and the existence of suitable resolutions. This non-constructive principle is avoided by adopting a direct definition: an R-module M is flat if the functor -\otimes_R M preserves all exact sequences of R-modules. This equivalence to \Tor_1^R(N, M) = 0 for all R-modules N holds only under additional axioms such as the axiom of choice, which is needed to construct free resolutions without which Tor groups cannot be defined for arbitrary modules.^[46] Over the ring of integers \mathbb{Z}, free modules remain flat in the constructive sense, as tensoring with a free module preserves exactness by explicit construction of isomorphisms and injections. Torsion-free \mathbb{Z}-modules are also flat constructively, since the proof exploits the principal ideal structure of \mathbb{Z} to show that multiplication by non-zero integers induces injective maps on tensors, without invoking excluded middle. However, modules that are classically flat via proofs depending on non-constructive case analysis may not be provably flat without such principles; for instance, certain torsion-free modules over more complex rings require decidability assumptions not generally available.^[46] Work by Thierry Coquand and collaborators on geometric logic provides a constructive framework for sheaves on sites, where flat functors—generalizing algebraic flatness—play a central role in preserving finite limits and ensuring the coherence of sheaf conditions without choice axioms. This approach interprets classical algebraic geometry intuitionistically, using flat functors to model descent and gluing in toposes, thus extending the utility of flat modules to constructive sheaf theory.^[47] A key implication is that flat covers, which exist classically for every module via Zorn's lemma, may not exist in constructive mathematics without axioms of choice, as their construction depends on maximal elements in partially ordered sets of flat extensions. This limitation underscores the need for alternative dynamical methods in constructive homological algebra to approximate or replace such covers.

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[28]
Section 15.59 (06XY): Derived tensor product—The Stacks project
In fact, it turns out that the boundedness assumptions are not necessary, provided we choose K-flat resolutions. Definition 15.59.1. Let R be a ring. A complex ...<|control11|><|separator|>
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[PDF] Homological Algebra - Purdue Math
flat, projective, injective). (Flat modules are defined in Definition 2.3, projective modules in. Section 5 and injective modules in Section 24.) An exact ...
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Section 10.75 (00LY): Tor groups and flatness—The Stacks project
10.75 Tor groups and flatness. In this section we use some of the homological algebra developed in the previous section to explain what Tor groups are.Missing: characterization | Show results with:characterization
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Can a quotient ring R/J ever be flat over R? - MathOverflow
Oct 8, 2009 · You have to be careful. If you put that A-module structure on A, it is not flat. In your construction, the quotient you consider is A/(x_1), ...
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Example for $R[X]/(rX)$ is a flat $R$-module - Math Stack Exchange
Nov 7, 2017 · Then R[X]/(rX) is not flat as an R algebra if R is an integral domain. But what can we say if R is not an integral domain ? EDIT: Consider a ...Prove that the polynomial ring $R[x]$ is a flat $R$-module.ring theory - Can it be that $R[[x]]$ is flat over $R$ but not over $R[x]$?More results from math.stackexchange.com
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[PDF] Introduction to Commutative Algebra - OSU Math
Apr 8, 2008 · of an A-module homomorphism is a local property. 8. Page 15. Chapter 1 ... Since A[x] is a free A-module, A[x] is a flat A module.
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https://stacks.math.columbia.edu/tag/00HJ
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https://stacks.math.columbia.edu/tag/01U2
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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.
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Lemma 10.39.17 (00HR)—The Stacks project
The Stacks project · bibliography · blog · Table of contents; Part 1 ... Lemma 10.39.17. A flat local ring homomorphism of local rings is faithfully ...
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Lemma 10.39.16 (00HQ)—The Stacks project
### Summary of Lemma 10.39.16 (00HQ) from The Stacks Project
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Lemma 10.97.3 (00MC)—The Stacks project
In particular, if (R, \mathfrak m) is a Noetherian local ring, then the completion \mathop{\mathrm{lim}}\nolimits _ n R/\mathfrak m^ n is faithfully flat.
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Section 10.164 (033D): Descending properties—The Stacks project
In this section we start proving some algebraic facts concerning the “descent” of properties of rings. It turns out that it is often “easier” to descend ...
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15.45 Permanence of properties under henselization - Stacks Project
The Stacks project · bibliography · blog · Table of contents; Part 1 ... As a flat local ring homomorphism is faithfully flat (Algebra, Lemma 10.39.17) ...
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https://academic.oup.com/blms/33.4.413
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Commutative algebra: Constructive methods.Finite projective modules
This book is an introductory course to basic commutative algebra with a par- ticular emphasis on finitely generated projective modules, ...