Faithful Flatness Research Articles

On the ring of D-valued R-polynomials over E

Let [Formula: see text] be an integral domain with quotient field [Formula: see text], [Formula: see text] be an overring of [Formula: see text], [Formula: see text] be an indeterminate over [Formula: see text] and [Formula: see text] be a subset of [Formula: see text]. We consider the ring of [Formula: see text]-valued[Formula: see text]-polynomials on[Formula: see text], denoted by [Formula: see text], formed by the polynomials [Formula: see text] such that [Formula: see text] for each [Formula: see text], that is, [Formula: see text]. In this paper, we study localization properties, local freeness and faithful flatness of [Formula: see text] over various classes of locally essential domains. In particular, we extend some known results to more general settings.

Flatness of Noetherian Hopf algebras over coideal subalgebras

It is proved in the paper that a Noetherian residually finite-dimensional Hopf algebra H is a flat module over any right Noetherian right coideal subalgebra A. In the case when A is a Hopf subalgebra we get faithful flatness. These results are obtained by verifying the existence of classical quotient rings of A and H. It is also proved that the antipode of either right or left Noetherian residually finite-dimensional Hopf algebra is bijective. As a consequence, such a Hopf algebra is right and left Noetherian simultaneously.

Note on integer-valued polynomials on a residually cofinite subset

Let D be an integral domain with quotient field K, E a subset of K and X an indeterminate over K. The ring $$\mathrm {Int}(E,D):=\{f\in K[X]\;:\; f(E)\subseteq D\}$$ , of integer-valued polynomials on E with respect to D, is known to be a D-algebra. Obviously, $$\mathrm {Int}(D,D)=\mathrm {Int}(D)$$ , is the classical ring of integer-valued polynomials over D. In this paper, we study the faithful flatness, the local freeness and the calculation of the Krull dimension of integer-valued polynomials on a residually cofinite subset over locally essential domains. In particular, we prove that $$\mathrm {Int}(E,D)$$ is faithfully flat as a D-module and is of Krull dimension less than that of D[X], when D is a locally essential domain and $$E\subseteq D$$ is residually cofinite with D. Also, we get stronger results in the case of domains that are either almost Krull or t-almost Dedekind.

Subgroups of quantum groups

Abstract We investigate the notion of a subgroup of a quantum group. We suggest a general definition, which takes into account the work that has been done for quantum homogeneous spaces. We further restrict our attention to reductive subgroups, where some faithful flatness conditions apply. Furthermore, we proceed with a categorical approach to the problem of finding quantum subgroups. We translate all existing results into the language of module and monoidal categories and give another characterization of the notion of a quantum subgroup.

Special properties of Hurwitz series rings

In this paper, we study some properties of the Hurwitz series ring HR (resp. Hurwitz polynomial ring hR), such as the flatness or the faithful flatness of HR / (f) (resp. hR / (f)), the strongly Hopfian property and the radical property of HR (resp. hR). We give some sufficient and necessary conditions for HR / (f) (resp. hR / (f)) to be flat or faithful flat. We also prove that the strongly Hopfian property transfer between R and HR (resp. hR), and some radicals of HR can be determined in terms of those of R, in case R satisfies some additional conditions.

On the flatness and the projectivity over Hopf subalgebras of Hopf algebras over Dedekind rings

We study the flatness and the projectivity of Hopf algebras, defined over a Dedekind ring, over their Hopf subalgebras. We give a criterion for the faithful flatness and use it to show the faithful flatness of an arbitrary flat Hopf algebra upon its finite normal Hopf subalgebras. For the projectivity over Hopf subalgebras of a Hopf algebra we need some finiteness condition in terms of the module of integrals. In particular we show that the module of integrals is projective of rank one.

On the Hopf (co)center of a Hopf algebra

The notion of Hopf center and Hopf cocenter of a Hopf algebra is investigated by the extension theory of Hopf algebras. We prove that each of them yields an exact sequence of Hopf algebras. Moreover the exact sequences are shown to satisfy the faithful (co)flatness condition. Hopf center and cocenter are computed for Uq(g) and the Hopf algebra Pol(Gq), where Gq is the Drinfeld–Jimbo quantization of a compact semisimple simply connected Lie group G and g is a simple complex Lie algebra.

Jordan-Hölder theorem for finite dimensional Hopf algebras

We show that a Jordan-Hölder theorem holds for appropriately defined composition series of finite dimensional Hopf algebras. This answers an open question of N. Andruskiewitsch. In the course of our proof we establish analogues of the Noether isomorphism theorems of group theory for arbitrary Hopf algebras under certain faithful (co)flatness assumptions. As an application, we prove an analogue of Zassenhaus’ butterfly lemma for finite dimensional Hopf algebras. We then use these results to show that a Jordan-Hölder theorem holds as well for lower and upper composition series, even though the factors of such series may not be simple as Hopf algebras.

Strong connections and the relative Chern-Galois character for corings

The Chern-Galois theory is developed for corings or coalgebras over non-commutative rings. As the first step the notion of an entwined extension as an extension of algebras within a bijective entwining structure over a non-commutative ring is introduced. A strong connection for an entwined extension is defined and it is shown to be closely related to the Galois property and to the equivariant projectivity of the extension. A generalisation of the Doi theorem on total integrals in the framework of entwining structures over a non-commutative ring is obtained, and the bearing of strong connections on properties such as faithful flatness or relative injectivity is revealed. A family of morphisms between the K0-group of the category of finitely generated projective comodules of a coring and even relative cyclic homology groups of the base algebra of an entwined extension with a strong connection is constructed. This is termed a relative Chern-Galois character. Explicit examples include the computation of a Chern-Galois character of depth 2 Frobenius split (or separable) extensions over a separable algebra R. Finitely generated and projective modules are associated to an entwined extension with a strong connection, the explicit form of idempotents is derived, the corresponding (relative) Chern characters are computed, and their connection with the relative Chern-Galois character is explained.

The monoidal center construction and bimodules

Let C be a cocomplete monoidal category such that the tensor product in C preserves colimits in each argument. Let A be an algebra in C . We show (under some assumptions including “faithful flatness” of A) that the center of the monoidal category ( A C A,⊗ A) of A– A-bimodules is equivalent to the center of C (hence in a sense trivial): Z( A C A)≅ Z( C) . Assuming A to be a commutative algebra in the center Z( C) , we compute the center Z( C A) of the category of right A-modules (considered as a subcategory of A C A using the structure of A∈ Z( C) . We find Z( C A)≅ dys Z( C) A , the category of dyslectic right A-modules in the braided category Z( C) .

Some Relationship between Anti-Integral Extensions of Noetherian Domains

Let R be a Noetherian domain with quotient field K and let α be an anti-integral element of degree d over R. Let β be an elemen of R(α) (resp. R(α,α-1)) such that β is an anti-integral element over R and that R(α) (resp. R(α,α-1)) is integral over R(β)). We shall investigate some properties descending from R(α) (resp. R(α,α-1)) to R(β), i. e., flatness and faithful flatness, and study the ideals J(α), J(β), J(α) and J(β). Let R be a Noetherian domain and R(X) a polynomial ring. Let α be an element of an algebraic extension field L of the quotient field K of R and let π: R(X) →R(α) be the R-algebra homomorphism, sending X to α. Let ψα(X) be the monic minimal polynomial of α over K with deg ψα(X)=d and write

Faithful Flatness of Hopf Algebras

Faithful Flatness of Hopf Algebras

Zariski rings and flatness of completion

We relate flatness and faithful flatness of the completion to the comparison and Zariskian property resp. of the filtration. Some applications have been included.

Ascent of properties for absolutely flat homomorphisms

We show that if A – B is an absolutely flat homomorphism of consultative rings and A is arithmetical, then B is such, thus generalising a previous result (see [9]) on valutation rings. To prove the above theorem we show first that it is true when A is a local ring and B is a local ind-étale homomorphism (in par ticular if B is a strict henselization of A ) , and we apply the following general fact:a local property which ascends to strict henselization and descends by faithful flatness ascends also by ab solutely flat homomorphlsms. This last result also applies to other properties, such as locally noetherian, geometrically unibranche, Rn,Sn, Cohen-Macaulay.

Brauer groups and Amitsur cohomology for general commutative ring extensions

Exact couples are interconnected families of long exact sequences extending the short exact sequences usually derived from spectral sequences. This is exploited to give a long exact sequence connecting Amitsur cohomology groups H> n( S R , U) (where U means the multiplicative group) and H n( S R , Pic) and a third sequence of groups H n ( J), for every faithfully flat commutative R-algebra S. This same sequence is derived in another way without assuming faithful flatness and H n ( J) is identified explicitly as a certain subquotient of a group of isomorphism classes of pairs ( P, α) with P a rank one, projective S n-module and α an isomorphism from the coboundary of P ( in Pic S n + 1 ) to S n + 1 . (Here S n denotes repeated tensor product of S over R.) This last formulation allows us to construct a homomorphism of the relative Brauer group B( S R ) to H 2( J) which is a monomorphism when S is faithfully flat over R, and an isomorphism when some S-module is faithfully projective over R. The first approach also identifies H 2( J) with Ker[ H 2( R, U)→ H 2( S, U)], where H 2( R, U) denotes the ordinary, Grothendieck cohomology (in the étale topology, for example).