1,721,014 research outputs found
Separable Cowreaths Over Clifford Algebras
No full textThe fundamental notion of separability for commutative algebras was interpreted in categorical setting where also the stronger notion of heavily separability was introduced. These notions were extended to (co)algebras in monoidal categories, in particular to cowreaths. In this paper, we consider the cowreath (A⊗H4op,H4,ψ), where H4 is the Sweedler 4-dimensional Hopf algebra over a field k and A= Cl(α, β, γ) is the Clifford algebra generated by two elements G, X with relations G2= α, X2= β and XG+ GX= γ, (α, β, γ∈ k) which becomes naturally an H4-comodule algebra. We show that, when char (k) ≠ 2 , this cowreath is always separable and h-separable as well
Heavily separable cowreaths
No full textMotivated by an example related to the tensor algebra, a stronger version of the notion of separable functor, called heavily separable (h-separable for short), was introduced and investigated in [1]. Here we study h-coseparable coalgebras in monoidal categories with special concern with the monoidal category TA♯ of right transfer morphisms through an algebra A in a monoidal category. We characterize the h-separability of the forgetful functor from the category of entwined modules associated to a cowreath to the base category using suitable Casimir morphisms. Even if there are non trivial examples of h-coseparable coalgebras over a field [1, Theorem 4.4], here we provide non trivial examples of h-coseparable coalgebras in the monoidal category T_{A⊗H^{op}}^ where H=H_4 is the Sweedler 4-dimensional Hopf algebra over a field k and A=Cl(α,β,γ) the Clifford algebra
Heavily separable functors
Full text linkPrompted by an example related to the tensor algebra, we introduce and investigate a stronger version of the notion of separable functor that we call heavily separable. We test this notion on several functors traditionally connected to the study of separability. (C) 2019 Elsevier Inc. All rights reserved
A Schneider type Theorem for Hopf Algebroids
No full textComodule algebras of a Hopf algebroid H with a bijective antipode, i.e. algebra extensions B ⊆ A by H, are studied. Assuming that a lifted canonical map is a split epimorphism of modules of the (noncommutative) base algebra of H, relative injectivity of the H-comodule algebra A is related to the Galois
property of the extension B ⊆ A and also to the equivalence of the category of relative Hopf modules to the category of B-modules. This extends a classical theorem by H.-J. Schneider on Galois extensions by a Hopf algebra. Our main tool is an observation that relative injectivity of a comodule algebra is equivalent to relative separability of a forgetful functor, a notion introduced and analysed hereby
Effects of tillage systems on energy and carbon balance in north-eastern Italy
Full text linkAn energy analysis of three cropping systems with different intensities of soil tillage (conventional tillage-CT, ridge tillage-RT and no-tillage-NT) was done in a loamy-silt soil (fulvi-calcaric Cambisol) at Legnaro, NE Italy (11°58’ E, 45°21’ N, 8 m a.s.l., average rainfall: 822 mm, average temperature: 11.7°C) . This and measurements of the evolution of the organic matter content in the soil also allowed the consequences to be evaluated in terms of CO2 emissions.
The weighted average energy input per hectare was directly proportional to tillage intensity (CT>RT>NT). Compared to CT, total energy savings per hectare were 10% with RT and 32% with NT. Average energy costs per unit production were fairly similar (between 4.5 and 5 MJ kg-1), with differences of 11%. The energy outputs per unit area were highest in CT for all crops, lowest in NT. The RT outputs were on average more similar to CT (-12%). The output/input ratio tended to increase when soil tillage operations were reduced and was 4.09, 4.18 and 4.57 for CT, RT and NT, respectively. As a consequence of fewer mechanical operations and a greater working capacity of the machines, there was lower fuel consumption and a consistently higher organic matter content in the soil with the conservation tillage methods.
These two effects result in a lower CO2 emission into the atmosphere (at 0°C and pressure of 101.3-103 kPa) with respect to CT, of 1190 m3 ha-1 yr-1 in RT and 1553 m3 ha-1 yr-1 in NT. However, the effect due to carbon sequestration as organic matter will decline to zero over a period of years
Liftable pairs of functors and initial objects
No full textLet A and B be monoidal categories and let R : A -> B be a lax monoidal functor. If R has a left adjoint L, it is well-known that the two adjoints induce functors (R) over bar = Alg(R) : Alg(A) -> Alg(B) and (L) under bar = Coalg(L) : Coalg(B) -> Coalg (A) respectively. The pair (L, R) is called liftable if the functor R has a left adjoint and if the functor (L) under bar has a right adjoint. A pleasing fact is that, when A, B and R are moreover braided, a liftable pair of functors as above gives rise to an adjunction at the level of bialgebras. In this note, sufficient conditions on the category A for (R) over bar to possess a left adjoint, are given. Natively these conditions involve the existence of suitable colimits that we interpret as objects which are simultaneously initial in four distinguished categories (among which the category of epiinduced objects), allowing for an explicit construction of (L) under bar, under the appropriate hypotheses. This is achieved by introducing a relative version of the notion of weakly coreflective subcategory, which turns out to be a useful tool to compare the initial objects in the involved categories. We apply our results to obtain an analogue of Sweedler's finite dual for the category of vector spaces graded by an abelian group G endowed with a bicharacter. When the bicharacter on G is skew-symmetric, a lifted adjunction as mentioned above is explicitly described, inducing an auto-adjunction on the category of bialgebras "colored" by G
A categorical proof of a useful result
No full textThe crucial result on vector spaces that can be found in Lemma 9.1.5 in [M. Sweedler, Hopf Algebras, Benjamin, New York, 1969] is proved in the general framework of abelian monoidal categories
On the structure of *-modules
No full textLet A and B be rings, A a subcategory of Mod-A closed under submodules and containing AA, and B a subcategory of Mod-B closed under direct sums and epimorphic images. Then any equivalence between A and B is represented by a bimodule APB with A=EndPB via the functors --⊗AP and HomB(P,–). This was proved by Menini and A. Orsatti [Rend. Sem. Mat. Univ. Padova 82 (1989), 203--231 (1990); MR1049594 (91h:16026)], and such a module is called a ∗-module. For instance, every quasi-progenerator and every tilting module is a ∗-module. The purpose of this paper is, in the authors' words, "to measure the gaps between the classes of ∗-modules, of quasi-progenerators and of tilting modules''. Among the results in this direction, it is shown that every finitely generated ∗-module over a commutative ring is a quasi-progenerator; and that this is not the case in general
Hochschild cohomology and “smoothness” in monoidal categories
No full textAbstractWe introduce and investigate the properties of Hochschild cohomology of algebras in an abelian monoidal category M. We show that the second Hochschild cohomology group of an algebra in M classifies extensions of A up to an equivalence. We characterize algebras of Hochschild dimension 0 (separable algebras), and of Hochschild dimension ≤1 (formally smooth algebras). Several particular cases and applications are included in the last section of the paper
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