scholarly journals Equivariant splitting of the Hodge–de Rham exact sequence

2021 ◽  
Author(s):  
Jędrzej Garnek
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Short Exact Sequence ◽  
Algebraic Curve ◽  
Hyperelliptic Curves ◽  
Witt Vectors ◽  
De Rham ◽  
A Finite Group

AbstractLet X be an algebraic curve with a faithful action of a finite group G over a field k. We show that if the Hodge–de Rham short exact sequence of X splits G-equivariantly then the action of G on X is weakly ramified. In particular, this generalizes the result of Köck and Tait for hyperelliptic curves. We discuss also converse statements and tie this problem to lifting coverings of curves to the ring of Witt vectors of length 2.

The mod ℭ Suspension Theorem

1969 ◽  
Vol 21 ◽  
pp. 684-701 ◽  
Author(s):  
Benson Samuel Brown
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Abelian Groups ◽  
Finitely Generated ◽  
Finite Abelian Groups ◽  
Cw Complexes ◽  
Image Position ◽  
A Finite Group

Our aim in this paper is to prove the general mod ℭ suspension theorem: Suppose that X and Y are CW-complexes,ℭ is a class offinite abelian groups, and that(i) πi(Y) ∈ℭfor all i < n,(ii) H*(X; Z) is finitely generated,(iii) Hi(X;Z) ∈ℭfor all i > k.Then the suspension homomorphismis a(mod ℭ) monomorphism for 2 ≦ r ≦ 2n – k – 2 (when r= 1, ker E is a finite group of order d, where Zd∈ ℭ and is a (mod ℭ) epimorphism for 2 ≦ r ≦ 2n – k – 2The proof is basically the same as the proof of the regular suspension theorem. It depends essentially on (mod ℭ) versions of the Serre exact sequence and of the Whitehead theorem.

scholarly journals Faithfulness of Actions on Riemann-Roch Spaces

2015 ◽  
Vol 67 (4) ◽  
pp. 848-869 ◽  
Author(s):  
Bernhard Köck ◽  
Joseph Tait
Keyword(s):  
Finite Group ◽  
Riemann Surface ◽  
Coding Theory ◽  
Algebraic Curve ◽  
Deformation Theory ◽  
Analogous Problem ◽  
Invariant Divisor ◽  
A Finite Group

AbstractGiven a faithful action of a finite groupGon an algebraic curveXof genusgX≥ 2, we giveexplicit criteria for the induced action ofGon the Riemann–Roch spaceH0(X,OX(D)) to be faithful,whereDis aG-invariant divisor on X of degree at least 2gX− 2. This leads to a concise answer to the question of when the action ofGon the spaceH0(X,Ωx⊗m) of global holomorphic polydifferentials of order m is faithful. IfXis hyperelliptic, we provide an explicit basis of H0(X,Ωx⊗m). Finally, we giveapplications in deformation theory and in coding theory and discuss the analogous problem for theaction ofGon the first homologyH1(X,ℤ/mℤ) ifXis a Riemann surface.

Natural maps of extension functors and a theorem of R. G. Swan

1961 ◽  
Vol 57 (3) ◽  
pp. 489-502 ◽  
Author(s):  
P. J. Hilton ◽  
D. Rees
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Group Ring ◽  
Projective Module ◽  
Additive Group ◽  
Integral Group Ring ◽  
Integral Group ◽  
Natural Maps ◽  
Image Position ◽  
A Finite Group

The present paper has been inspired by a theorem of Swan(5). The theorem can be described as follows. Let G be a finite group and let Γ be its integral group ring. We shall denote by Z an infinite cyclic additive group considered as a left Γ-module by defining gm = m for all g in G and m in Z. By a Tate resolution of Z is meant an exact sequencewhere Xn is a projective module for − ∞ < n < + ∞, and.

Explicit homotopy equivalences in dimension two

2002 ◽  
Vol 133 (3) ◽  
pp. 411-430 ◽  
Author(s):  
F. E. A. JOHNSON
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Finitely Generated ◽  
Stable Class ◽  
A Finite Group

Let G be a finite group; by an algebraic 2-complex over G we mean an exact sequence of Z[G]-modules of the form:E = (0 → J → E2 → E1 → E0 → Z → 0)where Er is finitely generated free over Z[G] for 0 [les ] r [les ] 2. The module J is determined up to stability by the fact of appearing in such an exact sequence; we denote its stable class by Ω3(Z); E is said to be minimal when rkZ(J) attains the minimum possible value within Ω3(Z).

scholarly journals Cohomology Theory for Non-Normal Subgroups and Non-Normal Fields

1954 ◽  
Vol 2 (2) ◽  
pp. 66-76 ◽  
Author(s):  
Iain T. Adamson
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Cohomology Group ◽  
Cohomology Theory ◽  
Normal Subgroups ◽  
Cohomology Groups ◽  
Coset Space ◽  
Image Position ◽  
A Finite Group

Let G be a finite group, H an arbitrary subgroup (i.e., not necessarily normal); we decompose G as a union of left cosets modulo H:choosing fixed coset representatives v. In this paper we construct a “coset space complex” and assign cohomology groups; Hr([G: H], A), to it for all coefficient modules A and all dimensions, -∞<r<∞. We show that ifis an exact sequence of coefficient modules such that H1U, A')= 0 for all subgroups U of H, then a cohomology group sequencemay be defined and is exact for -∞<r<∞. We also provide a link between the cohomology groups Hr([G: H], A) and the cohomology groups of G and H; namely, we prove that if Hv(U, A)= 0 for all subgroups U of H and for v = 1, 2, …, n–1, then the sequenceis exact, where the homomorphisms of the sequence are those induced by injection, inflation and restriction respectively.

scholarly journals Projective characters of degree one and the inflation-restriction sequence

1989 ◽  
Vol 46 (2) ◽  
pp. 272-280 ◽  
Author(s):  
R. J. Higgs
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Exact Sequence ◽  
Schur Multiplier ◽  
Original Result ◽  
Image Position ◽  
A Finite Group ◽  
Invariant Element

AbstractLet G be a finite group, α be a fixed cocycle of G and Proj (G, α) denote the set of irreducible projective characters of G lying over the cocycle α.Suppose N is a normal subgroup of G. Then the author shows that there exists a G- invariant element of Proj(N, αN) of degree 1 if and only if [α] is an element of the image of the inflation homomorphism from M(G/N) into M(G), where M(G) denotes the Schur multiplier of G. However in many situations one can produce such G-invariant characters where it is not intrinsically obvious that the cocycle could be inflated. Because of this the author obtains a restatement of his original result using the Lyndon-Hochschild-Serre exact sequence of cohomology. This restatement not only resolves the apparent anomalies, but also yields as a corollary the well-known fact that the inflation-restriction sequence is exact when N is perfect.

scholarly journals Extensions of G-posets and Quillen's complex

1994 ◽  
Vol 57 (1) ◽  
pp. 60-75 ◽  
Author(s):  
Yoav Segev ◽  
Peter Webb
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Exact Sequence ◽  
Partially Ordered Sets ◽  
Topological Spaces ◽  
Ordered Sets ◽  
Wedge Product ◽  
Partially Ordered ◽  
Steinberg Module ◽  
A Finite Group

AbstractWe develop techniques to compute the homology of Quillen's complex of elementary abelian p-subgroups of a finite group in the case where the group has a normal subgroup of order divisible by p. The main result is a long exact sequence relating the homologies of these complexes for the whole group, the normal subgroup, and certain centralizer subgroups. The proof takes place at the level of partially-ordered sets. Notions of suspension and wedge product are considered in this context, which are analogous to the corresponding notions for topological spaces. We conclude with a formula for the generalized Steinberg module of a group with a normal subgroup, and give some examples.

scholarly journals BLOCKS WITH NORMAL ABELIAN DEFECT AND ABELIAN p′ INERTIAL QUOTIENT

2019 ◽  
Vol 70 (4) ◽  
pp. 1437-1448
Author(s):  
David Benson ◽  
Radha Kessar ◽  
Markus Linckelmann
Keyword(s):  
Finite Group ◽  
Semidirect Product ◽  
Group Algebra ◽  
Explicit Description ◽  
Closed Field ◽  
Characteristic P ◽  
Witt Vectors ◽  
Frobenius Numbers ◽  
A Finite Group ◽  
Key Steps

Abstract Let $k$ be an algebraically closed field of characteristic $p$, and let ${\mathcal{O}}$ be either $k$ or its ring of Witt vectors $W(k)$. Let $G$ be a finite group and $B$ a block of ${\mathcal{O}} G$ with normal abelian defect group and abelian $p^{\prime}$ inertial quotient $L$. We show that $B$ is isomorphic to its second Frobenius twist. This is motivated by the fact that bounding Frobenius numbers is one of the key steps towards Donovan’s conjecture. For ${\mathcal{O}}=k$, we give an explicit description of the basic algebra of $B$ as a quiver with relations. It is a quantized version of the group algebra of the semidirect product $P\rtimes L$.

scholarly journals Extensions of tensor categories by finite group fusion categories

2019 ◽  
pp. 1-29
Author(s):  
SONIA NATALE ◽  
Keyword(s):  
Finite Group ◽  
Exact Sequence ◽  
Fusion Category ◽  
Matched Pair ◽  
Tensor Categories ◽  
Semisimple Hopf Algebra ◽  
Fusion Categories ◽  
A Finite Group ◽  
Exact Sequences ◽  
Group Fusion

Abstract We study exact sequences of finite tensor categories of the form Rep G → 𝒞 → 𝒟, where G is a finite group. We show that, under suitable assumptions, there exists a group Γ and mutual actions by permutations ⊳ : Γ × G → G and ⊲ : Γ × G→ Γ that make (G, Γ) into matched pair of groups endowed with a natural crossed action on 𝒟 such that 𝒞 is equivalent to a certain associated crossed extension 𝒟(G,Γ) of 𝒟. Dually, we show that an exact sequence of finite tensor categories Vec G → 𝒞 → 𝒟 induces an Aut(G)-grading on 𝒞 whose neutral homogeneous component is a (Z(G), Γ)-crossed extension of a tensor subcategory of 𝒟. As an application we prove that such extensions 𝒞 of 𝒟 are weakly group-theoretical fusion categories if and only if 𝒟 is a weakly group-theoretical fusion category. In particular, we conclude that every semisolvable semisimple Hopf algebra is weakly group-theoretical.

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