scholarly journals Ordinary Grothendieck Groups of a Frobenius P-Category

2011 ◽  
Vol 18 (01) ◽  
pp. 1-76 ◽  
Author(s):  
Lluis Puig
Keyword(s):  
Characteristic Zero ◽  
Inverse Limit ◽  
Grothendieck Group ◽  
Characteristic P ◽  
Direct Sum ◽  
Grothendieck Groups ◽  
Frobenius Categories

In [7] we have introduced the Frobenius categories [Formula: see text] over a finite p-group P, and we have associated to [Formula: see text] — suitably endowed with some central k*-extensions — a “Grothendieck group” as an inverse limit of Grothendieck groups of categories of modules in characteristic p obtained from [Formula: see text], determining its rank. Our purpose here is to introduce an analogous inverse limit of Grothendieck groups of categories of modules in characteristic zero obtained from [Formula: see text], determining its rank and proving that its extension to a field is canonically isomorphic to the direct sum of the corresponding extensions of the “Grothendieck groups” above associated with the centralizers in [Formula: see text] of a suitable set of representatives of the [Formula: see text]-classes of elements of P.

Grothendieck Groups of Bass Orders

1971 ◽  
Vol 23 (1) ◽  
pp. 103-115
Author(s):  
Klaus W. Roggenkamp
Keyword(s):  
Special Class ◽  
Grothendieck Group ◽  
Commutative Rings ◽  
Gorenstein Rings ◽  
Direct Sum ◽  
Maximal Orders ◽  
Grothendieck Groups ◽  
Epimorphic Image ◽  
Carry Over ◽  
Separable Algebras

Commutative Bass rings, which form a special class of Gorenstein rings, have been thoroughly investigated by Bass [1]. The definitions do not carry over to non-commutative rings. However, in case one deals with orders in separable algebras over fields, Bass orders can be defined. Drozd, Kiricenko, and Roïter [3] and Roïter [6] have clarified the structure of Bass orders, and they have classified them. These Bass orders play a key role in the question of the finiteness of the non-isomorphic indecomposable lattices over orders (cf. [2; 8]). We shall use the results of Drozd, Kiricenko, and Roïter [3] to compute the Grothendieck groups of Bass orders locally. Locally, the Grothendieck group of a Bass order (with the exception of one class of Bass orders) is the epimorphic image of the direct sum of the Grothendieck groups of the maximal orders containing it.

scholarly journals Auslander–Reiten Triangles and Grothendieck Groups of Triangulated Categories

2021 ◽  
Author(s):  
Johanne Haugland
Keyword(s):  
Grothendieck Group ◽  
Triangulated Category ◽  
Triangulated Categories ◽  
Isomorphism Classes ◽  
Grothendieck Groups ◽  
Frobenius Categories

AbstractWe prove that if the Auslander–Reiten triangles generate the relations for the Grothendieck group of a Hom-finite Krull–Schmidt triangulated category with a (co)generator, then the category has only finitely many isomorphism classes of indecomposable objects up to translation. This gives a triangulated converse to a theorem of Butler and Auslander–Reiten on the relations for Grothendieck groups. Our approach has applications in the context of Frobenius categories.

Grothendieck groups of quadratic forms and G-equivalence of fields

1973 ◽  
Vol 73 (1) ◽  
pp. 29-36
Author(s):  
K. Szymiczek
Keyword(s):  
Abelian Group ◽  
Quadratic Forms ◽  
Multiplicative Group ◽  
Grothendieck Group ◽  
Grothendieck Groups

Let k be a field of characteristic other than 2 and let g(k) denote the multiplicative group k* of the field k modulo squares, i.e. g(k) = k*/k*2. This is an abelian group of exponent 2 and its order, if finite, is a power of 2. We denote by G(k) the Grothendieck group of quadratic forms over k.

Grothendieck groups for categories of complexes

2008 ◽  
Vol 3 (1) ◽  
pp. 165-203 ◽  
Author(s):  
Hans-Bjørn Foxby ◽  
Esben Bistrup Halvorsen
Keyword(s):  
Local Ring ◽  
Finite Length ◽  
Full Subcategory ◽  
Grothendieck Group ◽  
Projective Dimension ◽  
Intersection Theorem ◽  
Projective Modules ◽  
Finitely Generated ◽  
Grothendieck Groups ◽  
Finitely Generated Modules

AbstractThe new intersection theorem states that, over a Noetherian local ring R, for any non-exact complex concentrated in degrees n,…,0 in the category P(length) of bounded complexes of finitely generated projective modules with finite-length homology, we must have n ≥ d = dim R.One of the results in this paper is that the Grothendieck group of P(length) in fact is generated by complexes concentrated in the minimal number of degrees: if Pd(length) denotes the full subcategory of P(length) consisting of complexes concentrated in degrees d,…0, the inclusion Pd(length) → P(length) induces an isomorphism of Grothendieck groups. When R is Cohen–Macaulay, the Grothendieck groups of Pd(length) and P(length) are naturally isomorphic to the Grothendieck group of the category M(length) of finitely generated modules of finite length and finite projective dimension. This and a family of similar results are established in this paper.

scholarly journals The inverse limit of some free algebras

1973 ◽  
Vol 16 (2) ◽  
pp. 129-145
Author(s):  
A. L. Allen ◽  
S. Moran
Keyword(s):  
Characteristic Zero ◽  
Inverse Limit ◽  
The Other ◽  
Natural Projection ◽  
Free Algebras ◽  
Fixed Field ◽  
Degree Function ◽  
Image Position

Let Ω[x1, x2, …, xn] denote the algebra of polynomials in variables x1, x2, …, xn with coefficients from a fixed field Ω of characteristic zero, where n = 1, 2,…. There exists a natural projection which maps xn onto 0 and all the other variables onto themselves, for n = 1, 2, …. This enables one to construct the corresponding inverse limit which we here denote by Ω[x]. The algebra Ω[x] has a natural degree function defined on it.

On arithmetically realizable classes

1995 ◽  
Vol 118 (3) ◽  
pp. 383-392 ◽  
Author(s):  
D. Burns
Keyword(s):  
Finite Group ◽  
Grothendieck Group ◽  
Number Fields ◽  
Additive Subgroup ◽  
Semi Group ◽  
Grothendieck Groups ◽  
Symplectic Characters ◽  
A Finite Group ◽  
Theoretic Description ◽  
Complex Characters

We fix a number field L and a finite group G, and write Cl (ℤL[G]) for the reduced Grothendieck group of the category of finitely generated projective ℤL[G]-modules. We let RG denote the ring of complex characters of G, with SG the additive subgroup which is generated by the irreducible symplectic characters. We shall say that an element c ∈ Cl (ℤL[G]) is ‘(arithmetically) realizable’ if there exists a tamely ramified Galois extension N/K of number fields with L ⊆ K and an identification Gal (N/K) →˜ G via which c is the class of some Gal (N/K)-stble ℤN-ideal. We let RL(G) denote the subgroup of Cl (ℤL[G]) which is generated by the realizable elements for varying N/K. Our interest in RL(G) arises from the fact that it is the largest subset of Cl (ℤL[G]) upon which the results of Chinburg and the author in [Bu, Ch] can be used to give an explicit module theoretic description of the action of the integral semi-group ring AL, G of the Adams-Cassou-Noguès-Taylor operators (ΨL, k): k ∈ ℤ, 2 × k if SG ≠ {0}}. Whilst the results of [Bu, Ch] can (at least partially) be understood ‘geometrically’ via the action of Bott cannibalistic elements on suitable Grothendieck groups (cf. [Ch, E, P, T], [Bu]), the underlying problem of finding an explicit module theoretic interpretation of the action of AL, G on all elements of Cl(ℤL[G]) is of course essentially algebraic in nature. It is in this context that we were originally motivated to investigate RL(G).

Operations in Grothendieck Rings and the Symmetric Group

1974 ◽  
Vol 26 (3) ◽  
pp. 543-550 ◽  
Author(s):  
John Burroughs
Keyword(s):  
Symmetric Group ◽  
Symmetric Groups ◽  
Integral Representations ◽  
Symmetric Powers ◽  
Direct Sum ◽  
Grothendieck Groups ◽  
Finite Integral ◽  
Principal Tool ◽  
Grothendieck Rings ◽  
Divided Powers

In [1] Atiyah described how to use the complex representations of the symmetric group, Sn, to define and investigate operations in complex topological K-theory. In this paper operations for more general Grothendieck groups are described in terms of the integral representations of Sn using the representations directly without passing to the dual as Atiyah did. The principal tool, which is proved in the first section, is the theorem that the direct sum of the Grothendieck groups of finite integral representations of Sn form a bialgebra isomorphic to a polynomial ring with a sequence of divided powers. A consequence of this theorem is that the only operations that can be constructed from the symmetric groups will be polynomials in the symmetric powers.

scholarly journals Length of Local Cohomology in Positive Characteristic and Ordinarity

2018 ◽  
Vol 2020 (7) ◽  
pp. 1921-1932 ◽  
Author(s):  
Thomas Bitoun
Keyword(s):  
Characteristic Zero ◽  
Local Cohomology ◽  
Differential Operators ◽  
Positive Characteristic ◽  
Isolated Singularity ◽  
Local Cohomology Module ◽  
Perfect Field ◽  
Tight Closure ◽  
Characteristic P

Abstract Let D be the ring of Grothendieck differential operators of the ring R of polynomials in d ≥ 3 variables with coefficients in a perfect field of characteristic p. We compute the D-module length of the 1st local cohomology module ${H^{1}_{f}}(R)$ with respect to a polynomial f with an isolated singularity, for p large enough. The expression we give is in terms of the Frobenius action on the top coherent cohomology of the exceptional fibre of a resolution of the singularity. Our proof rests on a tight closure computation of Hara. Since the above length is quite different from that of the corresponding local cohomology module in characteristic zero, we also consider a characteristic zero D-module whose length is expected to equal that above, for ordinary primes.

scholarly journals Commutative group algebras and Prufer groups

2010 ◽  
Vol 41 (1) ◽  
pp. 85-95
Author(s):  
P. V. Danchev
Keyword(s):  
Group Ring ◽  
Quotient Group ◽  
Commutative Group ◽  
Group Algebras ◽  
Direct Factor ◽  
Perfect Field ◽  
Characteristic P ◽  
Arbitrary Characteristic ◽  
Direct Sum ◽  
Normalized Units

Suppose $G$ is a multiplicatively written abelian $p$-group, where $p$ is a prime, and $F$ is a field of arbitrary characteristic. The main results in this paper are that none of the Sylow $p$-group of all normalized units $S(FG)$ in the group ring $FG$ and its quotient group $S(FG)/G$ cannot be Prufer groups. This contrasts a classical conjecture for which $S(FG)/G$ is a direct factor of a direct sum of generalized Prufer groups whenever $F$ is a perfect field of characteristic $p$.

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