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1
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
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Moduled categories and adjusted modules over traced rings

100%
Simson D.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS 1. Introduction.......................................................................................5 2. Traced rings and adjusted modules..................................................9 3. Moduled categories.........................................................................21 4. Triangular adjustments....................................................................32 5. Categories of matrices and $_{A}M_{B}$-matrix modules...............43 6. Trace and cotrace reductions.........................................................47 7. Concluding remarks........................................................................60 References.........................................................................................64
2
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Path coalgebras of profinite bound quivers, cotensor coalgebras of bound species and locally nilpotent representations

94%
Simson D.
Colloquium Mathematicum
|
2007
|
tom 109
|
nr 2
307-343
EN
We prove that the study of the category C-Comod of left comodules over a K-coalgebra C reduces to the study of K-linear representations of a quiver with relations if K is an algebraically closed field, and to the study of K-linear representations of a K-species with relations if K is a perfect field. Given a field K and a quiver Q = (Q₀,Q₁), we show that any subcoalgebra C of the path K-coalgebra K^{◻}Q containing $K^{◻}Q₀ ⊕ K^{◻}Q₁$ is the path coalgebra $K^{◻}(Q,𝔅)$ of a profinite bound quiver (Q,𝔅), and the category C-Comod of left C-comodules is equivalent to the category $Rep_{K}^{ℓnℓf}(Q,𝔅)$ of locally nilpotent and locally finite K-linear representations of Q bound by the profinite relation ideal $𝔅 ⊂ \widehat{KQ}$. Given a K-species $ℳ = (F_{j},_{i}M_{j})$ and a relation ideal 𝔅 of the complete tensor K-algebra $T̂(ℳ ) = \widehat {T_F(M)}$ of ℳ, the bound species subcoalgebra $T^{◻}(ℳ,𝔅)$ of the cotensor K-coalgebra $T^{◻}(ℳ ) = T^{◻}_{F}(M)$ of ℳ is defined. We show that any subcoalgebra C of $T^{◻}(ℳ )$ containing $T^{◻}(ℳ )₀ ⊕ T^{◻}(ℳ )$₁ is of the form $T^{◻}(ℳ, 𝔅)$, and the category C-Comod is equivalent to the category $Rep_{K}^{ℓnℓf}(ℳ, 𝔅 )$ of locally nilpotent and locally finite K-linear representations of ℳ bound by the profinite relation ideal 𝔅. The question when a basic K-coalgebra C is of the form $T^{◻}_{F}(M,𝔅 )$, up to isomorphism, is also discussed.
3
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Coalgebras, comodules, pseudocompact algebras and tame comodule type

94%
Simson D.
Colloquium Mathematicum
|
2001
|
tom 90
|
nr 1
101-150
EN
We develop a technique for the study of K-coalgebras and their representation types by applying a quiver technique and topologically pseudocompact modules over pseudocompact K-algebras in the sense of Gabriel [17], [19]. A definition of tame comodule type and wild comodule type for K-coalgebras over an algebraically closed field K is introduced. Tame and wild coalgebras are studied by means of their finite-dimensional subcoalgebras. A weak version of the tame-wild dichotomy theorem of Drozd [13] is proved for a class of K-coalgebras. By applying [17] and [19] it is shown that for any length K-category 𝔄 there exists a basic K-coalgebra C and an equivalence of categories 𝔄 ≅ C-comod. This allows us to define tame representation type and wild representation type for any abelian length K-category. Hereditary coalgebras and path coalgebras KQ of quivers Q are investigated. Tame path coalgebras KQ are completely described in Theorem 9.4 and the following K-coalgebra analogue of Gabriel's theorem [18] is established in Theorem 9.3. An indecomposable basic hereditary K-coalgebra C is left pure semisimple (that is, every left C-comodule is a direct sum of finite-dimensional C-comodules) if and only if the quiver $_{C}Q*$ opposite to the Gabriel quiver $_{C}Q$ of C is a pure semisimple locally Dynkin quiver (see Section 9) and C is isomorphic to the path K-coalgebra $K(_{C}Q)$. Open questions are formulated in Section 10.
4
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Incidence coalgebras of intervally finite posets, their integral quadratic forms and comodule categories

94%
Simson D.
Colloquium Mathematicum
|
2009
|
tom 115
|
nr 2
259-295
EN
The incidence coalgebras $C = K^{□} I$ of intervally finite posets I and their comodules are studied by means of their Cartan matrices and the Euler integral bilinear form $b_{C}: ℤ^{(I)} × ℤ^{(I)} → ℤ$. One of our main results asserts that, under a suitable assumption on I, C is an Euler coalgebra with the Euler defect $∂_{C}: ℤ^{(I)} × ℤ^{(I)} → ℤ$ zero and $b_{C}(lgth M,lgth} N) = χ_{C}(M,N)$ for any pair of indecomposable left C-comodules M and N of finite K-dimension, where $χ_{C}(M,N)$ is the Euler characteristic of the pair M, N and $lgth M∈ ℤ^{(I)}$ is the composition length vector. The structure of minimal injective resolutions of simple left C-comodules is described by means of the inverse $ℭ_{I}^{-1} ∈ 𝕄 ^{⪯}_{I}(ℤ)$ of the incidence matrix $ℭ_{I} ∈ 𝕄_{I}(ℤ)$ of the poset I. Moreover, we describe the Bass numbers $μₘ^{I}(S_{I}(a),S_{I}(b))$, with m ≥ 0, for any simple $K^{□} I$-comodules $S_{I}(a)$, $S_{I}(b)$ by means of the coefficients of the bth row of $ℭ_{I}^{-1}$. We also show that, for any poset I of width two, the Grothendieck group $K₀(K^{□} I-Comod_{fc})$ of the category of finitely copresented $K^{□} I$-comodules is generated by the classes $[S_{I}(a)]$ of the simple comodules $S_{I}(a)$ and the classes $[E_{I}(a)]$ of the injective covers $E_{I}(a)$ of $S_{I}(a)$, with a ∈ I.
5
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Tame three-partite subamalgams of tiled orders of polynomial growth

94%
Simson D.
Colloquium Mathematicum
|
1999
|
tom 81
|
nr 2
237-262
EN
Assume that K is an algebraically closed field. Let D be a complete discrete valuation domain with a unique maximal ideal p and residue field D/p ≌ K. We also assume that D is an algebra over the field K . We study subamalgam D-suborders $Λ^•$ (1.2) of tiled D-orders Λ (1.1). A simple criterion for a tame lattice type subamalgam D-order $Λ^•$ to be of polynomial growth is given in Theorem 1.5. Tame lattice type subamalgam D-orders $Λ^•$ of non-polynomial growth are completely described in Theorem 6.2 and Corollary 6.3.
6
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Representations of bounded stratified posets, coverings and socle projective modules

83%
Simson D.
Banach Center Publications
|
1990
|
tom 26
|
nr 1
499-533
7
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Stable derived functors of the second symmetric power functor, second exterior power functor and Whitehead gamma functor

71%
Simson D.
Colloquium Mathematicum
|
1974-1975
|
tom 32
|
nr 1
49-55
8
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Connected sequences of stable derived functors and their applications

66%
Simson D., Tyc A.
Instytut Matematyczny Polskiej Akademii Nauk
EN
CONTENTS 1. Introduction........................................................................................................................................................................................................ 5 2. Category of complexes.................................................................................................................................................................................... 7 3. Left stable derived functors of covariant functors....................................................................................................................................... 11 4. Functors with, extensions............................................................................................................................................................................... 24 5. On the exactness of connected sequences................................................................................................................................................ 37 6. Right and left stable derived functors of contravariant functors. Right stable derived functors of covariant functors................... 39 7. Symmetric power functor $SP^n$ and exterior power functor $Λ^n$..................................................................................................... 43 8. On J. H. C. Whitehead's functor Γ.................................................................................................................................................................. 48 9. Computation of the modules $L^s_qSP^2(R)$, $L^s_qΛ^2(R)$ and $L^s_qΓ(R)$........................................................................... 53 10. Computation of the functors $L^s_qSP^2$, $L^s_qΛ^2$ and $L^s_qΓ$............................................................................................ 59 11. Eilenberg-MacLane's stable homology and cohomology functors...................................................................................................... 64 References............................................................................................................................................................................................................ 67
9
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Incidence coalgebras of interval finite posets of tame comodule type

60%
Leszczyński Z., Simson D.
Colloquium Mathematicum
|
2015
|
tom 141
|
nr 2
261-295
EN
The incidence coalgebras $K^{□} I$ of interval finite posets I and their comodules are studied by means of the reduced Euler integral quadratic form $q^{•}: ℤ^{(I)} → ℤ$, where K is an algebraically closed field. It is shown that for any such coalgebra the tameness of the category $K^{□} I-comod$ of finite-dimensional left $K^{□} I$-modules is equivalent to the tameness of the category $K^{□} I-Comod_{fc}$ of finitely copresented left $K^{□} I$-modules. Hence, the tame-wild dichotomy for the coalgebras $K^{□} I$ is deduced. Moreover, we prove that for an interval finite 𝔸̃ *ₘ-free poset I the incidence coalgebra $K^{□} I$ is of tame comodule type if and only if the quadratic form $q^{•}$ is weakly non-negative. Finally, we give a complete list of all infinite connected interval finite 𝔸̃ *ₘ-free posets I such that $K^{□} I$ is of tame comodule type. In this case we prove that, for any pair of finite-dimensional left $K^{□} I$-comodules M and N, $b̅_{K^{□} I} (dim M,dim N) = ∑_{j=0}^{∞} (-1)^{j} dim_{K} Ext_{K^{□} I}^{j}(M,N)$, where $b̅_{K^{□} I}: ℤ^{(I)} × ℤ^{(I)} → ℤ$ is the Euler ℤ-bilinear form of I and dim M, dim N are the dimension vectors of M and N.
10
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Bipartite coalgebras and a reduction functor for coradical square complete coalgebras

60%
Kosakowska J., Simson D.
Colloquium Mathematicum
|
2008
|
tom 112
|
nr 1
89-129
EN
Let C be a coalgebra over an arbitrary field K. We show that the study of the category C-Comod of left C-comodules reduces to the study of the category of (co)representations of a certain bicomodule, in case C is a bipartite coalgebra or a coradical square complete coalgebra, that is, C = C₁, the second term of the coradical filtration of C. If C = C₁, we associate with C a K-linear functor $ℍ_{C}: C-Comod → H_{C}-Comod$ that restricts to a representation equivalence $ℍ_{C}: C-comod → H_{C}-comod^{•}_{sp}$, where $H_{C}$ is a coradical square complete hereditary bipartite K-coalgebra such that every simple $H_{C}$-comodule is injective or projective. Here $H_{C}-comod^{∙}_{sp}$ is the full subcategory of $H_{C}-comod$ whose objects are finite-dimensional $H_{C}$-comodules with projective socle having no injective summands of the form $[S(i') \atop 0]$ (see Theorem 5.11). Hence, we conclude that a coalgebra C with C = C₁ is left pure semisimple if and only if $H_{C}$ is left pure semisimple. In Section 6 we get a diagrammatic characterisation of coradical square complete coalgebras C that are left pure semisimple. Tameness and wildness of such coalgebras C is also discussed.
11
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Embeddings of Kronecker modules into the category of prinjective modules and the endomorphism ring problem

60%
Göbel R., Simson D.
Colloquium Mathematicum
|
1998
|
tom 75
|
nr 2
213-244
12
Content available remote

A computation of positive one-peak posets that are Tits-sincere

60%
Gąsiorek M., Simson D.
Colloquium Mathematicum
|
2012
|
tom 127
|
nr 1
83-103
EN
A complete list of positive Tits-sincere one-peak posets is provided by applying combinatorial algorithms and computer calculations using Maple and Python. The problem whether any square integer matrix $A ∈ 𝕄 ₙ(ℤ)$ is ℤ-congruent to its transpose $A^{tr}$ is also discussed. An affirmative answer is given for the incidence matrices $C_{I}$ and the Tits matrices $Ĉ_{I}$ of positive one-peak posets I.
13
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On pure semisimplicity and representation-finite piecewise prime rings

60%
Klemp B., Simson D.
Banach Center Publications
|
1990
|
tom 26
|
nr 1
327-339
14
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
Content available

Preface, Contents, List of Participants

43%
Balcerzak S., Józefiak T., Krempa J., Simson D., Vogel W.
Banach Center Publications
|
1990
|
tom 26
|
nr 2
1-10
15
Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik
Content available

Preface, Contents, List of Participants

43%
Balcerzyk S., Józefiak T., Krempa J., Simson D., Vogel W.
Banach Center Publications
|
1990
|
tom 26
|
nr 1
1-10
16
Content available remote

On pure global dimension of locally finitely presented Grothendieck categories

42%
Simson D.
Fundamenta Mathematicae
|
1977
|
tom 96
|
nr 2
91-116
17
Content available remote

On the Auslander-Reiten valued quiver of right peak rings

36%
Klemp B., Simson D.
Fundamenta Mathematicae
|
1990
|
tom 136
|
nr 2
91-114
18
Content available remote

On pure semi-simple Grothendieck categories I

30%
Simson D.
Fundamenta Mathematicae
|
1978
|
tom 100
|
nr 3
211-222
19
Content available remote

On pure semi-simple Grothendieck categories II

30%
Simson D.
Fundamenta Mathematicae
|
1980
|
tom 110
|
nr 2
107-116
20
Content available remote

Peak reductions and waist reflection functors

30%
Simson D.
Fundamenta Mathematicae
|
1991
|
tom 137
|
nr 2
115-144
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