Proceedings of the 44th Symposium on Ring Theory and ...

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Proposition 10. The homomorphism δ : C ⊥ → C ◦ sending x to δ x is an isomorphism <strong>of</strong> R-modules. Theorem 11. Let R be a finite commutative QF ring. For a submodule C ⊆ R n , (C ⊥ ) ⊥ = C. By Theorem 1 and Theorem 4, we can get <strong>the</strong> following corollary. Corollary 12. Let R be a finite commutative QF ring. Then C is a polycyclic code if and only if C ⊥ is a sequential code. Theorem 13. Let R be a finite commutative QF ring. If C ⊆ R n is a free R-module <strong>of</strong> finite rank, <strong>the</strong>n C ⊥ is a free R-module <strong>of</strong> rankC ⊥ = n − rankC. We determine <strong>the</strong> parity check matrix <strong>of</strong> a constacyclic code. Proposition 14. Let R be a finite commutative QF ring and f = X n − α ∈ R[X]. Suppose f = hg ∈ R[X] where g and h are polynomials <strong>of</strong> degree n−k and k, respectively. Let C be <strong>the</strong> linear [n, k]-code corresponding to <strong>the</strong> ideal generated by g in R[X]/(X n − α) and h(X) = h k X k + h k−1 X k−1 + · · · + h 1 X + h 0 . Then C has <strong>the</strong> (n − k) × n parity check matrix H given by ⎛ ⎞ h k · · · h 1 h 0 0 · · · 0 0 h k · · · h 1 h 0 · · · 0 . H = 0 . . . . . . . . . . . . . . 0 . ⎜ ⎟ ⎝ . . ⎠ 0 · · · 0 h k · · · h 1 h 0 Definition 15. Let R be a finite commutative QF ring. For a sequential code C ⊆ R n , C is called a principal sequential code if C ⊥ is a principal polycyclic code. And C is called a principal sequential code with an invertible constant term if C ⊥ is a principal polycyclic code with an invertible constant term. Now we can get <strong>the</strong> main <strong>the</strong>orem. Theorem 16. Let R be a finite commutative QF ring. Suppose C is a free codes <strong>of</strong> R n . Then <strong>the</strong> following conditions are equivalent: (1) Both C and C ⊥ are principal polycyclic codes with invertible constant terms. (2) Both C and C ⊥ are principal sequential codes with invertible constant terms. (3) C is a principal polycyclic and sequential code with an invertible constant term. (4) C ⊥ is a principal polycyclic and sequential code with an invertible constant term. (5) C = (g)/(X n − α) is a constacyclic code with an invertible α. (6) C ⊥ = (q)/(X n − β) is a constacyclic code with an invertible β. Acknowledgement. The author wishes to thank Pr<strong>of</strong>. Y. Hirano, Naruto University <strong>of</strong> Education, for his helpful suggestions and valuable comments. –109–
References [1] D. Boucher and P. Solé, Skew constacyclic codes over Galois rings, Advances in Ma<strong>the</strong>matics <strong>of</strong> Communications, Volume 2, No. 3 (2008), 273–292. [2] M. Greferath, M. E. O’Sullivan, On bounds for codes over Frobenius rings under homogeneous weights, Discrete Math, 289 (2004), 11–24. [3] Y. Hirano, On admissible rings, Indag. Math. 8 (1997), 55–59. [4] S. Ikehata, On separable polynomials and Frobenius polynomials in skew polynomial rings, Math. J. Okayama. Univ. 22 (1980), 115–129. [5] T. Y. Lam, Lectures on Modules and Rings, Graduate Texts in Ma<strong>the</strong>matics, Vol. 189, Springer-Verlag, New York, 1999. [6] S. R. López-Permouth, B. R. Parra-Avila and S. Szabo, Dual generalizations <strong>of</strong> <strong>the</strong> concept <strong>of</strong> cyclicity <strong>of</strong> codes, Advances in Ma<strong>the</strong>matics <strong>of</strong> Communications, Volume 3, No. 3 (2009), 227–234. [7] M. Matsuoka, θ-polycyclic codes and θ-sequential codes over finite fields, International Journal <strong>of</strong> Algebra, Vol. 5 (2011), no. 2, 65–70. [8] B. R. McDonald, Finite Rings With Identity, Pure and Applied Ma<strong>the</strong>matics, Vol. 28, Marcel Dekker, Inc., New York, 1974. [9] J. A. Wood, Duality for modules over finite rings and applications to coding <strong>the</strong>ory, Amer. J. Math, 121 (1999), 555–575. Kuwanakita-Highscool 2527 Shim<strong>of</strong>ukayabe Kuwana Mie 511-0808 Japan E-mail address: e-white@hotmail.co.jp –110–
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Proceedings <stron
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Organizing Committee of</st
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Polycyclic codes and sequential cod
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Preface The 44th <
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8:40-9:30 Dan ZachariaSyracuse Univ
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The 44th S
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Tuesday September 27 8:40-9:30 Atsu
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Conversely, let (T , F) be a stable
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Then T = gen(X) and X is Ext-projec
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DIMENSIONS OF DERIVED CATEGORIES TA
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Notation 4. (1) Let A be an abelian
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of finitely genera
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is the map given b
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(2) Let R be a commutative ring whi
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QUIVER PRESENTATIONS OF GROTHENDIEC
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Proposition 3. Let C be a category,
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Example 11. Let Q be the</s
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and Gr(X ′ ) is given by
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particular, the de
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Proposition 1 and 2 leave us with d
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4. Examples Example 6. Let M = M(A
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EXAMPLE OF CATEGORIFICATION OF A CL
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The aim of <strong
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X ∈ mod Π Q S 1 S 2 S 3 1 ❁
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The previous proposition has a dual
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Computing inductively all t
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Example 23. We have ⎛ ⎞ ⎛ ⎞
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classification of
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quiver Q over K, and let D b (H) <s
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Then Q ∈ Q 17 . Suppose char K
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Moreover the Hochs
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DERIVED AUTOEQUIVALENCES AND BRAID
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3. Periodic Twists We now describe
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We now specialise to the</s
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and τ − n := Ext n Λ(DΛ, −)
Page 73 and 74: where r v ij := a i a j −a j a i
Page 75 and 76: ON A DEGENERATION PROBLEM FOR COHEN
Page 77 and 78: We make several othe</stron
Page 79 and 80: Let A be a commutative Gorenstein r
Page 81 and 82: 3. Extended orders In the</
Page 83 and 84: WEAK GORENSTEIN DIMENSION FOR MODUL
Page 85 and 86: 1.2. Introduction. The notion <stro
Page 87 and 88: e 2 A z 1 −→ X → 0 in mod-A,
Page 89 and 90: 5. Gorenstein algebras In this sect
Page 91 and 92: ON Ω-PERFECT MODULES AND SEQUENCES
Page 93 and 94: when R is a Nakayama algebra <stron
Page 95 and 96: says that components of</st
Page 97 and 98: then we obtain a c
Page 99 and 100: Proposition 16. Let C s be a stable
Page 101 and 102: 3.1. ZD ∞ -components. We assume
Page 103 and 104: Theorem 26. Let R be a local artini
Page 105 and 106: where each f i is an irreducible ep
Page 107 and 108: QUANTUM UNIPOTENT SUBGROUP AND DUAL
Page 109 and 110: {F i } i∈I and q-Serre relations
Page 111 and 112: Theorem 8. (1)Let U − q (w) → U
Page 113 and 114: E-mail address: ykimura@kurims.kyot
Page 115 and 116: Indeed, (1, 2, 3, 4) {1,2} −−
Page 117 and 118: By comparing this the</stro
Page 119 and 120: (a) (b) Proposition 17. Let G be a
Page 121 and 122: POLYCYCLIC CODES AND SEQUENTIAL COD
Page 123: ⎛ ⎞ 0 0 c 0 1 c t D c = ⎜ 1
Page 127 and 128: 2. Dimension of tr
Page 129 and 130: APR TILTING MODULES AND QUIVER MUTA
Page 131 and 132: (1) Q ′ is a quiver obtained from
Page 133 and 134: 1 arrows of ˜µ L
Page 135 and 136: [11] S. Fomin, A. Zelevinsky, Clust
Page 137 and 138: Theorem. Assume r is a common prime
Page 139 and 140: References [1] Apostol, T. M., The
Page 141 and 142: e(n, s) n = 6 7 8 s = 1 36 63 120 2
Page 143 and 144: Now we will show that the</
Page 145 and 146: is equal to ( n 2s−1 ); hence we
Page 147 and 148: THE NOETHERIAN PROPERTIES OF THE RI
Page 149 and 150: Proposition 4 (Paper III, Propositi
Page 151 and 152: HOCHSCHILD COHOMOLOGY OF QUIVER ALG
Page 153 and 154: (4) In [10], Green, Snashall and So
Page 155 and 156: 4. Quiver algebras defined by two c
Page 157 and 158: [6] L. Evens, The cohomology ring <
Page 159 and 160: Then there is an i
Page 161 and 162: • ( ˜F , ˜m) = ({ (1, 3), (2, 3
Page 163 and 164: Corollary 7 ([16, Theorem 3.4]). Th
Page 165 and 166: We have the direct
Page 167 and 168: We say a CW complex is regular, if
Page 169 and 170: SHARP BOUNDS FOR HILBERT COEFFICIEN
Page 171 and 172: Let us briefly note how this paper
Page 173 and 174: whence the set Λ
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Proof of</
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We have Q·H j m(A/(a 1 , a 2 , ·
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Let B = A/U(a d−1 ). We t
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• quiver Γ = (I, Ω) I Ω
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B = (B τ ) relations ρ 1 , · ·
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Definition 1. double quiver ˜Γ p
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◦ side Γ = (I, Ω) cycle (ac
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Definition 6. n × n A(Γ Dyn ) =
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(2) P Lie theory
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Example 15. Γ loop quiver λ ∈
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P (Γ)-module i-th radical B →
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B ∈ Λ(d) ε ∗ i (B) := dim C
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I-graded vector space V (d) (B τ )
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A ∗(i) l (a) = ∑n+1 t=l+1 (a i,
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wt(a) = (−d 1 , −d 2 , −d 3 )
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Ω Q Ω ( B Ω ; wt, ε i , ϕ
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“Λ(d) G(d)-” Definition 25.
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(( ( ) ( ) 0 s 0 0 0 , , , s) ( u 0
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MATRIX FACTORIZATIONS, ORBIFOLD CUR
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( ∂f Jac(f t t ) := C[x 1 , . . .
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Definition 14. CM L f (R f ) Ob(C
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4. Dolgachev Proposition 24 ([1]
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✤ ✤ ✤ ✤ ✤ ✤ γ 1 +γ 2
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(f, G f ) Dolgachev A Gf f Berg
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ON A GENERALIZATION OF COSTABLE TOR
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(2) P/t(P ) is σ-projective for an
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(3) P is a maximal σ-coessential e
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(2)→(1): Let σ be epi-preserving
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GRADED FROBENIUS ALGEBRAS AND QUANT
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Definition 2. A graded algebra A is
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In this case, ν ∈ Aut k E induce
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References [1] X. W. Chen, Graded s
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The topics covered are ring-<strong
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(2.3) soc( R R) ∼ = k⊕ s i T i
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principal ideal Rr ⊂ ker ϖ. Thus
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By Example 3, the
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weight wt(x) = d(x, 0) of</
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4.1. Fourier Transform. Gleason’s
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5. The Extension Problem In this se
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Because R is assumed to be Frobeniu
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is injective for every finite left
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linear code defined by η;
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ψ : ε(A) → Â. In this way, C
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REALIZING STABLE CATEGORIES AS DERI
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2.1. Positively graded self-injecti
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By the above resul
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Next we consider the</stron
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(2) There exists a triangle-equival
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ALGEBRAIC STRATIFICATIONS OF DERIVE
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The leaves of <str
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Step 2: Let A be a finite-dimension
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RECOLLEMENTS GENERATED BY IDEMPOTEN
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(a) the adjoint tr
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Cluster-tilting the</strong
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INTRODUCTION TO REPRESENTATION THEO
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is exact. Since E n ∼ = En+r , Mi
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field of V . We de
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(Proof) ( ) p 0 0
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we have a sequence of</stro
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R : CM(R⊗ k V ) → CM(R) defined
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P = P ′ t by t is a projective R
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SUBCATEGORIES OF EXTENSION MODULES
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Lemma 6. Let S 1 and S 2 be Serre s
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In the first row <
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