Cohen-Macaulay modules over the plane curve singularity of type T₃₆

For a wide class of Cohen–Macaulay modules over the local ring of the plane curve singularity of type T₃₆ we describe explicitly the corresponding matrix factorizations. The calculations are based on the technique of matrix problems, in particular, representations of bunches of chains.

Збережено в:

Бібліографічні деталі
Дата:	2019
Автори:	Drozd, Y.A., Tovpyha, O.
Формат:	Стаття
Мова:	English
Опубліковано:	Інститут прикладної математики і механіки НАН України 2019
Назва видання:	Algebra and Discrete Mathematics
Онлайн доступ:	http://dspace.nbuv.gov.ua/handle/123456789/188490
Теги:	Додати тег Немає тегів, Будьте першим, хто поставить тег для цього запису!
Назва журналу:	Digital Library of Periodicals of National Academy of Sciences of Ukraine
Цитувати:	Cohen-Macaulay modules over the plane curve singularity of type T₃₆ / Y.A. Drozd, O. Tovpyha // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 2. — С. 213–223. — Бібліогр.: 14 назв. — англ.

Репозитарії

Digital Library of Periodicals of National Academy of Sciences of Ukraine

id	irk-123456789-188490
record_format	dspace
spelling	irk-123456789-1884902023-03-03T01:27:02Z Cohen-Macaulay modules over the plane curve singularity of type T₃₆ Drozd, Y.A. Tovpyha, O. For a wide class of Cohen–Macaulay modules over the local ring of the plane curve singularity of type T₃₆ we describe explicitly the corresponding matrix factorizations. The calculations are based on the technique of matrix problems, in particular, representations of bunches of chains. 2019 Article Cohen-Macaulay modules over the plane curve singularity of type T₃₆ / Y.A. Drozd, O. Tovpyha // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 2. — С. 213–223. — Бібліогр.: 14 назв. — англ. 1726-3255 2010 MSC: 13C14, 14H20. http://dspace.nbuv.gov.ua/handle/123456789/188490 en Algebra and Discrete Mathematics Інститут прикладної математики і механіки НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
description	For a wide class of Cohen–Macaulay modules over the local ring of the plane curve singularity of type T₃₆ we describe explicitly the corresponding matrix factorizations. The calculations are based on the technique of matrix problems, in particular, representations of bunches of chains.
format	Article
author	Drozd, Y.A. Tovpyha, O.
spellingShingle	Drozd, Y.A. Tovpyha, O. Cohen-Macaulay modules over the plane curve singularity of type T₃₆ Algebra and Discrete Mathematics
author_facet	Drozd, Y.A. Tovpyha, O.
author_sort	Drozd, Y.A.
title	Cohen-Macaulay modules over the plane curve singularity of type T₃₆
title_short	Cohen-Macaulay modules over the plane curve singularity of type T₃₆
title_full	Cohen-Macaulay modules over the plane curve singularity of type T₃₆
title_fullStr	Cohen-Macaulay modules over the plane curve singularity of type T₃₆
title_full_unstemmed	Cohen-Macaulay modules over the plane curve singularity of type T₃₆
title_sort	cohen-macaulay modules over the plane curve singularity of type t₃₆
publisher	Інститут прикладної математики і механіки НАН України
publishDate	2019
url	http://dspace.nbuv.gov.ua/handle/123456789/188490
citation_txt	Cohen-Macaulay modules over the plane curve singularity of type T₃₆ / Y.A. Drozd, O. Tovpyha // Algebra and Discrete Mathematics. — 2019. — Vol. 28, № 2. — С. 213–223. — Бібліогр.: 14 назв. — англ.
series	Algebra and Discrete Mathematics
work_keys_str_mv	AT drozdya cohenmacaulaymodulesovertheplanecurvesingularityoftypet36 AT tovpyhao cohenmacaulaymodulesovertheplanecurvesingularityoftypet36
first_indexed	2025-07-16T10:34:29Z
last_indexed	2025-07-16T10:34:29Z
_version_	1837799385915719680
fulltext	“adm-n4” — 2020/1/24 — 13:02 — page 213 — #63 Algebra and Discrete Mathematics RESEARCH ARTICLE Volume 28 (2019). Number 2, pp. 213–223 c© Journal “Algebra and Discrete Mathematics” Cohen-Macaulay modules over the plane curve singularity of type T36 Yuriy A. Drozd and Oleksii Tovpyha Abstract. For a wide class of Cohen–Macaulay modules over the local ring of the plane curve singularity of type T36 we describe explicitly the corresponding matrix factorizations. The calculations are based on the technique of matrix problems, in particular, representations of bunches of chains. 1. Introduction Let ❦ be an algebraically closed field, S = ❦[[x, y]]. Recall that the complete local ring of the plane curve singularity of type T36 is R = S/(F ), where F = x(x− y2)(x−λy2) and λ ∈ ❦ \ {0, 1}. In this paper we present explicit description of a wide class of maximal Cohen–Macaulay modules over the ring R called modules of the first level. Note that T36 is one of the critical singularities of tame Cohen-Macaulay representation type [7]. Till now, only for the singularities of type T44 matrix factorizations have been described [8, 9]. 2. Matrix problem and the first reduction So, let R = ❦[[x, y]]/(F ), where F = x(x−y2)(x−λy2) (λ ∈ ❦\{0, 1}). We consider R as the subring of the direct product R̃ = R1 ×R2 ×R3, where all Ri = ❦[[t]], generated by the elements x = (0, t2, λt2) and y = (t, t, t). We denote by R12 the projection of R onto R1 ×R2. It is 2010 MSC: 13C14, 14H20. Key words and phrases: Cohen–Macaulay modules, matrix factorizations, bi- module problems, bunches of chains. “adm-n4” — 2020/1/24 — 13:02 — page 214 — #64 214 Cohen-Macaulay modules over T36 generated by (t, t, 0) and (0, t2, 0) and R12 ≃ ❦[[x, y]]/x(x − y2). It is a singularity of type A2, so all indecomposable R12-modules are R1, R2, R12 and R ′ 12 = R12[t1], where t1 = (t, 0, 0). We also set t2 = (0, t, 0). Denote M̃ = R̃M = M1⊕M2⊕M3, where Mi is an Ri-module. There is an exact sequence: 0 → M12 → M → M3 → 0, where M12 = M ∩ (M1 ⊕M2) is an R12-module, hence M12 = m1R1 ⊕ m2R2 ⊕m12R12 ⊕m′ 12R ′ 12. So M gives an element ξ ∈ Ext1 R (M3,M12). There is an exact sequence 0 → (x− λy2)R → R → R3 → 0, whence Ext1R(M3,M12) = M12/(x− λy2)M12 In the table below we present bases of the modules Ext1 R (R3, N), where N ∈ R1,R2,R12,R ′ 12. In this table each element actually denotes its residue class modulo x−λy2, 1s (s ∈ {1, 2, 12}) is the identity element of Rs, t12 = t1 + t2 and µ = (1− λ)/λ. R1 11, t11 R2 12, t2 R ′ 12 112, t1, t2, t21 = µt22 R12 112, t12, t21 = µt22, t31 = µt32 Homomorphisms R12 → Ri induce the maps Ext1 R (R3,R12) → Ext1 R (R3,Ri) which map 112 7→ 1i and t12 7→ ti. The embedding R ′ 12 → R12 which maps 112 7→ t12 induces the map Ext1 R (R3,R ′ 12) → Ext1 R (R3,R12) that coincide with the multiplication by t12. The embeddings Ri → R ′ 12 such that 1i 7→ ti induce the maps Ext1 R (R3,Ri) → Ext1 R (R3,R ′ 12) that coincide with the multiplication by ti. In particular, if we only consider free terms, we obtain representations of the partially ordered set of width 2: R12 R ′ 12 R1 R2 “adm-n4” — 2020/1/24 — 13:02 — page 215 — #65 Y. Drozd, O. Tovpyha 215 (see [12]), hence the matrix A0 of free terms can be reduced to the form: A0 =                  0 0 0 I1 0 0 0 I1 0 0 0 0 0 0 0 0 0 0 0 0 I2 0 0 0 0 I2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 I12 0 0 0 0 0 0 0 0 0 0 0 0 I12 0 0 0 0 0 0                  , where Is means 1sI for the identity matrices I of the appropriate size (maybe different for different matrices). In what follows we always suppose that A0 is of this form. 3. Modules of the first level Let now A = A0 + tA1 where A1 is also divided into blocks in the same way as A0. Using automorphisms of M3 we can make zero the 1st, 4th, 7th and 9th rows of A1, as well as one of the 2rd or 5th rows. Using automorphisms of M12 we can also make zero all columns in A1, except the 1st one and the parts of the 2nd, 3rd and 4th columns in the 10th row, where we can only delete all terms containing t2 or t3. In particular, the terms 112 from the last vertical stripe become direct summands of the whole matrix A. So in what follows we can omit this column. We always suppose that A1 has this form. Let A0 1 be the free term of the matrix A1. The non-zero part of its 10th row can be considered as representation of the partially ordered set: 2 3 4 1 Hence it can be reduced to the form     0 I 0 0 0 0 0 0 0 0 I 0 0 0 0 0 0 0 0 0 0 I 0 I 0 0 0 0 0 0 0 0     , “adm-n4” — 2020/1/24 — 13:02 — page 216 — #66 216 Cohen-Macaulay modules over T36 where I is again an identity matrix of the appropriate size (maybe different for different matrices). Then the whole matrix A modulo t2 can be reduced to the form: 1 2 3 4 5 0 0 0 0 0 0 I1 0 0 0 0 0 0 0 0 0 I1 0 1 A11t ∗ 0 I1 0 0 0 0 0 0 2 A21t ∗ 0 0 I1 0 0 0 0 0 3 A31t1 0 0 0 0 0 0 0 0 0 0 0 0 I2 0 0 0 0 0 0 0 0 0 I2 0 0 0 0 0 I2 0 0 0 0 0 0 0 0 0 I2 0 0 0 0 0 4 A41t2 0 0 0 0 0 0 0 0 5 A51t ∗ A52t ∗ 0 0 0 0 0 0 I12 6 A61 A62t ∗ 0 0 A63t1 0 A64t2 A65t ∗ 0 0 t12I 0 0 0 0 0 0 0 0 0 t12I 0 0 0 0 0 0 0 0 0 0 0 t12I 0 t12I 0 0 0 0 0 0 0 0 0 0 Here the symbol t∗ means that in this block t1 = −t2, and A61 is a matrix pencil X1t1 +X2t2. The horizontal lines show the division of A into the stripes such that the 1st stripe corresponds to R1, the 2nd to R2, the 3rd to R′ 12 and the 4th to R12. Moreover, as in the matrix A we have t21 = µt22 with µ 6= −1, one can delete all terms with t2i and t3i everywhere except the last block of the first column. The endomorphisms of M3 and M12 which do not destroy the shape of the matrices A0 and A0 1 induce the transfromations of columns that can be described by the scheme 2 1 66 // (( 3 // 5 4 66 and the transformations of rows that can be described by the scheme 5 6 66 // (( 3 // 2 // 1 4 66 “adm-n4” — 2020/1/24 — 13:02 — page 217 — #67 Y. Drozd, O. Tovpyha 217 For the matrix A61 it means that we can add the rows of X1 to those of Ai1 for i ∈ {1, 2, 3} and the rows of X2 to the rows of Ai1 for i ∈ {1, 2, 4}. In the same way, the columns of X1 can be added to those of A6j for j ∈ {3, 5}, while the columns of X2 can be added to those of A6j for j ∈ {4, 5}. The indecomposable matrix pencils (representations of the Kronecker quiver) are described in [11, 13]. In [13] the morphisms between indecom- posable representations are also described. It implies that the matrix A61 is a direct sum of the following matrices: A(n) =       t1 t2 0 . . . 0 0 t1 t2 . . . 0 . . . . . . . . . . . . . . . . . . 0 0 0 . . . t2 0 0 0 . . . t1       , B(n) =       t2 t1 0 . . . 0 0 t2 t1 . . . 0 . . . . . . . . . . . . . . . . . . 0 0 0 . . . t1 0 0 0 . . . t2       , C(n) =     t1 t2 0 . . . 0 0 0 t1 t2 . . . 0 0 . . . . . . . . . . . . . . . . . . . . . . 0 0 0 . . . t1 t2     , D(n) = C(n)⊤ It is easy to see that: • If A61 = A(n) than we can make zero all matrices above A61 except the 1st column of A41, and all matrices to the right of A61 except the last row of A64. • If A61 = B(n) than we can make zero all matrices above A61 except the 1st column of A31, and all matrices to the right of A61 except the last row of A63. • If A61 = C(n) than we can make zero all matrices to the right of A61, and all matrices above A61 except the last column of A31, the 1st column of A41 and one (any chosen) of the columns of the matrix A51. • If A61 = D(n) than we can make zero all matrices above A61, and all matrices to the right of A61 except the last row of A63, the 1st row of A64 and one (any chosen) of the rows of the matrix A62. Hence in the non-zero part of A51 (and, respectively, A62) we can left one non-zero element above each block of C(n) (and, respectively, D(n)). Therefore, except the summands A(n), B(n), C(n), D(n) in the blocks Aij , (i = 5, 6, j = 1, 2), we will also have the summands of the form C ′(n), with one additional element in A51-part as compared to C(n), and D′(n), with one additional element in A62-part as compared to D(n)). So we can “adm-n4” — 2020/1/24 — 13:02 — page 218 — #68 218 Cohen-Macaulay modules over T36 suppose that C ′(n) looks like B(n)⊤, but with the 1st row from A51, and D′(n) looks like A(n)⊤, but with the last column from A62. One can see that now we can make zero all elements of the matrix A52 except those which are in the zero rows of A51 and zero columns of A62. The remaining part of A51 can be reduced to the form ( I 0 0 0 ) . It gives direct summands of the whole matrix A of the form ( t1 t12 ) (certainly, t1 can be replace here by t2). Therefore, in what follows we can suppose that A52 = 0. Analogously, we can suppose that the matrices A11, A12 and A65 are also zero. Otherwise we obtain direct summands of A. For instance, if A65 6= 0, all non-zero elements are in the rows which do not belong to the non-zero parts of A66 and A65. So they give direct summands of the form     0 11 12 0 0 t2 t12 t12     . The description of homomorphisms between the representations of the Kronecker quuiver [13] show that we can add the non-zero columns over A(n) (respectively, B(n) ) to those over A(m) (respectively, B(m) ) for m > n, and the same for the non-zero rows to the right of A(n) or B(n). We can also add the non-zero columns over C(n) (respectively, non-zero rows to the right of D(n) ) to those of C(m) (respectively, of D(m)), where n < m, as well as to those of A(k) and B(k) for any k. It means that the possible transformations of these columns and rows can be considered as representations of a bunch of chains in the sense of [1] or [2, Appendix B] (we use the formulation of the second paper). Namely, we have the next pairs of chains: • E1 = {ai, di, d ′ i \| i ∈ N}, F1 = {c3} • E2 = {bi, d̃i, d̃ ′ i \| i ∈ N}, F2 = {c4} • E3 = {r3}, F3 = {ãi, ci, c ′ i \| i ∈ N} • E4 = {r4}, F4 = {b̃i, c̃i, c̃ ′ i \| i ∈ N} with the relation ∼: ai ∼ ãi, bi ∼ b̃i, ci ∼ c̃i, di ∼ d̃i, c′i ∼ c̃′i, d′i ∼ d̃′i (i ∈ N). Here r3, r4 corresponds to A31, A41 respectively and c3, c4 corresponds to A63, A64 respectively. “adm-n4” — 2020/1/24 — 13:02 — page 219 — #69 Y. Drozd, O. Tovpyha 219 Now we use the description of the indecomposable representations of this bunch of chains from [1, 2]. In our case they correspond to the fol- lowing words in the alphabet {ai, ãi, bi, b̃i, ci, c̃i, di, d̃i, c ′ i, c̃ ′ i, d ′ i, d̃ ′ i, c3, r4, c4, r3,−,∼}: • 4 type of words with ai, i ∈ N: wa(i) = r3 − ãi ∼ ai − c3 and 3 shorter words: r3 − ãi ∼ ai, ãi ∼ ai − c3, ãi ∼ ai; • 4 type of words with bi, i ∈ N: wb(i) = r4 − b̃i ∼ bi − c4 and 3 shorter words: r4 − b̃i ∼ bi, b̃i ∼ bi − c4, b̃i ∼ bi; • 4 type of words with ci, i ∈ N: wc(i) = r4 − c̃i ∼ ci − r3 and 3 shorter words: r4 − c̃i ∼ ci, c̃i ∼ ci − r3, c̃i ∼ ci; • 4 type of words with di, i ∈ N: wd(i) = c4 − d̃i ∼ di − c3 and 3 shorter words: c4 − d̃i ∼ di, d̃i ∼ di − c3, d̃i ∼ di; • 4 type of words with c′i, i ∈ N: w ′ c(i) = r4 − c̃′i ∼ c′i − r3 and 3 shorter words: r4 − c̃′i ∼ c′i, c̃ ′ i ∼ c′i − r3, c̃ ′ i ∼ c′i; • 4 type of words with d′i, i ∈ N: w′ d(i) = c4 − d̃′i ∼ d′i − c3 and 3 shorter words: c4 − d̃′i ∼ d′i, d̃ ′ i ∼ d′i − c3, d̃ ′ i ∼ d′i; Following the construction of indecomposable representations from [1], we construct the matrices corresponding to these words: Pa(n) =   0 1 t2e1 0 A(n) t2e ⊤ n   , Pb(n) =   t1e1 0 0 1 B(n) t1e ⊤ n   , Pc(n) =   t1e1 t2e1 C(n)   , Pd(n) =   0 1 0 0 0 1 D(n) t2e ⊤ n+1 t1e ⊤ n+1   , P ′ c(n) =   t1e1 0 t2e1 0 C ′(n) t12e ⊤ 1   , P ′ d(n) =     0 1 0 0 0 1 D′(n) t2e ⊤ n+1 t1e ⊤ n+1 t12en+1 0 0     . Here tr = y1r and en = (0, 0, . . . , 0, 1), e1 = (1, 0, . . . , 0) and ⊤ means the transposition. 4. Generators and relations. Example Now we calculate matrix factorizations of the polynomial F = x(x− y2)(x− λy2) corresponding to the indecomposable Cohen-Macaulay mod- ules over R. In other words, we find minimal sets of generators for these modules and minimal sets of relations for these generators. “adm-n4” — 2020/1/24 — 13:02 — page 220 — #70 220 Cohen-Macaulay modules over T36 In order to make smaller the arising matrices, we denote z = x− y2 an z′ = x− λy2. Thus F = xzz′. We do detailed calculations for the word wa(2) = r3 − ã2 ∼ a2 − c3. Since all calculations are similar, for other words we just write the resulting matrices. Pa(2) =     0 0 1 ye2 0 0 ye1 ye2 0 0 ye1 ye2     Here the first two stripes belongs to R1 and R2 respectively and the last stripe belongs to R′ 12. So we have generators: v1, v2, v3 ∈ R3, u1 ∈ R1, u2 ∈ R2, u121 , ū121 , u122 , ū122 ∈ R′ 12. (∗) Note that ye1u 12 i = ū12i and ye2u 12 i = yu12i − ū12i for u12i ∈ R′ 12, i = 1, 2. Then we have the following relations for these generators: z′v1 = ye2u 2 + ye1u 12 1 , z′v2 = ye2u 12 1 + ye1u 12 2 , z′v3 = u1 + ye2u 12 2 It implies that ū121 = z′v1 − yu2, ū122 = z′v1 − yu2 + z′v2 − yu112, u1 = z′v1 − yu2 + z′v2 − yu112 + z′v3 − yu122 Now we can exclude generators ū121 , ū122 , u1. It is important to note that ū12i , i = 1, 2 are annihilated by x. Since u1 ∈ R1 is also annihilated by x, u2 ∈ R2 is annihilated by z and u121 , u122 ∈ R′ 12 are annihilated by xz, we have the following relations for v1, v2, v3, u 2, u121 , u122 from (∗): zu2 = 0, xzu121 = 0, xzu122 = 0, xz′v1 − xyu2 = 0, xz′v2 − xyu121 = 0, xz′v3 − xyu122 = 0. It gives the following matrix factorization with columns corresponding to u2, u121 , u122 , v1, v2, v3, in this order: Qa(2) =         z 0 0 0 0 0 0 xz 0 0 0 0 0 0 xz 0 0 0 −xy 0 0 xz′ 0 0 0 −xy 0 0 xz′ 0 0 0 −xy 0 0 xz′         “adm-n4” — 2020/1/24 — 13:02 — page 221 — #71 Y. Drozd, O. Tovpyha 221 For the other three words with ai, namely r3 − ãi ∼ ai, ãi ∼ ai − c3, ãi ∼ ai we obtain the matrix factorizations by excluding some generators and the appropriate srows and columns: • Excluding u2 from the list of generators (∗) and deleting the first row and 1st column from the matrix Qa(2) we get the matrix factorization for ãi ∼ ai − c3. • Excluding v3 from the list of generators (∗) and deleting the last row and the last column from the matrix Qa(2) we get the matrix factorization for r3 − ãi ∼ ai. • Excluding both u2, v3 from the list of generators (∗) and deleting the first and the last rows and the first and the last columns from the matrix Qa(2) we get the matrix factorization for ãi ∼ ai. Now one can easily see how the matrix factorization Qa(i) for the word wa(i) = r3 − ãi ∼ ai − c3 looks like for i > 2. 5. Generators and relations. Other words For other modules of the first level the corresponding matrix factor- izations are calculated in a similar way. We only present the results for n = 2, since otherwise we obtain too cumbersome matrices. For the word wb(2) = r4 − b̃2 ∼ b2 − c4 we have the matrix of correspondences with columns corresponding to u1, u121 , u122 , v1, v2, v3: Qb(2) =         x 0 0 0 0 0 0 xz 0 0 0 0 0 0 xz 0 0 0 −xy −xy 0 xz′ 0 0 0 0 xy 0 xz′ 0 −zy −zy zy zz′ zz′ zz′         For the word wc(2) = r4 − c̃2 ∼ c2 − r3 we have the matrix of correspondences with columns corresponding to u2, u121 , u122 , v3, v2, v1: Qc(2) =         z 0 0 0 0 0 0 xz 0 0 0 0 0 0 xz 0 0 0 0 0 −xy xz′ 0 0 0 −xy 0 0 xz′ 0 −xy 0 0 0 0 xz′         “adm-n4” — 2020/1/24 — 13:02 — page 222 — #72 222 Cohen-Macaulay modules over T36 For the word wd(2) = c4 − d̃2 ∼ d2 − c3 we have the matrix of corre- spondences with columns corresponding to u2, u121 , u122 , u123 , v4, v3, v2, v1: Qd(2) =             z 0 0 0 0 0 0 0 0 xz 0 0 0 0 0 0 0 0 xz 0 0 0 0 0 0 0 0 xz 0 0 0 0 −x 0 0 0 xz′ 0 0 0 0 0 0 −xy 0 xz′ 0 0 0 0 0 −xy 0 0 xz′ 0 0 0 −xy 0 0 0 0 xz′             For the word w ′ c(2) = r4 − c̃′2 ∼ c′2 − r3 we have the matrix of corre- spondences with columns corresponding to u1, u2, u121 , u122 , v4, v3, v2, v1: Q′ c(2) =             x 0 0 0 0 0 0 0 0 z 0 0 0 0 0 0 0 0 xz 0 0 0 0 0 0 0 0 xz 0 0 0 0 0 0 0 0 xzz′ 0 0 0 0 0 0 −xy 0 xz′ 0 0 0 0 −xy 0 0 0 xz′ 0 −xy −xy 0 0 −xyz′ 0 0 xz′             For the word w ′ d(2) = c4−d̃′2 ∼ d′2−c3 we have the matrix of correspon- dences with columns corresponding to u2, u121 , u122 , u123 , v5, v4, v3, v2, v1, namely Q′ d(2) equals: Q′ d(2)               z 0 0 0 0 0 0 0 0 0 xz 0 0 0 0 0 0 0 0 0 xz 0 0 0 0 0 0 0 0 0 xz 0 0 0 0 0 −x 0 0 0 xz′ 0 0 0 −x 0 0 0 xzz′ −xzz′ 0 0 0 0 0 0 −xy 0 0 xz′ 0 0 0 0 0 −xy 0 0 0 xz′ 0 0 0 −xy 0 0 0 0 0 xz′               For the truncated words (without the first or the last letter) we apply the procedure analogous to that described at the end of the preceding section. In this way we obtain all matrix factorizations of the polynom F corresponding to the modules of the first level. “adm-n4” — 2020/1/24 — 13:02 — page 223 — #73 Y. Drozd, O. Tovpyha 223 References [1] Bondarenko, V.M., Representations of bundles of semichained sets and their applications, Algebra i Analiz, 3, no. 5 (1991) 38–61. [2] Burban, I., Drozd, Y., Derived categories of nodal algebras, J. Algebra, 272 (2004) 46–94. [3] Burban, I., Iyama, O., Keller, B., Reiten, I., Cluster tilting for one-dimensional hypersurface singularities. Adv. Math., 217 (2008) 2443–2484. [4] Dieterich, E., Lattices over curve singularities with large conductor. Invent math. 114 (1993) 399–433. [5] Drozd, Y., Cohen–Macaulay Modules over Cohen–Macaulay Algebras, Represen- tation Theory of Algebras and Related Topics, CMS Conference Proceedings, 19 (1996) 25–53. [6] Drozd, Y., Reduction algorithm and representations of boxes and algebras, C. R. Math. Acad. Sci., Soc. R. Can. 23, No.4 (2001) 97–125. [7] Drozd, Y., Greuel, G.-M., Cohen–Macaulay module type, Compositio Math., 89, no. 3 (1993) 315–338. [8] Drozd, Y., Tovpyha, O., On Cohen-Macaulay modules over the plane curve singu- larity of type T44, Arch. Math. 108 (2017) 569–579. [9] Drozd, Y., Tovpyha, O., On Cohen-Macaulay modules over the plane curve singu- larity of type T44, II, Algebra Discrete Math. 28 (2019) 75–93. [10] Eisenbud, D., Homological algebra on a complete intersection, with an application to group representations, Trans. Amer. Math. Soc. 260 (1980) 35–64. [11] Gantmacher, F.R.: The Theory of Matrices. Fizmatlit, Moscow, 2004. [12] Nazarova, L.A., Roiter, A.V., Representations of the partially ordered sets, Zap. Nauchn. Sem. LOMI, 28, "Nauka" (1972) 5–31. [13] Ringel, C.M.: Tame Algebras. Lecture Notes in Math. 1099, Springer–Verlag, 1984. [14] Yoshino, Y., Cohen–Macaulay Modules over Cohen–Macaulay Rings. Cambridge University Press, 1990. Contact information Y. Drozd, O. Tovpyha Institute of Mathematics, National Academy of Sciences, 01601 Kyiv, Ukraine E-Mail(s): y.a.drozd@gmail.com, drozd@imath.kiev.ua, tovpyha@gmail.com Web-page(s): http://www.imath.kiev.ua/∼drozd Received by the editors: 07.09.2019.

Cohen-Macaulay modules over the plane curve singularity of type T₃₆

Репозитарії

Схожі ресурси