Organization
Graduate School of Mathematics Division of Mathematics Fundamental Mathematics Professor
Graduate School
Graduate School of Mathematics
Undergraduate School
School of Science Department of Mathematics
Title
Professor
TAKAHASHI Ryo
Degree 3
-
博士（理学） （ 2004.3 岡山大学 ）
-
修士（理学） （ 2002.3 岡山大学 ）
-
学士（総合人間学） （ 2000.3 京都大学 ）
Research Areas 1
-
Natural Science / Algebra / Commutative Algebra
Current Research Project and SDGs 1
-
representation theory of commutative rings
Education 2
-
Okayama University Graduate School, Division of Science and Technology
- 2004.3
Country： Japan
-
Kyoto University Faculty of Integrated Human Studies
- 2000.3
Country： Japan
Professional Memberships 1
-
日本数学会
Awards 2
-
2020.3 The Mathematical Society of Japan Subcategories of module categories of commutative rings
Ryo Takahashi
-
2004.9 The Mathematical Society of Japan
Ryo Takahashi
Papers 131
-
Proxy small thick subcategories of derived categories Reviewed
Takahashi, R
JOURNAL OF ALGEBRA Vol. 662 page： 465 - 481 2025.1
-
Cofiniteness of local cohomology modules and subcategories of modules Reviewed
Takahashi, R; Wakasugi, N
COLLECTANEA MATHEMATICA Vol. 76 ( 1 ) page： 1 - 10 2025.1
-
On a generalization of Ulrich modules and its applications Reviewed
Ela Celikbas; Olgur Celikbas; Justin Lyle; Ryo Takahashi; Yongwei Yao
Tohoku Mathematical Journal 2025
-
Vanishing of DHKK complexities for singularity categories and generation of syzygy modules Reviewed
Tokuji Araya; Kei-ichiro Iima; Ryo Takahashi
Kyoto Journal of Mathematics 2025
-
Ring homomorphisms and local rings with quasi-decomposable maximal ideal Reviewed
Nasseh, S; Sather-Wagstaff, K; Takahashi, R
COMMUNICATIONS IN ALGEBRA Vol. 52 ( 10 ) page： 4295 - 4313 2024.10
-
On a Verdier quotient of a derived category of a local ring Reviewed
Yuki Mifune; Ryo Takahashi
Journal of the Mathematical Society of Japan 2024.8
-
ANNIHILATORS AND DECOMPOSITIONS OF SINGULARITY CATEGORIES Reviewed
Esentepe, Ö; Takahashi, R
PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY Vol. 67 ( 3 ) page： 699 - 713 2024.8
-
Auslander-Reiten conjecture and finite injective dimension of Hom Reviewed
Ghosh, D; Takahashi, R
KYOTO JOURNAL OF MATHEMATICS Vol. 64 ( 1 ) 2024.2
-
Rings with q-torsionfree canonical modules Reviewed International coauthorship
Naoki Endo; Laura Ghezzi; Shiro Goto; Jooyoun Hong; Shin-ichiro Iai; Toshinori Kobayashi; Naoyuki Matsuoka; Ryo Takahashi
The Mathematical Legacy of Wolmer V. Vasconcelos, De Gruyter Proc. Math., De Gruyter, Berlin 2024
-
Solid generators in module categories and applications Reviewed
Takahashi, R
RENDICONTI DEL SEMINARIO MATEMATICO DELLA UNIVERSITA DI PADOVA Vol. 152 page： 229 - 243 2024
-
When Is a Subcategory Serre or Torsion-Free? Reviewed
Iima, K; Matsui, H; Shimada, K; Takahashi, R
PUBLICATIONS OF THE RESEARCH INSTITUTE FOR MATHEMATICAL SCIENCES Vol. 60 ( 4 ) page： 831 - 857 2024
-
Remarks on complexities and entropies for singularity categories Reviewed
Takahashi, R
COMPTES RENDUS MATHEMATIQUE Vol. 361 ( 1 ) page： 1611 - 1623 2023.11
-
On the categorical entropy of the Frobenius pushforward functor Reviewed
Matsui, H; Takahashi, R
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY Vol. 55 ( 3 ) page： 1358 - 1372 2023.6
-
Dominant Local Rings and Subcategory Classification Reviewed
Takahashi, R
INTERNATIONAL MATHEMATICS RESEARCH NOTICES Vol. 2023 ( 9 ) page： 7259 - 7318 2023.5
-
Faltings' annihilator theorem and <i>t</i>-structures of derived categories Reviewed
Takahashi, R
MATHEMATISCHE ZEITSCHRIFT Vol. 304 ( 1 ) 2023.5
-
On the subcategories of <i>n</i>-torsionfree modules and related modules Reviewed
Dey, S; Takahashi, R
COLLECTANEA MATHEMATICA Vol. 74 ( 1 ) page： 113 - 132 2023.1
-
Maximal Cohen-Macaulay tensor products and vanishing of Ext modules Reviewed
Kimura, K; Otake, Y; Takahashi, R
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY Vol. 54 ( 6 ) page： 2456 - 2468 2022.12
-
Spanier-Whitehead Categories of Resolving Subcategories and Comparison with Singularity Categories Reviewed
Bahlekeh, A; Salarian, S; Takahashi, R; Toosi, Z
ALGEBRAS AND REPRESENTATION THEORY Vol. 25 ( 3 ) page： 595 - 613 2022.6
-
Comparisons Between Annihilators of Tor and Ext Reviewed
Dey, S; Takahashi, R
ACTA MATHEMATICA VIETNAMICA Vol. 47 ( 1 ) page： 123 - 139 2022.3
-
Filtrations in Module Categories, Derived Categories, and Prime Spectra Reviewed
Matsui, H; Takahashi, R
INTERNATIONAL MATHEMATICS RESEARCH NOTICES Vol. 2022 ( 5 ) page： 3457 - 3492 2022.2
-
ON REDUCING HOMOLOGICAL DIMENSIONS OVER NOETHERIAN RINGS Reviewed
Araya, T; Takahashi, R
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Vol. 150 ( 2 ) page： 469 - 480 2022.2
-
Intersections of resolving subcategories and intersections of thick subcategories Reviewed
Takahashi, R
EUROPEAN JOURNAL OF MATHEMATICS Vol. 7 ( 4 ) page： 1767 - 1790 2021.12
-
Grothendieck groups, convex cones and maximal Cohen-Macaulay points Reviewed
Takahashi, R
MATHEMATISCHE ZEITSCHRIFT Vol. 299 ( 1-2 ) page： 53 - 82 2021.10
-
Dao, HL; Kobayashi, T; Takahashi, R
JOURNAL OF PURE AND APPLIED ALGEBRA Vol. 225 ( 9 ) 2021.9
-
Classification of dominant resolving subcategories by moderate functions Reviewed
Takahashi, R
ILLINOIS JOURNAL OF MATHEMATICS Vol. 65 ( 3 ) page： 597 - 618 2021.9
-
Some criteria for detecting large homomorphisms of local rings Reviewed
Gheibi, M; Takahashi, R
COMMUNICATIONS IN ALGEBRA Vol. 49 ( 8 ) page： 3490 - 3500 2021.7
-
QUASI-PROJECTIVE DIMENSION Reviewed
Gheibi, M; Jorgensen, DA; Takahashi, R
PACIFIC JOURNAL OF MATHEMATICS Vol. 312 ( 1 ) page： 113 - 147 2021.5
-
Homogeneous prime elements in normal two-dimensional graded rings Reviewed
Singh, AK; Takahashi, R; Watanabe, K
JOURNAL OF ALGEBRA Vol. 571 page： 339 - 349 2021.4
-
The Jacobian ideal of a commutative ring and annihilators of cohomology Reviewed
Iyengar, SB; Takahashi, R
JOURNAL OF ALGEBRA Vol. 571 page： 280 - 296 2021.4
-
COHOMOLOGICAL DIMENSIONS OF SPECIALIZATION-CLOSED SUBSETS AND SUBCATEGORIES OF MODULES Reviewed
Matsui, H; Nam, TT; Takahashi, R; Tri, NM; Yen, DN
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Vol. 149 ( 2 ) page： 481 - 496 2021.2
-
POWERS OF THE MAXIMAL IDEAL AND VANISHING OF (CO)HOMOLOGY Reviewed
Celikbas, O; Takahashi, R
GLASGOW MATHEMATICAL JOURNAL Vol. 63 ( 1 ) page： 1 - 5 2021.1
-
Generation in module categories and derived categories of commutative rings Reviewed
Ryo Takahashi
Commutative algebra, Expository Papers Dedicated to David Eisenbud on the Occasion of his 75th Birthday, Springer, Cham page： 723 - 751 2021
-
Resolving subcategories whose finitely presented module categories are abelian Reviewed
Takahashi, R
COMPTES RENDUS MATHEMATIQUE Vol. 359 ( 5 ) page： 577 - 592 2021
-
Resolving Subcategories Closed Under Certain Operations and a Conjecture of Dao and Takahashi Reviewed
Sadeghi, A; Takahashi, R
MICHIGAN MATHEMATICAL JOURNAL Vol. 70 ( 2 ) page： 341 - 367 2021
-
FLAT RING EPIMORPHISMS AND UNIVERSAL LOCALIZATIONS OF COMMUTATIVE RINGS Reviewed
Hügel, LA; Marks, F; Stovícek, J; Takahashi, R; Vitória, J
QUARTERLY JOURNAL OF MATHEMATICS Vol. 71 ( 4 ) page： 1489 - 1520 2020.12
-
Construction of spectra of triangulated categories and applications to commutative rings Reviewed
Matsui, H; Takahashi, R
JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN Vol. 72 ( 4 ) page： 1283 - 1307 2020.10
-
Maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum Reviewed
Kobayashi, T; Lyle, J; Takahashi, R
JOURNAL OF PURE AND APPLIED ALGEBRA Vol. 224 ( 7 ) 2020.7
-
A TOPOLOGY ON THE SET OF ISOMORPHISM CLASSES OF MAXIMAL COHEN-MACAULAY MODULES Reviewed
Hiramatsu, N; Takahashi, R
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Vol. 148 ( 6 ) page： 2359 - 2369 2020.6
-
On the radius of the category of extensions of matrix factorizations Reviewed
Shimada, K; Takahashi, R
JOURNAL OF ALGEBRA Vol. 546 page： 566 - 579 2020.3
-
Local rings with quasi-decomposable maximal ideal Reviewed
Nasseh, S; Takahashi, R
MATHEMATICAL PROCEEDINGS OF THE CAMBRIDGE PHILOSOPHICAL SOCIETY Vol. 168 ( 2 ) page： 305 - 322 2020.3
-
GORENSTEIN MODIFICATIONS AND Q-GORENSTEIN RINGS Reviewed
Dao, H; Iyama, O; Takahashi, R; Wemyss, M
JOURNAL OF ALGEBRAIC GEOMETRY Vol. 29 ( 4 ) page： 729 - 751 2020
-
Burch ideals and Burch rings Reviewed
Dao, HL; Kobayashi, T; Takahashi, R
ALGEBRA & NUMBER THEORY Vol. 14 ( 8 ) page： 2121 - 2150 2020
-
LITTLE DIMENSION AND THE IMPROVED NEW INTERSECTION THEOREM Reviewed
Nakamura, T; Takahashi, R; Yassemi, S
MATHEMATICA SCANDINAVICA Vol. 126 ( 2 ) page： 209 - 220 2020
-
Generation in singularity categories of hypersurfaces of countable representation type Reviewed
Araya, T; Iima, K; Ono, M; Takahashi, R
ARCHIV DER MATHEMATIK Vol. 113 ( 6 ) page： 603 - 615 2019.12
-
Rings whose ideals are isomorphic to trace ideals Reviewed
Kobayashi, T; Takahashi, R
MATHEMATISCHE NACHRICHTEN Vol. 292 ( 10 ) page： 2252 - 2261 2019.10
-
ON THE IDEAL CASE OF A CONJECTURE OF HUNEKE AND WIEGAND Reviewed
Celikbas, O; Goto, S; Takahashi, R; Taniguchi, N
PROCEEDINGS OF THE EDINBURGH MATHEMATICAL SOCIETY Vol. 62 ( 3 ) page： 847 - 859 2019.8
-
ON THE SECOND RIGIDITY THEOREM OF HUNEKE AND WIEGAND Reviewed
Celikbas, O; Takahashi, R
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Vol. 147 ( 7 ) page： 2733 - 2739 2019.7
-
On the transitivity of degeneration of modules Reviewed
Takahashi, R
MANUSCRIPTA MATHEMATICA Vol. 159 ( 3-4 ) page： 431 - 444 2019.7
-
Ulrich modules over Cohen-Macaulay local rings with minimal multiplicity Reviewed
Kobayashi, T; Takahashi, R
QUARTERLY JOURNAL OF MATHEMATICS Vol. 70 ( 2 ) page： 487 - 507 2019.6
-
Hiramatsu, N; Takahashi, R; Yoshino, Y
JOURNAL OF ALGEBRA Vol. 525 page： 374 - 389 2019.5
-
Remarks on dimensions of triangulated categories Reviewed
Aihara, T; Takahashi, R
JOURNAL OF ALGEBRA Vol. 521 page： 235 - 246 2019.3
-
Applications and homological properties of local rings with decomposable maximal ideals Reviewed
Nasseh, S; Sather-Wagstaff, S; Takahashi, R; VandeBogert, K
JOURNAL OF PURE AND APPLIED ALGEBRA Vol. 223 ( 3 ) page： 1272 - 1287 2019.3
-
Openness of the Regular Locus and Generators for Module Categories Reviewed
Iyengar, SB; Takahashi, R
ACTA MATHEMATICA VIETNAMICA Vol. 44 ( 1 ) page： 207 - 212 2019.3
-
Totally reflexive modules and Poincare series Reviewed
Gheibi, M; Takahashi, R
JOURNAL OF ALGEBRA Vol. 520 page： 440 - 459 2019.2
-
Noncommutative resolutions using syzygies Reviewed
Dao, H; Iyama, O; Iyengar, SB; Takahashi, R; Wemyss, M; Yoshino, Y
BULLETIN OF THE LONDON MATHEMATICAL SOCIETY Vol. 51 ( 1 ) page： 43 - 48 2019.2
-
ASSOCIATED PRIMES AND SYZYGIES OF LINKED MODULES Reviewed
Celikbas, O; Dibaei, MT; Gheibi, M; Sadeghi, A; Takahashi, R
JOURNAL OF COMMUTATIVE ALGEBRA Vol. 11 ( 3 ) page： 301 - 323 2019
-
Two generalizations of Auslander-Reiten duality and applications Reviewed
Arash Sadeghi, Ryo Takahashi
Vol. 63 page： 335-351 2019
-
ON THE VANISHING OF SELF EXTENSIONS OVER COHEN-MACAULAY LOCAL RINGS Reviewed
Araya, T; Celikbas, O; Sadeghi, A; Takahashi, R
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Vol. 146 ( 11 ) page： 4563 - 4570 2018.11
-
Structure of Irreducible Homomorphisms to/from Free Modules Reviewed
Nasseh, S; Takahashi, R
ALGEBRAS AND REPRESENTATION THEORY Vol. 21 ( 2 ) page： 471 - 485 2018.4
-
On the ideal case of a conjecture of Auslander and Reiten Reviewed
Celikbas, O; Iima, K; Sadeghi, A; Takahashi, R
BULLETIN DES SCIENCES MATHEMATIQUES Vol. 142 page： 94 - 107 2018.2
-
Thick tensor ideals of right bounded derived categories of commutative rings Reviewed
Takahashi, R
REPRESENTATIONS OF ALGEBRAS Vol. 705 page： 225 - 250 2018
-
THICK SUBCATEGORIES OVER ISOLATED SINGULARITIES Reviewed
Takahashi, R
PACIFIC JOURNAL OF MATHEMATICS Vol. 291 ( 1 ) page： 183 - 211 2017.11
-
ON THE AUSLANDER-REITEN CONJECTURE FOR COHEN MACAULAY LOCAL RINGS Reviewed
Goto, S; Takahashi, R
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Vol. 145 ( 8 ) page： 3289 - 3296 2017.8
-
When are -syzygy modules -torsionfree? Reviewed
Matsui, H; Takahashi, R; Tsuchiya, Y
ARCHIV DER MATHEMATIK Vol. 108 ( 4 ) page： 351 - 355 2017.4
-
Singularity categories and singular equivalences for resolving subcategories Reviewed
Matsui, H; Takahashi, R
MATHEMATISCHE ZEITSCHRIFT Vol. 285 ( 1-2 ) page： 251 - 286 2017.2
-
Thick tensor ideals of right bounded derived categories Reviewed
Matsui, H; Takahashi, R
ALGEBRA & NUMBER THEORY Vol. 11 ( 7 ) page： 1677 - 1738 2017
-
Ulrich ideals and almost Gorenstein rings Reviewed
Shiro Goto, Ryo Takahashi, Naoki Taniguchi
Vol. 144 page： 2811-2823 2016
-
Annihilation of cohomology, generation of modules and finiteness of derived dimension Reviewed
Abdolnaser Bahlekeh, Ehsan Hakimian, Shokrollah Salarian, Ryo Takahashi
Vol. 67 page： 387-404 2016
-
Ulrich ideals and modules over two-dimensional rational singularities Reviewed
Shiro Goto, Kazuho Ozeki, Ryo Takahashi, Kei-ichi Watanabe, Ken-ichi Yoshida
Vol. 221 page： 69-110 2016
-
Extension closedness of syzygies and local Gorensteinness of commutative rings Reviewed
Shiro Goto, Ryo Takahashi
Vol. 19 page： 511-521 2016
-
Perfect linkage of Cohen-Macaulay modules over Cohen-Macaulay rings Reviewed
Kei-ichiro Iima, Ryo Takahashi
Vol. 458 page： 134-155 2016
-
Annihilation of cohomology and strong generation of module categories Reviewed
Srikanth B. Iyengar, Ryo Takahashi
International Mathematics Research Notices. IMRN Vol. 2016 page： 499-535 2016
-
Huneke-Wiegand conjecture and change of rings Reviewed
Shiro Goto, Ryo Takahashi, Naoki Taniguchi, Hoang Le Truong
J. Algebra Vol. 422 page： 33-52 2015
-
The dimension of a subcategory of modules Reviewed
Hailong Dao, Ryo Takahashi
Forum of Mathematics. Sigma Vol. 3 2015
-
Generators and dimensions of derived categories of modules Reviewed
Takuma Aihara, Ryo Takahashi
Communications in Algebra Vol. 43 page： 5003-5029 2015
-
Endofunctors of singularity categories characterizing Gorenstein rings Reviewed
Takuma Aihara, Ryo Takahashi
Proceedings of the American Mathematical Society Vol. 143 page： 3777-3779 2015
-
Non-commutative resolutions and Grothendieck groups Reviewed
Hailong Dao, Osamu Iyama, Ryo Takahashi, Charles Vial
J. Noncommut. Geom. Vol. 9 page： 21-34 2015
-
Maximal Cohen-Macaulay approximations and Serre's condition Reviewed
Hiroki Matsui, Ryo Takahashi
Acta Math. Vietnam. Vol. 40 page： 197-203 2015
-
Building modules from the singular locus Reviewed
Jesse Burke, Lars Winther Christensen, Ryo Takahashi
Math. Scand. Vol. 116 page： 23-33 2015
-
Upper bounds for dimensions of singularity categories Reviewed
Hailong Dao, Ryo Takahashi
C. R. Math. Acad. Sci. Paris Vol. 353 page： 297-301 2015
-
Almost Gorenstein rings -- towards a theory of higher dimension -- Reviewed
Shiro Goto, Ryo Takahashi, Naoki Taniguchi
J. Pure Appl. Algebra Vol. 219 page： 2666-2712 2015
-
Bounds on depth of tensor products of modules Reviewed
Olgur Celikbas, Arashi Sadeghi, Ryo Takahashi
J. Pure Appl. Algebra Vol. 219 page： 1670-1684 2015
-
Classification of resolving subcategories and grade consistent functions Reviewed
Hailong Dao, Ryo Takahashi
Int. Math. Res. Not. IMRN Vol. 2015 page： 119-149 2015
-
Annihilation of cohomology and decompositions of derived categories Reviewed
Srikanth B. Iyengar, Ryo Takahashi
Homology, Homotopy Appl. Vol. 16 ( 2 ) page： 231-237 2014
-
Modules that detect finite homological dimensions Reviewed
Olgur Celikbas, Hailong Dao, Ryo Takahashi
Kyoto J. Math. Vol. 54 ( 2 ) page： 295-310 2014
-
Brauer-Thrall for totally reflexive modules over local rings of higher dimension Reviewed
Olgur Celikbas, Mohsen Gheibi, Ryo Takahashi
Algebr. Represent. Theory Vol. 17 ( 3 ) page： 997-1008 2014
-
Reconstruction from Koszul homology and applications to module and derived categories Reviewed
Ryo Takahashi
Pacific J. Math. Vol. 268 ( 1 ) page： 231-248 2014
-
The radius of a subcategory of modules Reviewed
Hailong Dao, Ryo Takahashi
Algebra Number Theory Vol. 8 ( 1 ) page： 141-172 2014
-
Dimensions of triangulated categories with respect to subcategories Reviewed
Takuma Aihara, Tokuji Araya, Osamu Iyama, Ryo Takahashi, Michio Yoshiwaki
J. Algebra Vol. 399 page： 205-219 2014
-
Ulrich ideals and modules Reviewed
Shiro Goto, Kazuho Ozeki, Ryo Takahashi, Kei-ichi Watanabe, Ken-ichi Yoshida
Math. Proc. Cambridge Philos. Soc. Vol. 156 ( 1 ) page： 137-166 2014
-
Classifying resolving subcategories over a Cohen-Macaulay local ring Reviewed
Ryo Takahashi
Math. Z. Vol. 273 ( 1 ) page： 569-587 2013
-
Thick subcategories over Gorenstein local rings that are locally hypersurfaces on the punctured spectra Reviewed
Ryo Takahashi
J. Math. Soc. Japan Vol. 65 ( 2 ) page： 357-374 2013
-
Tilting and cluster tilting for quotient singularities Reviewed
Osamu Iyama, Ryo Takahashi
Math. Ann. Vol. 356 ( 3 ) page： 1065-1105 2013
-
Auslander-Reiten conjecture and Auslander-Reiten duality Reviewed
O. Celikbas, R. Takahashi
J. Algebra Vol. 382 page： 100-114 2013
-
On the left perpendicular category of the modules of finite projective dimension Reviewed
Tokuji Araya, Kei-ichiro Iima, Ryo Takahashi
Comm. Algebra Vol. 40 ( 8 ) page： 2693-2699 2012
-
On the structure of Cohen-Macaulay modules over hypersurfaces of countable Cohen-Macaulay representation type Reviewed
T. Araya, K.-i. Iima, R. Takahashi
J. Algebra Vol. 361 page： 213-224 2012
-
Contravariantly finite resolving subcategories over commutative rings Reviewed
R. Takahashi
Amer. J. Math. Vol. 133 ( 2 ) page： 417-436 2011
-
When is there a nontrivial extension-closed subcategory? Reviewed
R. Takahashi
J. Algebra Vol. 331 ( 1 ) page： 388-399 2011
-
Homological aspects of semidualizing modules Reviewed
R. Takahashi, D. White
Math. Scand. Vol. 106 ( 1 ) page： 5-22 2010
-
On the existence of embeddings into modules of finite homological dimensions Reviewed
R. Takahashi, S. Yassemi, Y. Yoshino
Proc. Amer. Math. Soc. Vol. 138 ( 7 ) page： 2265-2268 2010
-
Classifying thick subcategories of the stable category of Cohen-Macaulay modules Reviewed
R. Takahashi
Adv. Math. Vol. 225 ( 4 ) page： 2076-2116 2010
-
On localizing subcategories of derived categories Reviewed
R. Takahashi
J. Math. Kyoto Univ. Vol. 49 ( 4 ) page： 771-783 2009
-
Local cohomology based on a nonclosed support defined by a pair of ideals Reviewed
R. Takahashi, Y. Yoshino, T. Yoshizawa
J. Pure Appl. Algebra Vol. 213 ( 4 ) page： 582-600 2009
-
Modules in resolving subcategories which are free on the punctured spectrum Reviewed
R. Takahashi
Pacific J. Math. Vol. 241 ( 2 ) page： 347-367 2009
-
A generalization of a theorem of Foxby Reviewed
T. Araya, R. Takahashi
Arch. Math. (Basel) Vol. 93 ( 2 ) page： 123-127 2009
-
Classifying subcategories of modules over a commutative noetherian ring Reviewed
R. Takahashi
J. Lond. Math. Soc. (2) Vol. 78 ( 3 ) page： 767-782 2008
-
Quasi-socle ideals in a Gorenstein local ring Reviewed
S. Goto, R. Takahashi, N. Matsuoka
J. Pure Appl. Algebra Vol. 212 ( 5 ) page： 969-980 2008
-
Direct summands of syzygy modules of the residue class field Reviewed
R. Takahashi
Nagoya Math. J. Vol. 189 page： 1-25 2008
-
D-modules over rings with finite F-representation type Reviewed
S. Takagi, R. Takahashi
Math. Res. Lett. Vol. 15 ( 3 ) page： 563-581 2008
-
Finite Gorenstein representation type implies simple singularity Reviewed
L. W. Christensen, G. Piepmeyer, J. Striuli, R. Takahashi
Adv. Math. Vol. 218 ( 4 ) page： 1012-1026 2008
-
On vanishing of certain Ext modules Reviewed
S. Goto, F. Hayasaka, R. Takahashi
J. Math. Soc. Japan Vol. 60 ( 4 ) page： 1045-1064 2008
-
On G-regular local rings Reviewed
R. Takahashi
Comm. Algebra Vol. 36 ( 12 ) page： 4472-4491 2008
-
An uncountably infinite number of indecomposable totally reflexive modules Reviewed
R. Takahashi
Nagoya Math. J. Vol. 187 page： 35-48 2007
-
A new approximation theory which unifies spherical and Cohen-Macaulay approximations Reviewed
R. Takahashi
J. Pure Appl. Algebra Vol. 208 ( 2 ) page： 617-634 2007
-
On the number of indecomposable totally reflexive modules Reviewed
R. Takahashi
Bull. London Math. Soc. Vol. 39 ( 3 ) page： 487-492 2007
-
Totally reflexive modules constructed from smooth projective curves of genus g >= 2 Reviewed
R. Takahashi, K.-i. Watanabe
Arch. Math. (Basel) Vol. 89 ( 1 ) page： 60-67 2007
-
A characterization of modules locally of finite injective dimension Reviewed
R. Takahashi
Proc. Amer. Math. Soc. Vol. 135 ( 11 ) page： 3461-3464 2007
-
Openness of FID-loci Reviewed
R. Takahashi
Glasg. Math. J. Vol. 48 ( 3 ) page： 431-435 2006
-
The existence of finitely generated modules of finite Gorenstein injective dimension Reviewed
R. Takahashi
Proc. Amer. Math. Soc. Vol. 134 ( 11 ) page： 3115-3121 2006
-
Remarks on modules approximated by G-projective modules Reviewed
R. Takahashi
J. Algebra Vol. 301 ( 2 ) page： 748-780 2006
-
Syzygy modules with semidualizing or G-projective summands Reviewed
R. Takahashi
J. Algebra Vol. 295 ( 1 ) page： 179-194 2006
-
A note on a theorem of F. Wang and G. Tang Reviewed
R. Takahashi
Bull. Austral. Math. Soc. Vol. 73 page： 293-297 2006
-
On the category of modules of Gorenstein dimension zero Reviewed
R. Takahashi
Math. Z. Vol. 251 ( 2 ) page： 249-256 2005
-
Homological invariants associated to semi-dualizing bimodules Reviewed
T. Araya, R. Takahashi, Y. Yoshino
J. Math. Kyoto Univ. Vol. 45 ( 2 ) page： 287-306 2005
-
Modules of G-dimension zero over local rings of depth two Reviewed
R. Takahashi
Illinois J. Math. Vol. 48 ( 3 ) page： 945-952 2004
-
Upper complete intersection dimension relative to a local homomorphism Reviewed
R. Takahashi
Tokyo J. Math. Vol. 27 ( 1 ) page： 209-219 2004
-
On the category of modules of Gorenstein dimension zero II Reviewed
R. Takahashi
J. Algebra Vol. 278 ( 1 ) page： 402-410 2004
-
Characterizing Cohen-Macaulay local rings by Frobenius maps Reviewed
R. Takahashi, Y. Yoshino
Proc. Amer. Math. Soc. Vol. 132 ( 11 ) page： 3177-3187 2004
-
Some characterizations of Gorenstein local rings in terms of G-dimension Reviewed
R. Takahashi
Acta Math. Hungar. Vol. 104 ( 4 ) page： 315-322 2004
-
Upper Cohen-Macaulay dimension Reviewed
T. Araya, R. Takahashi, Y. Yoshino
Math. J. Okayama Univ. Vol. 46 page： 17-30 2004
-
Nagata criterion for Serre's (Rn) and (Sn)-conditions Reviewed
R. Takahashi
Math. J. Okayama Univ. Vol. 41 page： 37-43 1999
KAKENHI (Grants-in-Aid for Scientific Research) 6
-
Grant number：23K03070 2023.4 - 2026.3
科学研究費助成事業 基盤研究(C)
高橋 亮
Authorship：Principal investigator
Grant amount：\4680000 （ Direct Cost: \3600000 、 Indirect Cost：\1080000 ）
圏（集合のようなもの）が与えられたとき、それの充満部分圏（部分集合のようなもの）のうち適切な条件をみたすものをすべて分類する・決定するという問題が考えられる。これは、数学の複数の分野にまたがって60年以上にわたり多くの研究者によって調べられてきた重要な問題である。本研究は、可換Noether環上の有限生成加群の有界複体全体のなす導来圏という特定の圏に対してこの問題を考察することを目的とする。
(1) 分解部分圏に関して以前得た結果を局所コホモロジーの余有限性問題に応用し、Bahmanpourの問いに一つの肯定的解答を与えた。(2) 可換ネーター環の各元が特異圏において零射で作用する対象全体のなす部分圏を調べた。特異圏の分解定理を示し、特異圏のRouquier次元の上限を与えた。(3) 新しいUlrich加群の概念を導入した。古典的なUlrich加群と異なりいつでも存在すること、それでいて古典的なUlrich加群の持つ性質を数多く維持することを示した。(4) 擬直可約極大イデアルをもつ環の図式による完全交差の特徴付けを与えた。また、有限単純グラフから自然に発生する完備局所環が擬直可約極大イデアルを持つことを示した。(5) 完全交差局所環に対し、その有限生成加群の有界導来圏のpreaisleのうち直和因子で閉じ環を含むものを完全に分類した。Stevensonの分類定理、Dao-Takahashiの分類定理を復元した。(6) Iyengar-Takahashiの問いの解答を与え、正標数のF有限なネーター環の強生成元をFrobenius順像を用いて具体的に与えた。(7) 特異圏のDimitrov-Haiden-Katzarkov-Kontsevich複雑量が零でない範囲が有界になることを、Krull次元が有限でGorensteinなJ-2環などの比較的弱い仮定をみたすいくつかのケースで示した。(8) Extの高次消滅が対称にならないGorenstein局所UFD、depth formulaをみたさないGorenstein局所UFD、環との正のExtが消滅する非全反射加群を持つCohen-Macaulay局所UFDを（等標数かつ孤立特異点を持つかたちで）構成した。(9) 局所環の導来圏を環と剰余体で割ってできる三角圏が加法生成元を持つための必要条件および十分条件を与えた。
「研究実績の概要」で述べたことはいずれも研究題目そのものに関する研究成果であるか、研究題目と密接に関係する研究成果である。たとえば、(2)および(7)で述べたように、可換ネーター環の特異圏の分解定理とRouquier次元の上限が得られ、Dimitrov-Haiden-Katzarkov-Kontsevich複雑量が零でない範囲の有界性が弱い仮定のもとで示されたが、これらの成果は特異圏の構造の理解に進展をもたらすものである。また(5)で述べたように、完全交差局所環の導来圏のpreaisleのうち直和因子で閉じ環を含むものを完全に分類したことで、導来圏の構造の理解が大いに進展したと言える。また(6)で述べたように、正標数のネーター環の加群圏の強生成元をFrobenius順像を用いて具体的に与えたが、加群圏の強生成元は導来圏の強生成元でもあるので、これは導来圏の構造の理解の進展をもたらすものである。さらに(9)で述べた、局所環の導来圏を環と剰余体で割ってできる三角圏が加法生成元を持つための必要条件および十分条件も、導来圏の構造の新しい理解を与えるものである。
今後も可換ネーター環上の有限生成加群の有界導来圏の構造解析およびそれに関連する研究に取り組む。超曲面局所環の一般化である支配的局所環（dominant local ring）に対して、導来圏のthick部分圏を考察する。具体的には、任意の素イデアルで局所化して支配的局所環になるような可換ネーター環、つまり局所支配環（locally dominant ring）の特異圏のthick部分圏は完全分類が既に終わっているので、それを用いて局所支配環の導来圏のthick部分圏の分類を試みるのが有効であると考えている。また、申請書の研究計画に記載したultimate dimensionの上界の研究にも本格的に着手する。具体的には、孤立特異点をもつ超曲面局所環に対してBallard-Favero-Katzarkovによるultimate dimensionの上界を得る手法の詳細な分析し、超曲面という条件がどれだけ本質的な役割を果たしているのか模索することから始める予定である。 -
Generation problems in module categories and derived categories of commutative noetherian rings
Grant number：19K03443 2019.4 - 2023.3
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (C)
Takahashi Ryo
Authorship：Principal investigator
Grant amount：\4290000 （ Direct Cost: \3300000 、 Indirect Cost：\990000 ）
I introduced a new class of commutative noetherian local rings which are called dominant local rings. I investigated basic properties of dominant local rings, and compare them with other classes of commutative noetherian local rings. Also, I gave various methods to get another one from a given dominant local ring. Moreover, under certain assumptions on the dominance of localizations, I classified suitable resolving subcategories of finitely generated modules, suitable thick subcategories of the bounded derived category of finitely generated modules, and the whole thick subcategories of the singularity category. This classification theorem recovers all the existing classification theorems in the same context.
-
Generation problems of subcategories of the module category of a Cohen-Macaulay local ring(Fostering Joint International Research)
Grant number：16KK0099 2017 - 2019
Fund for the Promotion of Joint International Research (Fostering Joint International Research)
Takahashi Ryo
Authorship：Principal investigator
Grant amount：\13260000 （ Direct Cost: \10200000 、 Indirect Cost：\3060000 ）
Various results are obtained. In particular, many properties of the structure of Cohen-Macaulay modules are clarified. A Cohen-Macaulay local ring is said to have finite CM+ representation type if there are only finitely many indecomposable Cohen-Macaulay modules whose nonfree loci have positive dimension. For a Gorenstein ring of dimension one, it is proved that having finite CM+ representation type is equivalent to having either an isolated singularity or countable CM representation type. Consider the Grothendieck group of finitely generated modules modulo the subgroup spanned by pseudo-zero modules. Tensor the real number field and consider the convex cone spanned by Cohen-Macaulay modules. Various topological properties on this convex cone are obtained.
-
Comprehensive research on representation theory of orders
Grant number：16H03923 2016.4 - 2021.3
Grants-in-Aid for Scientific Research Grant-in-Aid for Scientific Research (B)
Iyama Osamu
Authorship：Coinvestigator(s)
We studied the representation theory of orders mainly from a point of view of tilting theory, CM (=Cohen-Macaulay) representations, and AR (=Auslander-Reiten) theory. In addition to the following main results, we obtained a number of new results.
(1) I gave an invited talk at the International Congress of Mathematicians 2018 on tilting theory and CM representations. We also published a book on Geigle-Lenzing complete intersections from AMS Memoir. (2) We made a great progress on a basic theory in tilting theory including torsion classes, reductions of triangulated categories, simple-minded collections, and so on. (3) We studied Auslander-Gorenstein algebras from a point of view of higher dimensional AR theory, and applied them to the study of non-commutative resolutions of singularities. We also made a systematic construction of d-representation-finite selfinjective algebras, and applied it to the study of the periodicity of trivial extension algebras. -
Thick subcategories and dimensions of derived categories of commutative rings
Grant number：16K05098 2016.4 - 2020.3
Takahashi Ryo
Authorship：Principal investigator
Grant amount：\4550000 （ Direct Cost: \3500000 、 Indirect Cost：\1050000 ）
The thick subcategories of the singularity category of a local ring with quasi-decomposable maximal ideal are completely classified. The cocompactly generated thick tensor ideals of the right bounded derived category of finitely generated modules over a commutative noetherian ring are completely classified. For the stable category of Cohen-Macaulay modules over an Iwanaga Gorenstein isolated singularity, being locally finite and having finite representation type turn out to be equivalent. It turns out to hold that if a finitely generated Krull-Schmidt triangulated category is locally finite, then it has dimension zero, and the converse is true if it is Ext-finite. It is found out that the residue field of a local hypersurface of countable representation type has level at most one with respect to each nonzero object.
-
Foundations of rigid geometry from birational viewpoint
Grant number：15H03607 2015.4 - 2019.3
FUJIWARA KAZUHIRO
Authorship：Other
Aimed for establishing foundational aspects of rigid geometry, "Foundations of rigid geometry I" (joint with F. Kato at titech) was published from EMS (2018). A new aspect, i.e., spectral theory of filtered rings, is added during the project. Also, perfectoid spaces, which have close connection to rigid geometry, are put into our plan. After P. Scholze's introduction, many applications of perfectoid spaces are known up to now. The principal investigator has tried to understand phenomena behind cohomological purity conjectures from perfectoid viewpoint, and has given a new proof of the absolute purity conjecture.