Atom spectra of symmetric monoidal abelian categories and classification of subcategories

Shunya Saito

arxiv: 2604.25511 · v1 · submitted 2026-04-28 · 🧮 math.RT · math.CT· math.RA

Atom spectra of symmetric monoidal abelian categories and classification of subcategories

Shunya Saito This is my paper

Pith reviewed 2026-05-07 14:25 UTC · model grok-4.3

classification 🧮 math.RT math.CTmath.RA

keywords symmetric monoidal abelian categoryatom spectrumtorsion-free classSerre subcategoryKanda atom spectrumorbit atom spectrumtensor structurenoetherian abelian category

0 comments

The pith

Torsion-free classes compatible with the tensor structure are classified by arbitrary subsets of the orbit atom spectrum.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper extends the classification of torsion classes and torsion-free classes, previously known for modules over commutative noetherian rings, to suitable symmetric monoidal closed noetherian abelian categories. It defines the orbit atom spectrum as the quotient of Kanda's atom spectrum by the action of tensoring with invertible objects. Under natural tensor-theoretic assumptions, the paper shows that torsion-free classes respecting the monoidal structure correspond exactly to arbitrary subsets of this orbit spectrum. This approach also causes various classes of subcategories to collapse into Serre subcategories or torsion-free classes.

Core claim

Under natural tensor-theoretic assumptions in a symmetric monoidal closed noetherian abelian category, torsion-free classes compatible with the tensor structure are classified by arbitrary subsets of the orbit atom spectrum, defined as the quotient of Kanda's atom spectrum by the action induced by tensoring with invertible objects. Several classes of subcategories collapse to Serre subcategories or torsion-free classes. The result recovers the classical classifications for commutative noetherian rings and yields analogues for graded modules, coherent sheaves, and dg modules.

What carries the argument

The orbit atom spectrum, the quotient of Kanda's atom spectrum by the action induced by tensoring with invertible objects, which labels the torsion-free classes compatible with the tensor structure.

If this is right

Torsion-free classes compatible with the tensor structure correspond bijectively to arbitrary subsets of the orbit atom spectrum.
Under the tensor assumptions, several classes of subcategories coincide with Serre subcategories or torsion-free classes.
The classification recovers the known results for finitely generated modules over commutative noetherian rings.
Analogous classifications hold for categories of graded modules, coherent sheaves, and dg modules.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

In any new symmetric monoidal noetherian abelian category where the orbit atom spectrum can be explicitly computed, the bijection immediately supplies all tensor-compatible torsion-free classes.
The same quotient construction could be tested in non-noetherian or non-closed monoidal settings by checking which tensor assumptions remain necessary.
The collapse of subcategory classes suggests that similar simplifications may occur when classifying other tensor-compatible structures such as thick subcategories in derived settings.

Load-bearing premise

The category must be a suitable symmetric monoidal closed noetherian abelian category in which the natural tensor-theoretic assumptions hold and the quotient orbit atom spectrum is well-defined.

What would settle it

Construct or exhibit a symmetric monoidal closed noetherian abelian category satisfying the tensor assumptions together with a tensor-compatible torsion-free class that does not arise from any subset of the orbit atom spectrum.

read the original abstract

We extend the classification results for torsion classes and torsion-free classes in the category of finitely generated modules over a commutative noetherian ring to suitable symmetric monoidal closed noetherian abelian categories. Our main tool is the orbit atom spectrum, defined as the quotient of Kanda's atom spectrum by the action induced by tensoring with invertible objects. We prove that, under natural tensor-theoretic assumptions, several classes of subcategories collapse to Serre subcategories or torsion-free classes. Moreover, torsion-free classes compatible with the tensor structure are classified by arbitrary subsets of the orbit atom spectrum. As applications, we recover the classical classifications for commutative noetherian rings and obtain analogues for graded modules, coherent sheaves, and dg modules.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Saito quotients Kanda's atom spectrum by invertible objects to classify tensor-compatible torsion-free classes in monoidal abelian categories.

read the letter

The paper's main contribution is defining the orbit atom spectrum as the quotient of Kanda's atom spectrum under the tensor action of invertible objects, then showing that tensor-compatible torsion-free classes in symmetric monoidal closed noetherian abelian categories correspond to arbitrary subsets of this spectrum. It recovers the classical case for modules over commutative noetherian rings and gives versions for graded modules, coherent sheaves, and dg-modules by specializing the general setup. Under the stated tensor assumptions, various subcategory notions collapse to Serre subcategories or torsion-free classes as needed for the classification to work cleanly. This is a direct extension rather than a rephrasing of prior results. The unification across these examples is the part that lands best, since the same bijection applies once the category satisfies the noetherian and closed monoidal conditions. The construction itself is straightforward if one accepts Kanda's spectrum as input, and the applications follow without extra machinery. The soft spots are limited. The bijection with arbitrary subsets depends on the quotient preserving the relevant lattice properties, which holds under the listed assumptions but could use more explicit verification when the group of invertible objects is non-trivial, as in some graded or sheaf settings. The collapse statements are presented as independent but rely on the same tensor conditions, so a reader might want to see one or two non-ring examples worked out in full detail to confirm there are no hidden restrictions. No circularity or missing reduction is apparent from the stated claims. This paper is for people already working with atom spectra, torsion theories, or classification of subcategories in abelian categories. A reader familiar with Kanda's work or with module categories over rings will get the most out of the applications. It deserves a serious referee because the core construction is new, the evidence for the generalization is concrete in the recovered cases, and the framework is set up for further use in representation theory or algebraic geometry.

Referee Report

0 major / 3 minor

Summary. The manuscript extends the classification of torsion classes and torsion-free classes from the category of finitely generated modules over a commutative noetherian ring to suitable symmetric monoidal closed noetherian abelian categories. The central tool is the orbit atom spectrum, defined as the quotient of Kanda's atom spectrum by the action of invertible objects under tensoring. Under natural tensor-theoretic assumptions, several classes of subcategories are shown to collapse to Serre subcategories or torsion-free classes, and tensor-compatible torsion-free classes are classified by arbitrary subsets of the orbit atom spectrum. Applications recover the classical classification for commutative noetherian rings and yield analogues for graded modules, coherent sheaves, and dg-modules.

Significance. If the central claims hold, the work supplies a useful generalization of a standard classification theorem in commutative algebra to the broader setting of symmetric monoidal abelian categories. The orbit construction is presented as preserving the bijection between arbitrary subsets and the relevant subcategories, thereby unifying prior results while enabling new applications. Explicit recovery of the classical case for commutative rings and the treatment of coherent sheaves and dg-modules are concrete strengths that increase the result's reach in algebraic geometry and homological algebra.

minor comments (3)

[Abstract] The abstract refers to 'natural tensor-theoretic assumptions' without enumerating them; a short explicit list or forward reference to the section containing the precise hypotheses would improve readability.
Notation for the orbit atom spectrum should be introduced with a clear distinction from Kanda's original atom spectrum (e.g., via a dedicated definition or diagram) to prevent any ambiguity in later statements about quotients and subsets.
[Applications] In the applications section, the precise way the general theorem specializes to graded modules or coherent sheaves could be illustrated with one or two explicit subset correspondences to make the recovery of known results more transparent.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, the clear summary of its contributions, and the recommendation for minor revision. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity; orbit spectrum quotient and subset bijection are independent extensions

full rationale

The paper defines the orbit atom spectrum as the quotient of Kanda's (external) atom spectrum by the action of invertible objects via tensoring. It then proves that, under stated tensor assumptions, certain subcategories collapse to Serre or torsion-free classes and that tensor-compatible torsion-free classes are in bijection with arbitrary subsets of this quotient spectrum. This extends the classical classification for commutative noetherian rings without the main bijection or collapse statements reducing to a self-definition, fitted parameter, or self-citation chain. The tensor-theoretic assumptions supply independent content that forces the collapse, and the applications to graded modules, coherent sheaves, and dg-modules are new. No load-bearing step equates the output classification to its inputs by construction. This is the normal non-circular case for an extension paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claim rests on the new orbit atom spectrum construction and standard domain assumptions about tensor behavior; no free parameters are introduced.

axioms (2)

domain assumption natural tensor-theoretic assumptions hold
Invoked to ensure collapse of subcategory classes to Serre or torsion-free and to obtain the classification by subsets.
domain assumption Kanda's atom spectrum admits a well-defined quotient under the action induced by tensoring with invertible objects
Required for the definition of the orbit atom spectrum to make sense.

invented entities (1)

orbit atom spectrum no independent evidence
purpose: Classify tensor-compatible torsion-free classes via arbitrary subsets
Newly defined as the quotient of Kanda's atom spectrum by the invertible objects action.

pith-pipeline@v0.9.0 · 5415 in / 1473 out tokens · 78191 ms · 2026-05-07T14:25:31.229424+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

48 extracted references · 48 canonical work pages

[1]

T.\ Adachi, O.\ Iyama, I.\ Reiten, -tilting theory, Compos.\ Math.\ 150 (2014), no.\ 3, 415--452

work page 2014
[2]

S.\ Asai, Semibricks, Int.\ Math.\ Res.\ Not.\ IMRN (2020), no.\ 16, 4993--5054

work page 2020
[3]

K.\ Ahmadi, R.\ Sazeedeh, Hereditary torsion theories of a locally Noetherian Grothendieck category, Bull.\ Aust.\ Math.\ Soc.\ 94 (2016), no.\ 3, 421--430

work page 2016
[4]

I.\ Assem, D.\ Simson, A.\ Skowro\' n ski, Elements of the representation theory of associative algebras. Vol.\ 1. Techniques of representation theory, London Math.\ Soc.\ Stud.\ Texts, 65 Cambridge University Press, Cambridge, 2006. x+458 pp

work page 2006
[5]

R.\ H.\ Bak, Dualizable and semi-flat objects in abstract module categories, Math.\ Z.\ 296 (2020), no.\ 1--2, 353--371

work page 2020
[6]

P.\ Balmer, The spectrum of prime ideals in tensor triangulated categories, J.\ Reine Angew.\ Math.\ 588 (2005), 149--168

work page 2005
[7]

P.\ Balmer, H.\ Krause, G.\ Stevenson, Tensor-triangular fields: ruminations, Selecta Math.\ 25 (2019), no.\ 1, Paper No.\ 13, 36 pp

work page 2019
[8]

M.\ Borelli, Divisorial varieties, Pacific J.\ Math.\ 13 (1963), 375--388

work page 1963
[9]

P.\ C oupek, J.\ S t ov\' i c ek, Cotilting sheaves on Noetherian schemes, Math.\ Z.\ 296 (2020), no.\ 1--2, 275--312

work page 2020
[10]

B.\ Day, On closed categories of functors, Reports of the Midwest Category Seminar, IV, pp.\ 1--38, Lecture Notes in Math., Vol.\ 137 Springer-Verlag, Berlin-New York, 1970

work page 1970
[11]

H.\ Enomoto, Rigid modules and ICE-closed subcategories in quiver representations, J.\ Algebra 594 (2022), 364--388

work page 2022
[12]

H.\ Enomoto, IE-closed subcategories of commutative rings are torsion-free classes, preprint (2023), arXiv:2304.03260v2

work page arXiv 2023
[13]

H.\ Enomoto, A.\ Sakai, ICE-closed subcategories and wide -tilting modules, Math.\ Z.\ 300 (2022), no.\ 1, 541--577

work page 2022
[14]

P.\ Etingof, S.\ Gelaki, D.\ Nikshych, V.\ Ostrik, Tensor categories, Math.\ Surveys Monogr.\ 205, American Mathematical Society, Providence, RI, 2015, xvi+343 pp

work page 2015
[15]

P.\ Gabriel, Des cat\' e gories ab\' e liennes , Bull.\ Soc.\ Math.\ France 90 (1962), 323--448

work page 1962
[16]

Schemes--with examples and exercises, Second edition, Springer Studium Mathematik--Master, Springer Spektrum, Wiesbaden, 2020, vii+625 pp

U.\ G\" o rtz, T.\ Wedhorn, Algebraic geometry I. Schemes--with examples and exercises, Second edition, Springer Studium Mathematik--Master, Springer Spektrum, Wiesbaden, 2020, vii+625 pp

work page 2020
[17]

U.\ G\" o rtz, T.\ Wedhorn, Algebraic geometry II: Cohomology of scheme--with examples and exercises, Springer Stud.\ Math.\ Master, Springer Spektrum, Wiesbaden, 2023, vii+869 pp

work page 2023
[18]

P.\ Gross, The resolution property of algebraic surfaces, Compos.\ Math.\ 148 (2012), no.\ 1, 209--226

work page 2012
[19]

I.\ Herzog, The Ziegler spectrum of a locally coherent Grothendieck category, Proc.\ London Math.\ Soc.\ (3) 74 (1997), no.\ 3, 503--558

work page 1997
[20]

Sci.\ China Math.\ 66 (2023), no.\ 11, 2471--2494

H.\ Holm, S.\ Odaba s , The tensor embedding for a Grothendieck cosmos. Sci.\ China Math.\ 66 (2023), no.\ 11, 2471--2494

work page 2023
[21]

H.\ Al Hwaeer, G.\ Garkusha, Grothendieck categories of enriched functors, J.\ Algebra 450 (2016), 204--241

work page 2016
[22]

C.\ Ingalls, H.\ Thomas, Noncrossing partitions and representations of quivers, Compos.\ Math.\ 145 (2009), no.\ 6, 1533--1562

work page 2009
[23]

K.\ Iima, H.\ Matsui, K.\ Shimada, R.\ Takahashi, When Is a Subcategory Serre or Torsion-Free?, Publ.\ Res.\ Inst.\ Math.\ Sci.\ 60 (2024), no.\ 4, 831--857

work page 2024
[24]

O.\ Iyama, Y.\ Kimura, Classifying subcategories of modules over Noetherian algebras, Adv.\ Math.\ 446 (2024), Paper No.\ 109631

work page 2024
[25]

A.\ Jerem\' i as L\' o pez, M.\ P.\ L\' o pez, E.\ Villanueva N\' o voa, Localization in symmetric closed Grothendieck categories, Third Week on Algebra and Algebraic Geometry (SAGA III), Bull.\ Soc.\ Math.\ Belg.\ S\' e r.\ A 45 (1993), no.\ 1--2, 197--221

work page 1993
[26]

R.\ Kanda, Classifying Serre subcategories via atom spectrum, Adv.\ Math.\ 231 (2012), no.\ 3--4, 1572--1588

work page 2012
[27]

R.\ Kanda, Specialization orders on atom spectra of Grothendieck categories, J.\ Pure Appl.\ Algebra 219 (2015), no.\ 11, 4907--4952

work page 2015
[28]

R.\ Kanda, Classification of categorical subspaces of locally Noetherian schemes, Doc.\ Math.\ 20 (2015), 1403--1465

work page 2015
[29]

xii+531 pp

C.\ Kassel, Quantum groups, Grad.\ Texts in Math., 155, Springer-Verlag, New York, 1995. xii+531 pp

work page 1995
[30]

G.\ M.\ Kelly, Basic concepts of enriched category theory, London Math.\ Soc.\ Lecture Note Ser.\ 64, Cambridge University Press, Cambridge-New York, 1982. 245 pp

work page 1982
[31]

B.\ Keller, Deriving DG categories , Ann.\ Sci.\ \' E cole Norm.\ Sup.\ (4) 27 (1994), no.\ 1, 63--102

work page 1994
[32]

T.\ Kobayashi, S.\ Saito, When are KE-closed subcategories torsion-free classes?, Math.\ Z.\ 307 (2024), no.\ 4, Paper No.\ 65

work page 2024
[33]

T.\ Kobayashi, S.\ Saito, Classifying KE-closed subcategories over a commutative noetherian ring, preprint (2025), arXiv:2509.05767

work page arXiv 2025
[34]

H.\ Krause, The spectrum of a locally coherent category, J.\ Pure Appl.\ Algebra 114 (1997), no.\ 3, 259--271

work page 1997
[35]

H.\ Krause, Thick subcategories of modules over commutative Noetherian rings (with an appendix by Srikanth Iyengar), Math.\ Ann.\ 340 (2008), no.\ 4, 733--747

work page 2008
[36]

H.\ Krause, Krull--Schmidt categories and projective covers, Expo.\ Math.\ 33 (2015), no.\ 4, 535--549

work page 2015
[37]

Grad.\ Texts in Math.\ 5, Springer-Verlag, New York, 1998

S.\ Mac Lane, Categories for the working mathematician, Second edition. Grad.\ Texts in Math.\ 5, Springer-Verlag, New York, 1998. xii+314 pp

work page 1998
[38]

F.\ Marks, J.\ S t'ov\' i c ek, Torsion classes, wide subcategories and localisations, Bull.\ Lond.\ Math.\ Soc.\ 49 (2017), no.\ 3, 405--416

work page 2017
[39]

xx+636 pp

J.\ C.\ McConnell, J.\ C.\ Robson, Noncommutative Noetherian rings, With the cooperation of L.\ W.\ Small.\ Revised edition, Grad.\ Stud.\ Math., 30, American Mathematical Society, Providence, RI, 2001. xx+636 pp

work page 2001
[40]

xiv+304 pp

C.\ N a st a sescu, F.\ Van Oystaeyen, Methods of graded rings, Lecture Notes in Math.\ 1836 Springer-Verlag, Berlin, 2004. xiv+304 pp

work page 2004
[41]

S.\ Posur, Atom spectra of graded rings and sheafification in toric geometry, J.\ Algebra 534 (2019), 207--227

work page 2019
[42]

S.\ Saito, Classifying torsionfree classes of the category of coherent sheaves and their Serre subcategories, J.\ Pure Appl.\ Algebra 229 (2025), no.\ 1, Paper No.\ 107799, 34 pp

work page 2025
[43]

D.\ Stanley, B.\ Wang, Classifying subcategories of finitely generated modules over a Noetherian ring, J.\ Pure Appl.\ Algebra 215 (2011), no.\ 11, 2684--2693

work page 2011
[44]

An introduction to methods of ring theory, Die Grundlehren der mathematischen Wissenschaften, Band 217 Springer-Verlag, New York-Heidelberg, 1975

B.\ Stenstr\" o m, Rings of quotients. An introduction to methods of ring theory, Die Grundlehren der mathematischen Wissenschaften, Band 217 Springer-Verlag, New York-Heidelberg, 1975. viii+309 pp

work page 1975
[45]

R.\ Takahashi, Classifying subcategories of modules over a commutative Noetherian ring, J.\ Lond.\ Math.\ Soc.\ (2) 78 (2008), no.\ 3, 767--782

work page 2008
[46]

B.\ Totaro, The resolution property for schemes and stacks, J.\ Reine Angew.\ Math.\ 577 (2004), 1--22

work page 2004
[47]

P.\ Berthelot, A.\ Grothendieck, L.\ Illusie, Th\' e orie des intersections et th\' e or\` e me de Riemann-Roch , Lecture Notes in Mathematics, 225, Springer-Verlag, 1971

work page 1971
[48]

The Stacks Project Authors , Stacks Project, https://stacks.math.columbia.edu

work page

[1] [1]

T.\ Adachi, O.\ Iyama, I.\ Reiten, -tilting theory, Compos.\ Math.\ 150 (2014), no.\ 3, 415--452

work page 2014

[2] [2]

S.\ Asai, Semibricks, Int.\ Math.\ Res.\ Not.\ IMRN (2020), no.\ 16, 4993--5054

work page 2020

[3] [3]

K.\ Ahmadi, R.\ Sazeedeh, Hereditary torsion theories of a locally Noetherian Grothendieck category, Bull.\ Aust.\ Math.\ Soc.\ 94 (2016), no.\ 3, 421--430

work page 2016

[4] [4]

I.\ Assem, D.\ Simson, A.\ Skowro\' n ski, Elements of the representation theory of associative algebras. Vol.\ 1. Techniques of representation theory, London Math.\ Soc.\ Stud.\ Texts, 65 Cambridge University Press, Cambridge, 2006. x+458 pp

work page 2006

[5] [5]

R.\ H.\ Bak, Dualizable and semi-flat objects in abstract module categories, Math.\ Z.\ 296 (2020), no.\ 1--2, 353--371

work page 2020

[6] [6]

P.\ Balmer, The spectrum of prime ideals in tensor triangulated categories, J.\ Reine Angew.\ Math.\ 588 (2005), 149--168

work page 2005

[7] [7]

P.\ Balmer, H.\ Krause, G.\ Stevenson, Tensor-triangular fields: ruminations, Selecta Math.\ 25 (2019), no.\ 1, Paper No.\ 13, 36 pp

work page 2019

[8] [8]

M.\ Borelli, Divisorial varieties, Pacific J.\ Math.\ 13 (1963), 375--388

work page 1963

[9] [9]

P.\ C oupek, J.\ S t ov\' i c ek, Cotilting sheaves on Noetherian schemes, Math.\ Z.\ 296 (2020), no.\ 1--2, 275--312

work page 2020

[10] [10]

B.\ Day, On closed categories of functors, Reports of the Midwest Category Seminar, IV, pp.\ 1--38, Lecture Notes in Math., Vol.\ 137 Springer-Verlag, Berlin-New York, 1970

work page 1970

[11] [11]

H.\ Enomoto, Rigid modules and ICE-closed subcategories in quiver representations, J.\ Algebra 594 (2022), 364--388

work page 2022

[12] [12]

H.\ Enomoto, IE-closed subcategories of commutative rings are torsion-free classes, preprint (2023), arXiv:2304.03260v2

work page arXiv 2023

[13] [13]

H.\ Enomoto, A.\ Sakai, ICE-closed subcategories and wide -tilting modules, Math.\ Z.\ 300 (2022), no.\ 1, 541--577

work page 2022

[14] [14]

P.\ Etingof, S.\ Gelaki, D.\ Nikshych, V.\ Ostrik, Tensor categories, Math.\ Surveys Monogr.\ 205, American Mathematical Society, Providence, RI, 2015, xvi+343 pp

work page 2015

[15] [15]

P.\ Gabriel, Des cat\' e gories ab\' e liennes , Bull.\ Soc.\ Math.\ France 90 (1962), 323--448

work page 1962

[16] [16]

Schemes--with examples and exercises, Second edition, Springer Studium Mathematik--Master, Springer Spektrum, Wiesbaden, 2020, vii+625 pp

U.\ G\" o rtz, T.\ Wedhorn, Algebraic geometry I. Schemes--with examples and exercises, Second edition, Springer Studium Mathematik--Master, Springer Spektrum, Wiesbaden, 2020, vii+625 pp

work page 2020

[17] [17]

U.\ G\" o rtz, T.\ Wedhorn, Algebraic geometry II: Cohomology of scheme--with examples and exercises, Springer Stud.\ Math.\ Master, Springer Spektrum, Wiesbaden, 2023, vii+869 pp

work page 2023

[18] [18]

P.\ Gross, The resolution property of algebraic surfaces, Compos.\ Math.\ 148 (2012), no.\ 1, 209--226

work page 2012

[19] [19]

I.\ Herzog, The Ziegler spectrum of a locally coherent Grothendieck category, Proc.\ London Math.\ Soc.\ (3) 74 (1997), no.\ 3, 503--558

work page 1997

[20] [20]

Sci.\ China Math.\ 66 (2023), no.\ 11, 2471--2494

H.\ Holm, S.\ Odaba s , The tensor embedding for a Grothendieck cosmos. Sci.\ China Math.\ 66 (2023), no.\ 11, 2471--2494

work page 2023

[21] [21]

H.\ Al Hwaeer, G.\ Garkusha, Grothendieck categories of enriched functors, J.\ Algebra 450 (2016), 204--241

work page 2016

[22] [22]

C.\ Ingalls, H.\ Thomas, Noncrossing partitions and representations of quivers, Compos.\ Math.\ 145 (2009), no.\ 6, 1533--1562

work page 2009

[23] [23]

K.\ Iima, H.\ Matsui, K.\ Shimada, R.\ Takahashi, When Is a Subcategory Serre or Torsion-Free?, Publ.\ Res.\ Inst.\ Math.\ Sci.\ 60 (2024), no.\ 4, 831--857

work page 2024

[24] [24]

O.\ Iyama, Y.\ Kimura, Classifying subcategories of modules over Noetherian algebras, Adv.\ Math.\ 446 (2024), Paper No.\ 109631

work page 2024

[25] [25]

A.\ Jerem\' i as L\' o pez, M.\ P.\ L\' o pez, E.\ Villanueva N\' o voa, Localization in symmetric closed Grothendieck categories, Third Week on Algebra and Algebraic Geometry (SAGA III), Bull.\ Soc.\ Math.\ Belg.\ S\' e r.\ A 45 (1993), no.\ 1--2, 197--221

work page 1993

[26] [26]

R.\ Kanda, Classifying Serre subcategories via atom spectrum, Adv.\ Math.\ 231 (2012), no.\ 3--4, 1572--1588

work page 2012

[27] [27]

R.\ Kanda, Specialization orders on atom spectra of Grothendieck categories, J.\ Pure Appl.\ Algebra 219 (2015), no.\ 11, 4907--4952

work page 2015

[28] [28]

R.\ Kanda, Classification of categorical subspaces of locally Noetherian schemes, Doc.\ Math.\ 20 (2015), 1403--1465

work page 2015

[29] [29]

xii+531 pp

C.\ Kassel, Quantum groups, Grad.\ Texts in Math., 155, Springer-Verlag, New York, 1995. xii+531 pp

work page 1995

[30] [30]

G.\ M.\ Kelly, Basic concepts of enriched category theory, London Math.\ Soc.\ Lecture Note Ser.\ 64, Cambridge University Press, Cambridge-New York, 1982. 245 pp

work page 1982

[31] [31]

B.\ Keller, Deriving DG categories , Ann.\ Sci.\ \' E cole Norm.\ Sup.\ (4) 27 (1994), no.\ 1, 63--102

work page 1994

[32] [32]

T.\ Kobayashi, S.\ Saito, When are KE-closed subcategories torsion-free classes?, Math.\ Z.\ 307 (2024), no.\ 4, Paper No.\ 65

work page 2024

[33] [33]

T.\ Kobayashi, S.\ Saito, Classifying KE-closed subcategories over a commutative noetherian ring, preprint (2025), arXiv:2509.05767

work page arXiv 2025

[34] [34]

H.\ Krause, The spectrum of a locally coherent category, J.\ Pure Appl.\ Algebra 114 (1997), no.\ 3, 259--271

work page 1997

[35] [35]

H.\ Krause, Thick subcategories of modules over commutative Noetherian rings (with an appendix by Srikanth Iyengar), Math.\ Ann.\ 340 (2008), no.\ 4, 733--747

work page 2008

[36] [36]

H.\ Krause, Krull--Schmidt categories and projective covers, Expo.\ Math.\ 33 (2015), no.\ 4, 535--549

work page 2015

[37] [37]

Grad.\ Texts in Math.\ 5, Springer-Verlag, New York, 1998

S.\ Mac Lane, Categories for the working mathematician, Second edition. Grad.\ Texts in Math.\ 5, Springer-Verlag, New York, 1998. xii+314 pp

work page 1998

[38] [38]

F.\ Marks, J.\ S t'ov\' i c ek, Torsion classes, wide subcategories and localisations, Bull.\ Lond.\ Math.\ Soc.\ 49 (2017), no.\ 3, 405--416

work page 2017

[39] [39]

xx+636 pp

J.\ C.\ McConnell, J.\ C.\ Robson, Noncommutative Noetherian rings, With the cooperation of L.\ W.\ Small.\ Revised edition, Grad.\ Stud.\ Math., 30, American Mathematical Society, Providence, RI, 2001. xx+636 pp

work page 2001

[40] [40]

xiv+304 pp

C.\ N a st a sescu, F.\ Van Oystaeyen, Methods of graded rings, Lecture Notes in Math.\ 1836 Springer-Verlag, Berlin, 2004. xiv+304 pp

work page 2004

[41] [41]

S.\ Posur, Atom spectra of graded rings and sheafification in toric geometry, J.\ Algebra 534 (2019), 207--227

work page 2019

[42] [42]

S.\ Saito, Classifying torsionfree classes of the category of coherent sheaves and their Serre subcategories, J.\ Pure Appl.\ Algebra 229 (2025), no.\ 1, Paper No.\ 107799, 34 pp

work page 2025

[43] [43]

D.\ Stanley, B.\ Wang, Classifying subcategories of finitely generated modules over a Noetherian ring, J.\ Pure Appl.\ Algebra 215 (2011), no.\ 11, 2684--2693

work page 2011

[44] [44]

An introduction to methods of ring theory, Die Grundlehren der mathematischen Wissenschaften, Band 217 Springer-Verlag, New York-Heidelberg, 1975

B.\ Stenstr\" o m, Rings of quotients. An introduction to methods of ring theory, Die Grundlehren der mathematischen Wissenschaften, Band 217 Springer-Verlag, New York-Heidelberg, 1975. viii+309 pp

work page 1975

[45] [45]

R.\ Takahashi, Classifying subcategories of modules over a commutative Noetherian ring, J.\ Lond.\ Math.\ Soc.\ (2) 78 (2008), no.\ 3, 767--782

work page 2008

[46] [46]

B.\ Totaro, The resolution property for schemes and stacks, J.\ Reine Angew.\ Math.\ 577 (2004), 1--22

work page 2004

[47] [47]

P.\ Berthelot, A.\ Grothendieck, L.\ Illusie, Th\' e orie des intersections et th\' e or\` e me de Riemann-Roch , Lecture Notes in Mathematics, 225, Springer-Verlag, 1971

work page 1971

[48] [48]

The Stacks Project Authors , Stacks Project, https://stacks.math.columbia.edu

work page