On the depth of tensor products over Cohen-Macaulay rings

Andrew J. Soto-Levins; Justin Lyle; Kaito Kimura

REVIEW 2 major objections 4 minor 27 references

On the depth of tensor products over Cohen-Macaulay rings

T0 review · 2 major / 4 minor · reviewed 2026-08-16 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that the derived left-hand depth inequality over a Cohen-Macaulay local ring is equivalent to the ring having its dimension as a uniform Auslander bound.

desk verdict A useful paper on the depth formula with a real proof gap in the main theorem: (3)⇒(4) of Theorem 4.4 applies Lemma 3.9 to f_M⊗f_N although the lemma requires f_M⊗N. read the letter →

arxiv 2505.00441 v1 pith:B662QT5I submitted 2025-05-01 math.AC

Kaito Kimura , Justin Lyle , Andrew J. Soto-Levins This is my paper · ORCID

classification math.AC MSC 13C1513D07

keywords depthtensorproductmaximalCohen-MacaulayuniformAuslanderconditionBuchweitzderivedcategoryTorExt

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper splits the classical depth formula for tensor products over a local ring into two inequalities, called (ldep) and (rdep), and asks when each can hold. Its main theorem is that, over a Cohen-Macaulay local ring $R$ of dimension $d$, the derived left inequality—$\operatorname{depth}_R(M\otimes^{\mathbf{L}}_R N)+\operatorname{depth}(R)\ge \operatorname{depth}_R(M)+\operatorname{depth}_R(N)$ whenever the derived tensor product has bounded homology—is equivalent to $d$ being the ring's uniform Auslander bound, a strong vanishing condition on Ext. In other words, one global number on Ext-vanishing completely controls whether the left half of the depth formula holds, and the same circle of conditions implies that maximal Cohen-Macaulay modules are Tor-independent whenever their Tor is finite. A dual theorem ties the right inequality to a new 'uniform Buchweitz condition.' The upshot is that depth conditions on tensor products detect representation-theoretic finiteness previously studied in other terms, and they yield a formula for the top nonvanishing Tor degree $q_R(M,N)$ as a supremum of local depth discrepancies.

What carries the argument

The load-bearing objects are the derived tensor product $M\otimes^{\mathbf{L}}_R N$, whose depth the paper defines through Ext into the residue field and local cohomology, and the $d$-th syzygy construction $\Omega^d_R(-)$, normalized so that an MCM complex is one whose local cohomology is concentrated in degree $\dim(R)$. The paper repeatedly uses $f_M := \Omega^d_R(M/xM)$ for a maximal regular sequence $x$: it is an MCM module supported on the punctured spectrum, and Lemma 3.9 converts the statement that $f_M \otimes N$ is MCM into Tor-vanishing between $M$ and $N$. The argument for Theorem 1.1 also passes to the completion to obtain a canonical module and applies local duality, reducing the equivalence to the Uniform Auslander Condition. For the right-hand condition, the new Uniform Buchweitz Condition (UBC) plays the dual role, saying every finite Ext-vanishing $b_R(M,N)$ is at least $\operatorname{codepth}_R(M)$; the proof reduces (rdep) to constant-rank modules on the punctured spectrum and then to control of nonfree loci via pushforwards along multiplication by a regular element.

What would settle it

Find a Cohen-Macaulay local ring $R$ of dimension $d>0$ with two maximal Cohen-Macaulay modules $M,N$ satisfying $q_R(M,N)=1$: nonzero Tor in degree 1 and vanishing beyond. Theorem 1.1 predicts that such a ring cannot satisfy derived (ldep), because condition (4) would force $q_R(M,N)=0$. So exhibiting such a ring that does satisfy derived (ldep)—or merely checking whether a ring known to satisfy condition (3) admits such a pair—would settle the equivalence; equivalently, for any candidate ring, computing the Auslander bound $b_R$ and checking whether $b_R=d$ gives a direct numerical test.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is Theorem 1.1: for a Cohen-Macaulay local ring $R$ of dimension $d$, the following are equivalent: derived (ldep); derived (ldep) for modules; the condition that $d$-th syzygies of finite-length modules have maximal Cohen-Macaulay (MCM) derived tensor products whenever their Tor is finite; the condition that MCM modules with finite Tor are Tor-independent with MCM tensor product; the bound $\operatorname{codepth}_R(M):=\operatorname{depth}(R)-\operatorname{depth}_R(M)$ on every finite Ext-vanishing $b_R(M,N)$; and the Uniform Auslander Condition with bound $d$. If this theorem is right, the left half of the depth formula is not a mild hypothesis—it is exactly the statement that the ring's Auslander bound equals its dimension. For Gorenstein rings of positive dimension the paper derives that (ldep) implies (rdep), so the full depth formula follows from the left inequality alone. The companion theorem for (rdep) shows that (1) implies (2), that (2), (3), and (4) are equivalent, that (4) implies (5) implies (6), that (3) implies (7) implies (8), that (2) implies (1) when $d>0$, and that (6) and (8) are equivalent when a canonical module exists. The paper also proves Theorem 1.3: if derived (dep) holds at every localization, then $q_R(M,N)=\sup\{\operatorname{depth}(R_p)-\operatorname{depth}_{R_p}(M_p)-\operatorname{depth}_{R_p}(N_p)\mid p\in\operatorname{Supp}(M)\cap\operatorname{Supp}(N)\}$.

Load-bearing premise

The load-bearing premise is that the reduction to a complete ring with a canonical module is free—specifically, the cited completion lemmas [KLOT23, Lemma 2.10] and [CH10, Remark 5.7] apply as stated—and that Lemma 3.9, proved for $f_M \otimes N$, also yields its conclusion for $f_M \otimes f_N$ in the step from (3) to (4).

Editorial extensions

If this is right

On a Cohen-Macaulay ring of dimension $d$, if the derived left depth inequality holds, then every pair of maximal Cohen-Macaulay modules with finite Tor is Tor-independent and has maximal Cohen-Macaulay tensor product.
For Gorenstein rings of positive dimension, (ldep) implies (rdep), so the full classical depth formula holds whenever its left half does.
The (ldep) and (rdep) conditions ascend and descend along completion and modding out by a regular sequence, but they need not localize; Example 5.15 exhibits a complete Cohen-Macaulay ring satisfying them globally whose localization at a prime fails them.
When derived (dep) holds at every localization, the top nonvanishing Tor degree $q_R(M,N)$ is exactly the supremum of $\operatorname{depth}(R_p)-\operatorname{depth}_{R_p}(M_p)-\operatorname{depth}_{R_p}(N_p)$ over the common support, generalizing Jorgensen's formula to modules beyond complete intersection dimension.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If Theorem 1.1 is correct, computational searches for rings satisfying (ldep) can be replaced by a single computation of the Auslander bound $b_R$; any ring with $b_R>\dim(R)$ automatically provides an explicit pair of modules for which the depth formula's left inequality fails.
The internal gap in the proof of (3) implies (4)—Lemma 3.9 is stated for $f_M \otimes N$ yet applied to $f_M \otimes f_N$—suggests a stable-isomorphism argument is needed; if no such argument exists, condition (3) may be strictly weaker than derived (ldep), which would be a precise place to look for a counterexample.
Since every Artinian ring satisfies derived (rdep) for modules, the right-hand condition only becomes restrictive in positive dimension; this suggests studying derived (rdep) for complexes, where the paper shows the Artinian obstruction is real, as the genuinely uniform version.
The paper's $q_R$ formula, combined with the observation that derived (dep) localizes for modules of finite complete intersection dimension but not in general, raises the question of exactly which module classes the equality can cover; each such class would give a new extension of Jorgensen's formula.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: this is a genuinely useful paper, but the proof of Theorem 1.1 has a load-bearing gap at (3)⇒(4). The reader's stress-test is right. In the proof, after showing f_M⊗f_N is MCM (where f_M = Ω^d(M/xM) and f_N = Ω^d(N/xN)), the text says "Lemma 3.9 gives Tor_{1≤i≤d}(M,N)=0 and M⊗N is MCM." But Lemma 3.9 as stated concerns f_M⊗N, not f_M⊗f_N. You would need f_N to be stably isomorphic to N, which is false in general — e.g., for a 1-dimensional CM domain and N=R, f_N is the canonical module's syzygy, essentially the maximal ideal, not free. The gap is not cosmetic: it is the step that connects condition (3) to the Tor-vanishing and MCM conclusions, so the equivalence of (1)–(6) is not established as written.

What the paper does well: it splits the depth formula into (ldep) and (rdep), proves that derived (ldep) is equivalent to UAC with b_R = dim(R), introduces a plausible dual condition UBC, extends Jorgensen's formula for q_R(M,N), and gives the first examples showing these conditions need not localize. The writing is clear and the surrounding lemmas are mostly well argued. The q_R formula in Section 6 and the non-localization examples are likely to survive independent of the gap.

Where I stand: the central idea is good and the paper deserves a serious referee. The gap is real and must be fixed before the main theorem can be trusted, but it does not look obviously fatal — the authors may be able to supply an extra argument or adjust condition (3). I would send this to review, with the explicit instruction that the proof of (3)⇒(4) be repaired. If it is fixable, this is a strong paper. If the authors cannot fix it, the equivalence claim should be weakened.

For a reading group, this is actually a good paper to work through — the definitions are clean and the gap provides a useful exercise in checking how Lemma 3.9 is being used. I would cite it once the gap is resolved, but not before.

Referee Report

2 major / 4 minor

Summary. The paper studies depth inequalities for derived tensor products over commutative Noetherian local rings. It introduces left- and right-hand depth formulas (ldep) and (rdep), together with derived variants, and proves that for Cohen-Macaulay rings derived (ldep) is equivalent to the uniform Auslander condition with bound equal to the dimension; it introduces a dual uniform Buchweitz condition for (rdep), proves transfer properties under regular sequences and completion, gives examples showing failure of localization, and extends a formula of Jorgensen for q_R(M,N). The main theorems are Theorem 1.1 (an eight-condition equivalence for (ldep)), Theorem 1.2 (an analogous statement for (rdep)), and Theorem 1.3 (a local-to-global formula for q_R(M,N)).

Significance. The paper is well organized and generally carefully written, with detailed proofs and several illuminating examples. The proposed equivalence between derived (ldep) and the uniform Auslander condition is a strong and natural structural result if it holds, and the introduction of (UBC) as a dual notion for (rdep) is a useful contribution. The extension of Jorgensen's formula is a genuine added value. However, the proof of the central equivalence contains a gap in the (3) implies (4) step of Theorem 4.4, so the main result is not yet fully established as written.

major comments (2)

[Theorem 4.4, proof of (3) implies (4) (pp. 15-16)] The proof asserts that once f_M tensor^L_R e_N is MCM and q_R(f_M,e_N)=0, Lemma 3.9 gives Tor^R_{1 <= i <= d}(M,N)=0 and M tensor_R N MCM. This does not follow from the stated Lemma 3.9: its condition (3) is the MCM property of f_M tensor_R N for the original pair (M,N), not of f_M tensor_R e_N. Passing from e_N to N requires a stable isomorphism Omega^d_R(N/xN) congruent to N plus a free module, which is false in general. For example, with R = k[[t^2,t^3]], d = 1, N = m, and x = t^2, one has N/xN isomorphic to k^2, so e_N = Omega^1_R(N/xN) is isomorphic to m direct sum m; if this were stably isomorphic to N = m, ranks would force m direct sum m direct sum R^a isomorphic to m direct sum R^{a+1}, while the minimal number of generators would give 4+a = a+3, a contradiction. Since this step is the only argument linking condition (3) to condition (4), the equivalence of (1)-(6) in Theorem 1.1 is not established as written.
[Theorem 4.4, proof of (5) implies (3) (p. 16)] The displayed identity b_R(M,N^vee) = b_R(A,N^vee) + d has the wrong sign: for M = Omega^d_R(A), dimension shifting gives Ext^i_R(M,N^vee) isomorphic to Ext^{i+d}_R(A,N^vee) for i > 0, so the shift is by -d, not +d. The intended conclusion b_R(M,N^vee) = 0 can nevertheless be recovered directly from condition (5), because M is MCM and hence codepth_R(M) = 0; the erroneous formula should be removed or corrected.

minor comments (4)

[Theorem 4.4, proof of (4) implies (7)] In the (4) implies (7) part, the text says 'the condition of (3) forces q_R(M,N^vee)=0 and M tensor_R N is MCM'; it should refer to condition (4), and the module obtained as MCM is M tensor_R N^vee, from which Proposition 3.8 then gives b_R(M,N)=0.
[Theorem 4.4, proof structure] The paragraph labeled 'Next we show (7) implies (4)' invokes condition (5) before (5) has been proved; since the authors later prove (7) implies (5), the proof can be reordered, but as written the paragraph is logically premature and should be relabeled or moved.
[Section 6, Lemmas 6.1-6.2 and Theorem 6.3] The phrase 'derived (rdep) holds on SpecR for the module M' should read 'derived (rdep) holds for the local rings R_p' or similar; the hypotheses are ring conditions, not module conditions, and the current wording is confusing.
[Example 5.13] The statement that 'neither R nor A satisfies derived (ldep) by Corollary 4.6' is inaccurate for R, since R is Artinian and Corollary 4.6 requires dim(R) > 0; the conclusion for A, and then for B, suffices for the argument.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main equivalences are proved from independently defined conditions; flagged proof gaps concern correctness, not circularity.

full rationale

The paper's central derivation chain is not circular. The conditions (ldep), (rdep), their derived variants, UAC, and UBC are independently defined, and Theorems 4.4 and 5.8 prove equivalences among them by depth-lemma arguments, local duality, change-of-rings lemmas, and external cited results, rather than by defining one condition in terms of another. There are no fitted parameters or data-based predictions being renamed as conclusions. Same-author citations such as [KLOT23, Lemma 2.10] and [KOT22, Theorem 2.2] are used as external lemmas with stated hypotheses, and the target equivalences do not reduce to those citations by construction. The paper also explicitly lists open questions and limitations (Question 7.1, Question 7.4, Remark 5.11), which is incompatible with a hidden circular derivation. Two non-circular issues should be noted for correctness rather than circularity: (i) in Theorem 4.4, step (3)⇒(4) invokes Lemma 3.9 for f_M⊗f_N, although Lemma 3.9 as stated concerns f_M⊗N; this is a proof gap, not a definitional or fitted circularity; and (ii) the proof of Theorem 4.4 says [CH10, Remark 5.7] gives ascent/descent of condition (5), while Question 7.4 later asks whether condition (5) ascends to completion, an internal tension bearing on correctness, not on circularity. Per the stated rules, these concerns do not raise the circularity score.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces new named conditions ((ldep), (rdep), UBC) but no new objects beyond definitions. It relies on several standard and published results, including completion descent lemmas from the same authors' prior papers, which lowers the independence of the proof but does not make the central claims circular.

assumptions (5)

domain assumption R is a Cohen-Macaulay local ring throughout Sections 3-5.
This is the main hypothesis of Theorems 1.1 and 1.2; many arguments use the existence of a maximal regular sequence and depth inequalities.
domain assumption Complete local Cohen-Macaulay rings admit a canonical module, and reductions to the completion preserve the relevant conditions.
Used in the proofs of Theorem 4.4 and 5.8 to assume R is complete so that a canonical module exists, citing [KLOT23, Lemma 2.10] and [CH10, Remark 5.7].
standard math Local duality for complexes (see [IMSW21, 3.4.1]) is valid for complexes with bounded homology.
Used in the proof of (1) implies (5) in Theorem 4.4 and (1) implies (7) in Theorem 5.8 to relate Ext and depth.
standard math The depth lemma and Auslander-Buchsbaum formula hold for complexes (Proposition 3.1, from [Iye99]).
Foundational tools used throughout; stated but not proved.
domain assumption Syzygies of finite length modules over CM rings are MCM and locally free on the punctured spectrum.
Used in Lemma 3.9 and Theorem 4.4 to transfer vanishing and MCM properties between f_M and M.

how reviews work

0 comments

Cite this review

Pith. "Pith review of On the depth of tensor products over Cohen-Macaulay rings." pith.science (2026). https://pith.science/paper/B662QT5I

@misc{pith2026250500441,
  author       = {Pith},
  title        = {Pith review of: On the depth of tensor products over Cohen-Macaulay rings},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B662QT5I}},
  note         = {Machine review of arXiv:2505.00441}
}

abstract

Inspired by classical work on the depth formula for tensor products of finitely generated $R$-modules, we introduce two conditions which we call $(\mathbf{ldep})$ and $(\mathbf{rdep})$ and their derived variations. We show for Cohen-Macaulay local rings that derived $(\mathbf{ldep})$ is equivalent to $\dim(R)$ being a uniform Auslander bound for $R$, and if $\dim(R)>0$ that both are equivalent to $(\mathbf{ldep})$. We introduce an analogous condition we call the \emph{uniform Buchweitz condition} and provide a corresponding theorem for the $(\mathbf{rdep})$ condition. As a consequence of these results, we show $(\mathbf{ldep})$ implies $(\mathbf{rdep})$ when $R$ is Gorenstein and that the $(\mathbf{ldep})$ and $(\mathbf{rdep})$ conditions behave well under modding out by regular sequences and completion, but we give a concrete example showing they need not localize. Using our methods, we extend work of Jorgensen by calculating the value $q_R(M,N):=\sup\{i \mid \operatorname{Tor}^R_i(M,N) \ne 0\}$ under certain conditions.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 24 canonical work pages

[1]

Avramov, Vesselin N

Luchezar L. Avramov, Vesselin N. Gasharov, and Irena V. Peeva. Complete intersection dimension. Inst. Hautes \'Etudes Sci. Publ. Math. , (86):67--114, 1997

work page 1997
[2]

Avramov, Srikanth B

Luchezar L. Avramov, Srikanth B. Iyengar, Saeed Nasseh, and Keri Sather-Wagstaff. Persistence of homology over commutative noetherian rings. J. Algebra , 610:463--490, 2022

work page 2022
[3]

Auslander

M. Auslander. Modules over unramified regular local rings. Illinois J. Math. , 5:631--647, 1961

work page 1961
[4]

Cohen- M acaulay rings , volume 39 of Cambridge Studies in Advanced Mathematics

Winfried Bruns and J\" u rgen Herzog. Cohen- M acaulay rings , volume 39 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 1993

1993
[5]

Bergh, David A

Petter A. Bergh, David A. Jorgensen, and W. Frank Moore. A converse to a construction of E isenbud- S hamash. J. Commut. Algebra , 12(4):467--477, 2020

work page 2020
[6]

Algebras that satisfy A uslander's condition on vanishing of cohomology

Lars Winther Christensen and Henrik Holm. Algebras that satisfy A uslander's condition on vanishing of cohomology. Math. Z. , 265(1):21--40, 2010

work page 2010
[7]

Jorgensen

Lars Winther Christensen and David A. Jorgensen. Vanishing of T ate homology and depth formulas over local rings. J. Pure Appl. Algebra , 219(3):464--481, 2015

work page 2015
[8]

Hom and E xt, revisited

Hailong Dao, Mohammad Eghbali, and Justin Lyle. Hom and E xt, revisited. J. Algebra , 571:75--93, 2021

work page 2021

Show all 27 references

[9]

Classification of resolving subcategories and grade consistent functions

Hailong Dao and Ryo Takahashi. Classification of resolving subcategories and grade consistent functions. International Mathematics Research Notices , 2015(1):119--149, 2015

2015
[10]

Depth and amplitude for unbounded complexes

Hans-Bj rn Foxby and Srikanth Iyengar. Depth and amplitude for unbounded complexes. In Commutative algebra ( G renoble/ L yon, 2001) , volume 331 of Contemp. Math. , pages 119--137. Amer. Math. Soc., Providence, RI, 2003

2001
[11]

Heitmann

Raymond C. Heitmann. Completions of local rings with an isolated singularity. J. Algebra , 163(2):538--567, 1994

1994
[12]

Indecomposable canonical modules and connectedness

Melvin Hochster and Craig Huneke. Indecomposable canonical modules and connectedness. In Commutative algebra: syzygies, multiplicities, and birational algebra ( S outh H adley, MA , 1992) , volume 159 of Contemp. Math. , pages 197--208. Amer. Math. Soc., Providence, RI, 1994

1992
[13]

Tensor products of modules and the rigidity of Tor

Craig Huneke and Roger Wiegand. Tensor products of modules and the rigidity of Tor . Math. Ann. , 299(3):449--476, 1994

1994
[14]

Iyengar, Linquan Ma, Karl Schwede, and Mark E

Srikanth B. Iyengar, Linquan Ma, Karl Schwede, and Mark E. Walker. Maximal C ohen- M acaulay complexes and their uses: a partial survey. In Commutative algebra , pages 475--500. Springer, Cham, [2021] 2021

2021
[15]

S. Iyengar. Depth for complexes, and intersection theorems. Math. Z. , 230(3):545--567, 1999

1999
[16]

Jorgensen and Liana M

David A. Jorgensen and Liana M. Sega. Nonvanishing cohomology and classes of G orenstein rings. Adv. Math. , 188(2):470--490, 2004

2004
[17]

Jorgensen and Liana M

David A. Jorgensen and Liana M. Sega. Independence of the total reflexivity conditions for modules. Algebr. Represent. Theory , 9(2):217--226, 2006

2006
[18]

Jorgensen

David A. Jorgensen. A generalization of the A uslander- B uchsbaum formula. J. Pure Appl. Algebra , 144(2):145--155, 1999

1999
[19]

On the vanishing of Ext modules over a local unique factorization domain with an isolated singularity

Kaito Kimura , Justin Lyle , Yuya Otake , and Ryo Takahashi . On the vanishing of Ext modules over a local unique factorization domain with an isolated singularity . arXiv e-prints , page arXiv:2310.16599, October 2023

2023 arXiv
[20]

Maximal cohen--macaulay tensor products and vanishing of ext modules

Kaito Kimura, Yuya Otake, and Ryo Takahashi. Maximal cohen--macaulay tensor products and vanishing of ext modules. Bulletin of the London Mathematical Society , 54(6):2456--2468, 2022

2022
[21]

A method for constructing bad N oetherian local rings

Christer Lech. A method for constructing bad N oetherian local rings. In Algebra, algebraic topology and their interactions ( S tockholm, 1983) , volume 1183 of Lecture Notes in Math. , pages 241--247. Springer, Berlin, 1986

1983
[22]

Extremal growth of B etti numbers and trivial vanishing of (co)homology

Justin Lyle and Jonathan Monta\ n o. Extremal growth of B etti numbers and trivial vanishing of (co)homology. Trans. Amer. Math. Soc. , 373(11):7937--7958, 2020

2020
[23]

Leuschke and Roger Wiegand

Graham J. Leuschke and Roger Wiegand. Cohen- M acaulay representations , volume 181 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 2012

2012
[24]

Applications and homological properties of local rings with decomposable maximal ideals

Saeed Nasseh, Sean Sather-Wagstaff, Ryo Takahashi, and Keller VandeBogert. Applications and homological properties of local rings with decomposable maximal ideals. J. Pure Appl. Algebra , 223(3):1272--1287, 2019

2019
[25]

Local rings with quasi-decomposable maximal ideal

Saeed Nasseh and Ryo Takahashi. Local rings with quasi-decomposable maximal ideal. Math. Proc. Cambridge Philos. Soc. , 168(2):305--322, 2020

2020
[26]

Modules in resolving subcategories which are free on the punctured spectrum

Ryo Takahashi. Modules in resolving subcategories which are free on the punctured spectrum. Pacific J. Math. , 241(2):347--367, 2009

2009
[27]

A functorial approach to modules of G -dimension zero

Yuji Yoshino. A functorial approach to modules of G -dimension zero. Illinois J. Math. , 49(2):345--367, 2005

2005

Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [1]

Avramov, Vesselin N

Luchezar L. Avramov, Vesselin N. Gasharov, and Irena V. Peeva. Complete intersection dimension. Inst. Hautes \'Etudes Sci. Publ. Math. , (86):67--114, 1997

work page 1997

[2] [2]

Avramov, Srikanth B

Luchezar L. Avramov, Srikanth B. Iyengar, Saeed Nasseh, and Keri Sather-Wagstaff. Persistence of homology over commutative noetherian rings. J. Algebra , 610:463--490, 2022

work page 2022

[3] [3]

Auslander

M. Auslander. Modules over unramified regular local rings. Illinois J. Math. , 5:631--647, 1961

work page 1961

[4] [4]

Cohen- M acaulay rings , volume 39 of Cambridge Studies in Advanced Mathematics

Winfried Bruns and J\" u rgen Herzog. Cohen- M acaulay rings , volume 39 of Cambridge Studies in Advanced Mathematics . Cambridge University Press, Cambridge, 1993

1993

[5] [5]

Bergh, David A

Petter A. Bergh, David A. Jorgensen, and W. Frank Moore. A converse to a construction of E isenbud- S hamash. J. Commut. Algebra , 12(4):467--477, 2020

work page 2020

[6] [6]

Algebras that satisfy A uslander's condition on vanishing of cohomology

Lars Winther Christensen and Henrik Holm. Algebras that satisfy A uslander's condition on vanishing of cohomology. Math. Z. , 265(1):21--40, 2010

work page 2010

[7] [7]

Jorgensen

Lars Winther Christensen and David A. Jorgensen. Vanishing of T ate homology and depth formulas over local rings. J. Pure Appl. Algebra , 219(3):464--481, 2015

work page 2015

[8] [8]

Hom and E xt, revisited

Hailong Dao, Mohammad Eghbali, and Justin Lyle. Hom and E xt, revisited. J. Algebra , 571:75--93, 2021

work page 2021