Abstract
The purpose of this note is to show that a finitely generated graded module M over \(S=k[x_1,\ldots ,x_n]\), k a field, is sequentially Cohen-Macaulay if and only if its arithmetic degree \({\text {adeg}}(M)\) agrees with \({\text {adeg}}(F/{\text {gin}}_\textrm{revlex}(U))\), where F is a graded free S-module and \(M \cong F/U\). This answers positively a conjecture of Lu and Yu from 2016.
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1 Introduction
Sequentially Cohen-Macaulay modules were first introduced in the graded setting by Stanley [20], in connection with the theory of Stanley-Reisner rings. Later, Schenzel [19] introduced the notion of Cohen-Macaulay filtered module in a more general setting, and showed that in the graded case it coincides with the one introduced by Stanley.
Let k be a field, and \(S=k[x_1,\ldots ,x_n]\) with the standard grading. Let M be a finitely generated \(\mathbb {Z}\)-graded S-module of Krull dimension d. A module M is sequentially Cohen-Macaulay if one of the following three equivalent conditions is satisfied:
-
(1)
(Stanley) There exists a filtration \(0 = M_0 \subsetneq M_1 \subsetneq M_2 \subsetneq \cdots \subsetneq M_t = M\) with \(M_i/M_{i-1}\) Cohen-Macaulay graded S-modules of dimension \(d_i\) satisfying \(d_i>d_{i-1}\) for all \(i=1,\ldots ,t\).
-
(2)
(Schenzel) If \(\delta _i(M)\) denotes the largest submodule of M of dimension less than or equal to i, then the filtration
$$\begin{aligned} 0 \subseteq \delta _0(M) \subseteq \delta _1(M) \subseteq \cdots \subseteq \delta _{d}(M) = M \end{aligned}$$is such that \(\delta _i(M)/\delta _{i-1}(M)\) is either zero or Cohen-Macaulay for all \(i=1,\ldots ,d\).
-
(3)
(Peskine) For every \(i=0,\ldots ,n\), the module \({\text {Ext}}^{n-i}_S(M,S)\) is either zero or Cohen-Macaulay of dimension i.
Over the years, sequentially Cohen-Macaulay modules have been studied extensively, even from very different perspectives (for instance, see [1, 2, 7,8,9, 11,12,13,14]). We invite the interested reader to consult [6] for further details.
For the purposes of this note, it is important to mention a result of Herzog and Sbarra, which characterizes sequentially Cohen-Macaulay modules. We state it here in a dual version:
Theorem 1.1
([15, Main theorem]). Let F be a finitely generated free S-module, and \(U \subseteq F\) be a homogeneous submodule. Then, the module F/U is sequentially Cohen-Macaulay if and only if for all \(i = 0,\ldots ,n\), we have \({\text {HF}}({\text {Ext}}^{n-i}_S(F/U,S)) = {\text {HF}}({\text {Ext}}^{n-i}_S(F/{\text {gin}}_{revlex} (U),S))\), where \({\text {HF}}(-)\) denotes the Hilbert function of a graded S-module.
This theorem has been extended in two different directions: weaker versions of sequential Cohen-Macaulayness can be characterized either by showing that equality of Hilbert functions holds only for certain cohomological indices [17], or by replacing revlex with certain partial revlex orders [5].
To state our main theorem, we recall the notion of arithmetic degree, introduced by Bayer and Mumford in [3] (see also [21, 22]). First, recall that if \(0\ne M\) is a finitely generated \(\mathbb {Z}\)-graded module of dimension d, and \({\text {HF}}(M;j) = \dim _k(M_j)\) is the Hilbert function of M in degree j, then for \(j \gg 0\), we have
for some \({\text {e}}(M) \in \mathbb {Z}_{>0}\), called multiplicity (or degree) of M. Given an integer \(r \geqslant d\), we let
Definition 1.2
Let \(S=k[x_1,\ldots ,x_n]\), and M be a finitely generated graded S-module. For all \(r=0,\ldots ,n\), we let \({\text {adeg}}_r(M) = {\text {e}}_r({\text {Ext}}^{n-r}_S(M,S))\). The arithmetic degree of M is defined as \({\text {adeg}}(M) = \sum _{r=0}^n {\text {adeg}}_r(M)\).
Recall that \(\dim ({\text {Ext}}_S^{n-r}(M,S)) \leqslant r\) always holds (see for instance [18, Section 3.1]), therefore the above definition makes sense. Now let F be a free S-module, and \(U \subseteq F\) be a graded submodule. Given any weight \(\omega \in \mathbb {Z}^n\), by upper semi-continuity, one has \({\text {adeg}}_r(F/U) \leqslant {\text {adeg}}_r(F/{\text {in}}_\omega (U))\) for all \(r=0,\ldots ,n\) (see [21] for the case of cyclic modules); in particular, \({\text {adeg}}(F/U) \leqslant {\text {adeg}}(F/{\text {gin}}_\textrm{revlex}(U))\) always holds. If F/U is sequentially Cohen-Macaulay, then Theorem 1.1 immediately gives that \({\text {adeg}}(F/U) = {\text {adeg}}(F/{\text {gin}}_\textrm{revlex}(U))\). This result was observed by Lu and Yu [16, Proposition 3.6], and in the same paper they conjecture that the converse holds as well. The purpose of this note is to prove their conjecture:
Main theorem
Let \(S=k[x_1,\ldots ,x_n]\), with the standard grading. Let F be a finitely generated graded free S-module and \(U \subseteq F\) be a homogeneous submodule. We have that F/U is sequentially Cohen-Macaulay if and only if \({\text {adeg}}(F/U) = {\text {adeg}}(F/{\text {gin}}_\textrm{revlex}(U))\).
Actually, we prove their conjecture in a bigger generality, by showing that it is not necessary to take general coordinates when computing initial ideals as long as \(x_n,\ldots ,x_1\) is a filter regular sequence for F/U (see Sect. 2). As a consequence of our proof, we also obtain an analogous generalization of Theorem 1.1.
2 Preliminaries and main result
Let k be a field, and \(S=k[x_1,\ldots ,x_n]\) be a polynomial ring with standard grading \(\deg (x_i)=1\) for all i. Let \(\mathfrak {m}=(x_1,\ldots ,x_n)\). Throughout, M will always denote a finitely generated \(\mathbb {Z}\)-graded S-module.
Consider the weight \(\omega =(0,\ldots ,0,-1) \in \mathbb {Z}^n\) and, for a homogeneous submodule U of a graded free S-module F, we let \({\text {in}}(U):= {\text {in}}_\omega (U) \subseteq F\) be the initial submodule. Since revlex can be obtained as \({\text {in}}_\Omega \) for the following matrix of weights
of which \(\omega \) is the first row, we call \({\text {in}}(U)\) a “partial revlex submodule”. See [5] for more details on this construction, where \({\text {in}}(-)\) is denoted as \({\text {in}}_{\textrm{rev}_{1}}(-)\).
Definition 2.1
Let M be a finitely generated \(\mathbb {Z}\)-graded S-module. A homogeneous element \(f \in S\) is called filter regular if \(0:_M f\) has finite length. A sequence of elements \(f_1,\ldots ,f_t\) is called a filter regular sequence if \(f_{i+1}\) is filter regular for \(M/(f_1,\ldots ,f_i)M\) for all \(i<t\).
Remark 2.2
Let U be a homogeneous submodule of a graded free S-module F. Since \({\text {in}}(U)\) is a partial deformation towards \({\text {in}}_\textrm{revlex}(U)\), one has that \({\text {adeg}}_r(F/U) \leqslant {\text {adeg}}_r(F/{\text {in}}(U)) \leqslant {\text {adeg}}_r(F/{\text {in}}_\textrm{revlex}(U))\) for all \(r=0,\ldots ,n\) by upper semicontinuity.
In the proof of our main theorem, we will need the following lemma, which is an immediate consequence of a result of Serre.
Lemma 2.3
Let M be a finitely generated \(\mathbb {Z}\)-graded S-module of dimension \(d>0\), and let \(\ell \in S_1\) be such that \(M/\ell M\) is Cohen-Macaulay of dimension \(d-1\), and \({\text {e}}_{d-1}(M/\ell M) = {\text {e}}_d(M)\). Then \(\ell \) is an M-regular element, and M is Cohen-Macaulay.
Proof
Without loss of generality, we may assume that k is infinite. Then, we can find linear forms \(\ell _2,\ldots ,\ell _{d}\) which form a regular sequence for \(M/\ell M\). Since \(M/\ell M\) is Cohen-Macaulay, our assumptions guarantee that \({\text {e}}_d(M) = {\text {e}}_{d-1}(M/\ell M) = \lambda (M/(\ell ,\ell _2,\ldots ,\ell _d)M)\), where \(\lambda (-)\) denotes the length of a module. By the graded version of [4, Theorem 4.7.10(b)], we conclude that M is Cohen-Macaulay and \(\ell ,\ell _2,\ldots ,\ell _d\) is a regular sequence on M. \(\square \)
For a module M, we let \({\text {Ass}}_r(M) = {\text {Ass}}(M) \cap \{\mathfrak {p}\in {\text {Spec}}(S) \mid \dim (S/\mathfrak {p}) = r\}\).
Remark 2.4
We recall that, if M is finitely generated and \(\mathbb {Z}\)-graded, then \({\text {Ass}}_r(M) = {\text {Ass}}_r({\text {Ext}}^{n-r}_S(M,S))\) for all \(r = 0,\ldots ,n\). This follows by noticing that, for a prime \(\mathfrak {p}\) such that \(\dim (S/\mathfrak {p}) = r\), we have that \(\mathfrak {p}\in {\text {Ass}}_r(M)\) if and only if \(H^0_{\mathfrak {p}S_\mathfrak {p}}(M_\mathfrak {p}) \ne 0\), if and only if \(\left( {\text {Ext}}^{n-r}_S(M,S)\right) _\mathfrak {p}\ne 0\). Since \(\dim ({\text {Ext}}^{n-r}_S(M,S))\leqslant r\) (see for instance [18, Section 3.1]), this is equivalent to \(\mathfrak {p}\in {\text {Min}}({\text {Supp}}({\text {Ext}}^{n-r}_S(M,S)))\), which in turn is equivalent to \(\mathfrak {p}\in {\text {Ass}}_r({\text {Ext}}^{n-r}_S(M,S))\).
Remark 2.5
Let F be a finitely generated free S-module, and \(U \subseteq F\) be a homogeneous submodule. By [15, Theorem 2.2], we have that \(F/{\text {gin}}_\textrm{revlex}(U)\) is sequentially Cohen-Macaulay. However, the same argument works for \(F/{\text {in}}_\textrm{revlex}(U)\) assuming that \(x_n,x_{n-1},\ldots ,x_1\) form a filter regular sequence for F/U (see also [5, Key example 2.15]). In particular, under these assumptions, for all \(r=0,\ldots ,n\), the module \({\text {Ext}}^{n-r}_S(F/{\text {in}}_\textrm{revlex}(U),S)\) is either zero or Cohen-Macaulay of dimension r.
We are now ready to prove the conjecture of Lu and Yu [16].
Theorem 2.6
Let \(S=k[x_1,\ldots ,x_n]\), with the standard grading. Let F be a finitely generated graded free S-module and \(U \subseteq F\) be a homogeneous submodule. Assume that \(x_n,x_{n-1},\ldots ,x_1\) is a filter regular sequence for F/U. The following are equivalent:
-
(1)
F/U is sequentially Cohen-Macaulay,
-
(2)
\({\text {HF}}({\text {Ext}}^{n-r}_S(F/U,S)) = {\text {HF}}({\text {Ext}}^{n-r}_S(F/{\text {in}}_\textrm{revlex}(U),S))\) for all \(r=0,\ldots ,n\),
-
(3)
\({\text {adeg}}(F/U) = {\text {adeg}}(F/{\text {in}}_\textrm{revlex}(U))\).
Proof
We prove the equivalence of the three conditions by induction on \(d=\dim (F/U)\). Let \(V={\text {in}}_\textrm{revlex}(U)\). First of all, observe that \(U^{{\text {{sat}}}} = U:x_n^\infty \), and that \({\text {in}}_\textrm{revlex}(U^{{\text {{sat}}}}) = V:x_n^\infty = V^{{\text {{sat}}}}\) by well-known properties of revlex-type orders (see [10, 15.7]). It follows that \({\text {HF}}(U^{{\text {{sat}}}}/U) = {\text {HF}}(V^{{\text {{sat}}}}/V)\). Since \(U^{{\text {{sat}}}}/U\) is the graded Matlis dual of \({\text {Ext}}^n_S(F/U,S)\) (and similarly for V), we conclude that \({\text {HF}}({\text {Ext}}^n_S(F/U,S)) = {\text {HF}}({\text {Ext}}^n_S(F/V,S))\). In particular, \({\text {adeg}}_0(F/U) = {\text {e}}_0({\text {Ext}}^n_S(F/U,S)) = {\text {e}}_0({\text {Ext}}^n_S(F/V,S)) = {\text {adeg}}_0(F/V)\).
Since modules of dimension zero are automatically sequentially Cohen-Macaulay, for \(d=0\), the three statements are trivially equivalent because they are all true.
Now assume that \(d>0\). Recall that F/U is sequentially Cohen-Macaulay if and only if so is \(F/U^{{\text {{sat}}}}\) [6, Corollary 2.8], and that \({\text {Ext}}^{n-r}_S(F/U,S) \cong {\text {Ext}}^{n-r}_S(F/U^{{\text {{sat}}}},S)\) for all \(r>0\). Similar considerations hold for V. In view of the previous equalities, we may assume that \({\text {depth}}(F/U)>0\) after possibly replacing U with \(U^{{\text {{sat}}}}\).
Assume (1). By Remark 2.4, we have \({\text {Ass}}_r(F/U)= {\text {Ass}}_r({\text {Ext}}^{n-r}_S(F/U,S))\). Also, \({\text {Ass}}_r({\text {Ext}}^{n-r}_S(F/U,S)) = {\text {Ass}}({\text {Ext}}^{n-r}_S(F/U,S))\) as \({\text {Ext}}^{n-r}_S(F/U,S)\) is either zero or Cohen-Macaulay of dimension r. Since \(x_n\) is filter regular for F/U, we conclude that it is regular for \({\text {Ext}}^{n-r}_S(F/U,S)\) for all \(r>0\). For all \(r>0\), we have short exact sequences
which give that \(F/(U+x_nF)\) is sequentially Cohen-Macaulay of dimension \(d-1\). Moreover, for all \(r>0\) and \(j \in \mathbb {Z}\), we obtain
Now let \(\overline{S} = k[x_1,\ldots ,x_{n-1}]\). We can identify \(F/(U+x_nF)\) with a sequentially Cohen-Macaulay \(\overline{S}\)-module \(\overline{F}/\overline{U}\), where \(\overline{F}\) is a graded free \(\overline{S}\)-module and \(\overline{U} \subseteq \overline{F}\) is a homogeneous submodule. Since \(\dim (\overline{F}/\overline{U}) = d-1\), by induction, we have that \({\text {HF}}\left( {\text {Ext}}^{n-r}_{\overline{S}}(\overline{F}/\overline{U},\overline{S})\right) = {\text {HF}}\left( {\text {Ext}}^{n-r}_{\overline{S}}(\overline{F}/{\text {in}}_\textrm{revlex}(\overline{U}),\overline{S})\right) \) for all \(r > 0\). By [4, Lemma 3.1.16], for all \(r>0\), we have that
and
Using that \({\text {in}}_\textrm{revlex}(U+x_nF) = V+x_nF\) (see [10, 15.7]), we obtain
By the proof of [5, Proposition 3.5] and by [5, Lemma 2.9], for all \(r>0\), we have short exact sequences
which give
Combining (2.1) and (2.3), together with the fact that the \({\text {Ext}}\) modules vanish for sufficiently negative degrees, we finally obtain
and this concludes the proof that (1) \(\Rightarrow \) (2). The fact that (2) \(\Rightarrow \) (3) is trivial.
We finally show that (3) \(\Rightarrow \) (1). Let \(\omega =(0,\ldots ,0,-1) \in \mathbb {Z}^n\), and let \(W={\text {in}}_\omega (U)\). By Remark 2.2 and our assumptions, we must have \({\text {adeg}}_r(F/U)={\text {adeg}}_r(F/W) = {\text {adeg}}_r(F/V)\) for all \(r=0,\ldots ,n\). Again using the proof of [5, Proposition 3.5] and [5, Lemma 2.9], for all \(r>0\), we obtain short exact sequences
where we use that \(W+x_nF = U+x_nF\) holds because \(\omega \) is a partial revlex order. The short exact sequences (2.4) and (2.2) give that \({\text {adeg}}_{r-1}(F/(U+x_nF)) = {\text {adeg}}_r(F/W) = {\text {adeg}}_r(F/V) = {\text {adeg}}_{r-1}(F/(V+x_nF))\). We therefore obtain that \({\text {adeg}}(F/(U+x_nF)) = {\text {adeg}}(F/(V+x_nF))\). As above, we can identify \(F/(U+x_nF)\) with an \(\overline{S}\)-module \(\overline{F}/\overline{U}\). In this way, \(F/(V+x_nF)\) can be identified with \(\overline{F}/{\text {in}}_\textrm{revlex}(\overline{U})\), and therefore we have that \({\text {adeg}}(\overline{F}/\overline{U}) = {\text {adeg}}(\overline{F}/{\text {in}}_\textrm{revlex}(\overline{U}))\). Since \(\dim (\overline{F}/\overline{U})=d-1\) and \(x_{n-1},\ldots ,x_1\) form a filter regular sequence for \(\overline{F}/\overline{U}\), by induction, we have that \(\overline{F}/\overline{U}\) is sequentially Cohen-Macaulay, and so is \(F/(U+x_nF)\).
Now we note that if \({\text {Ext}}^{n-r}_S(F/U,S) \ne 0\), then \({\text {Ext}}^{n-r}_S(F/V,S) \ne 0\) by upper semi-continuity. Since we have \({\text {e}}_r({\text {Ext}}^{n-r}_S(F/U,S)) = {\text {adeg}}_r(F/U) = {\text {adeg}}_r(F/V) = {\text {e}}_r({\text {Ext}}^{n-r}_S(F/V,S))\), and the latter is positive by Remark 2.5, it follows that \(\dim ({\text {Ext}}^{n-r}_S(F/U,S))=r\). To complete the proof, we show by induction on \(r>0\) that if \({\text {Ext}}^{n-r}_S(F/U,S) \ne 0\), then it is a Cohen-Macaulay module, and that \(x_n\) is a regular element for it.
First note that for \(r>0\), since \({\text {Ass}}_r(F/U) = {\text {Ass}}_r({\text {Ext}}^{n-r}_S(F/U,S))\) by Remark 2.4, and because \(x_n\) is filter regular for F/U, we have that \(x_n\) avoids all minimal primes of \({\text {Supp}}({\text {Ext}}^{n-r}_S(F/U,S))\) of dimension r. In particular, \(\dim ({\text {Ext}}^{n-r}_S(F/U,S) \otimes _S S/(x_n)) = r-1\).
For \(r=1\), we have an exact sequence
which gives that \({\text {Ext}}^n_S(F/(U+x_nF),S) \cong {\text {Ext}}^{n-1}_S(F/U,S) \otimes _S S/(x_n)\). Using the short exact sequence (2.4), and recalling that \(U+x_nF = W + x_nF\), we obtain
By Lemma 2.3, we conclude that \({\text {Ext}}^{n-1}_S(F/U,S)\) is Cohen-Macaulay, and \(x_n\) is regular for it.
Now assume that \({\text {Ext}}^{n-i}_S(F/U,S)\) is Cohen-Macaulay for all \(i=1,\ldots ,r-1\) and that \(x_n\) is a regular element for all such modules. In particular, it is regular for \({\text {Ext}}^{n-r+1}_S(F/U,S)\), and therefore we have an exact sequence
We conclude that \({\text {Ext}}^{n-r+1}_S(F/(U+x_nF),S) \cong {\text {Ext}}^{n-r}(F/U,S) \otimes _S S/(x_n)\). As before, we have
Finally, by Lemma 2.3, we conclude that \({\text {Ext}}^{n-r}_S(F/U,S)\) is Cohen-Macaulay and \(x_n\) is regular for it, and the proof is complete. \(\square \)
Remark 2.7
We would like to point out that, if the field k is infinite, then the assumption that \(x_n,\ldots ,x_1\) form a filter regular sequence on F/U is not at all restrictive. In fact, by prime avoidance, we can always find a sequence of linearly independent linear forms \(\ell _n,\ldots ,\ell _1\) which form a filter regular sequence on F/U. After performing the change of coordinates \(x_i:= \ell _i\) for \(i=1,\ldots ,n\), we are in the assumptions of Theorem 2.6.
References
Adiprasito, K.A., Björner, A., Goodarzi, A.: Face numbers of sequentially Cohen-Macaulay complexes and Betti numbers of componentwise linear ideals. J. Eur. Math. Soc. (JEMS) 19(12), 3851–3865 (2017)
Àlvarez Montaner, J.: Lyubeznik table of sequentially Cohen-Macaulay rings. Comm. Algebra 43(9), 3695–3704 (2015)
Bayer, D., Mumford, D.: What can be computed in algebraic geometry? In: Computational Algebraic Geometry and Commutative Algebra (Cortona, 1991), pp. 1–48. Sympos. Math., XXXIV. Univ. Press, Cambridge (1993)
Bruns, W., Herzog, J.: Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics, 39. Cambridge University Press, Cambridge (1993)
Caviglia, G., De Stefani, A.: Decomposition of local cohomology tables of modules with large E-depth. J. Pure Appl. Algebra 225(6), 23 (2021)
Caviglia, G., De Stefani, A., Sbarra, E., Strazzanti, F.: On the notion of sequentially Cohen-Macaulay modules. Res. Math. Sci. 9(3), 27 (2022)
Cuong, N.T., Cuong, D.T.: On sequentially Cohen-Macaulay modules. Kodai Math. J. 30(3), 409–428 (2007)
Cuong, N.T., Truong, H.L.: Parametric decomposition of powers of parameter ideals and sequentially Cohen-Macaulay modules. Proc. Amer. Math. Soc. 137(1), 19–26 (2009)
Duval, A.M.: Algebraic shifting and sequentially Cohen-Macaulay simplicial complexes. Electron. J. Combin. 3(1), Research Paper 21, 14 pp. (1996)
Eisenbud, D.: Commutative Algebra. With a View Toward Algebraic Geometry. Graduate Texts in Mathematics, 150. Springer, New York (1995)
Faridi, S.: Monomial ideals via square-free monomial ideals. In: Commutative Algebra, pp. 85–114. Lect. Notes Pure Appl. Math., 244. Chapman & Hall/CRC, Boca Raton (2006)
Goodarzi, A.: Dimension filtration, sequential Cohen-Macaulayness and a new polynomial invariant of graded algebras. J. Algebra 456, 250–265 (2016)
Herzog, J., Hibi, T.: Componentwise linear ideals. Nagoya Math. J. 153, 141–153 (1999)
Herzog, J., Reiner, V., Welker, V.: Componentwise linear ideals and Golod rings. Michigan Math. J. 46(2), 211–223 (1999)
Herzog, J., Sbarra, E.: Sequentially Cohen-Macaulay modules and local cohomology. In: Algebra, Arithmetic and Geometry, Part I, II (Mumbai, 2000), pp. 327–340. Tata Inst. Fund. Res. Stud. Math., 16. Tata Inst. Fund. Res., Bombay (2002)
Lu, D., Yu, J.: Bounds for arithmetic degrees. Comm. Algebra 44(5), 1971–1980 (2016)
Sbarra, E., Strazzanti, F.: A rigidity property of local cohomology modules. Proc. Amer. Math. Soc. 145(10), 4099–4110 (2017)
Schenzel, P.: Dualisierende Komplexe in der lokalen Algebra und Buchsbaum-Ringe. With an English Summary. Lecture Notes in Mathematics, 907. Springer, Berlin-New York (1982)
Schenzel, P.: On the dimension filtration and Cohen-Macaulay filtered modules. In: Commutative Algebra and Algebraic Geometry (Ferrara), pp. 245–264. Lecture Notes in Pure and Appl. Math., 206. Dekker, New York (1999)
Stanley, R.P.: Combinatorics and Commutative Algebra. Progress in Mathematics, 41. Birkhäuser Boston Inc., Boston (1983)
Sturmfels, B., Trung, N.V., Vogel, W.: Bounds on degrees of projective schemes. Math. Ann. 302(3), 417–432 (1995)
Vasconcelos, W.V.: Computational Methods in Commutative Algebra and Algebraic Geometry. With Chapters by David Eisenbud, Daniel R. Grayson, Jürgen Herzog and Michael Stillman. Algorithms and Computation in Mathematics, 2. Springer, Berlin (1998)
Acknowledgements
We thank the anonymous referee for providing several suggestions which improved the quality of the exposition. The first author was partially supported by a grant from the Simons Foundation (41000748, G.C.). The second author was partially supported by the PRIN 2020 project 2020355B8Y “Squarefree Gröbner degenerations, special varieties and related topics”, the MIUR Excellence Department Project CUP D33C23001110001, and INdAM-GNSAGA.
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Caviglia, G., De Stefani, A. A criterion for sequential Cohen-Macaulayness. Arch. Math. 123, 137–145 (2024). https://doi.org/10.1007/s00013-024-02011-y
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DOI: https://doi.org/10.1007/s00013-024-02011-y