Comptes Rendus Mathématique

Uri Abraham

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The chain covering number of a poset with no infinite antichains

Henning Krause

Comptes Rendus
Mathématique
Henning Krause
On the symmetry of the finitistic dimension
Volume 361 (2023), p. 1449-1453
https://doi.org/10.5802/crmath.481
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
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e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 1449-1453
https://doi.org/10.5802/crmath.481
Representation theory / Théorie des représentations
On the symmetry of the finitistic dimension
Henning Krause a
a Fakultät für Mathematik Universität Bielefeld D-33501 Bielefeld Germany
E-mail: hkrause@math.uni-bielefeld.de
Dedicated to the memory of Helmut Lenzing.
Abstract. For any ring we propose the construction of a cover which increases the finitistic dimension on one
side and decreases the finitistic dimension to zero on the opposite side. This complements recent work of
Cummings.
2020 Mathematics Subject Classification. 16E10.
Manuscript received 20 November 2022, revised 25 January 2023, accepted 20 February 2023.
The finitistic dimension is a homological invariant of a ring which is conjectured to be finite
when the ring is a finite dimensional algebra over a field [2]. In recent work [3] Cummings
introduces for any finite dimensional algebra a related algebra; its purpose is to increase the
finitistic dimension on one side and to decrease the finitistic dimension to zero on the opposite
side. In this note we propose the construction of such an asymmetric cover for any ring and we
establish the same properties. This specialises to Cummings’ construction for finite dimensional
algebras over an algebraically closed field and yields examples of rings such that the finitistic
dimension is infinite while the finitistic dimension of the opposite ring is zero. We need to
distinguish between the small and the big finitistic dimension but our results cover both cases.
Let A be an associative ring. We consider the category of (right) A-modules and identify (left)
A-modules with modules over the opposite ring Aop. For a module M we write rad M for its
radical and soc M for its socle. We set top M = M/rad M . The functor
(−)∗ := HomA(−, A)
yields a duality between right and left A-modules. We consider the trivial extension
T (A) := A⋉ A♮
which is given by the bimodule A♮ := A A A . This ring is by definition the abelian group T (A) =
A⊕ A♮ with multiplication given by the formula
(x, y) · (x ′, y ′)= (xx ′, x y ′+ y x ′) .
Note that A♮ is a two-sided ideal with T (A)/A♮ ∼−→ A.
Lemma 1. Let A be a semisimple ring. Then
radT (A) = A♮ = socT (A) and topT (A) ∼= socT (A).
Proof. The first assertion is clear. Left multiplication with (0,1) gives a map T (A) → T (A) which
induces an isomorphism topT (A) ∼−→ socT (A) □
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1450 Henning Krause
For a ring A we denote by Σ(A) the set of isomorphism classes of simple A-modules. Set
S̄ := ∐
S∈Σ(A)
S and Ā := ∏
S∈Σ(A)
T (EndA(S)).
We view S̄ as an Ā-A-bimodule, with left action via T (EndA(S))↠ EndA(S) for each S inΣ(A), and
consider the triangular matrix ring
Ã :=
[
A 0
S̄ Ā
]
.
The idempotents e = [1 00 0] and f = [0 00 1] provide an Ã-module decomposition Ã = P ⊕Q, where
P := e Ã ∼= A and Q := f Ã ∼= S̄ ⊕ Ā.
We call Ã a cover of A because the idempotent e ∈ Ã yields an isomorphism
EndÃ(P )
∼= e Ãe ∼= A.
The following lemma expresses the distinct property of the cover Ã, namely that every simple
Ã-module embeds into Ã.
Lemma 2. We have S∗ ̸= 0 for every simple Ã-module S.
Proof. We claim that each simple Ã-module arises as the image of a morphism Ã = P ⊕Q → Q,
using that
HomÃ(P,Q)
∼= e Ã f = S̄ and HomÃ(Q,Q) ∼= f Ã f = Ā.
With Lemma 1 we compute socQ and obtain a decomposition into simples:
socQ = S̄ ⊕ soc Ā ∼=
∐
S∈Σ(A)
(
S ⊕EndA(S)♮
)
.
A simple Ã-module T comes either with a nonzero map P → T or a nonzero map Q → T . In the
first case T identifies with a simple A-module via the inclusion A → Ã given by x 7→ [ x 00 1], and
therefore with a summand of S̄ ⊆ socQ. In the second case T identifies with a simple Ā-module
via the inclusion Ā → Ã given by x 7→ [1 00 x ], and therefore with a summand of soc Ā ⊆ socQ. In
any case one obtains a monomomorphism T →Q ,→ Ã. □
Let P(A) denote the class of A-modules M that admit a finite resolution
0 −→ Pn −→ ·· · −→ P1 −→ P0 −→ M −→ 0
with all Pi finitely generated projective. We denote by
fin.dim A := sup{proj.dim M ∣∣M ∈P(A)}
the small finitistic dimension of A; this is a slight variation of the usual definition which seems
natural as the modules in P(A) are precisley the ones which become compact (or perfect) when
viewed as an object in the derived category of A.
The following lemma is [5, Lemma 7.2.8] and its proof is sketched for the convenience of the
reader; cf. the discussion in [2, § 5].
Lemma 3. For a ring A we have fin.dim A = 0 if and only if M∗ ̸= 0 for every finitely presented
Aop-module M.
Proof. Let proj A denote the category of finitely generated projective A-modules. The condition
fin.dim A = 0 means that every monomorphism in proj A splits. The duality
(−)∗ : (proj A)op ∼−−→ proj(Aop)
translates this into the condition on finitely presented Aop-modules. □
Henning Krause 1451
Theorem 4. For a ring A we have
fin.dim Ã ≥ fin.dim A and fin.dim Ãop = 0.
Proof. The idempotent e ∈ Ã with e Ãe ∼= A gives rise to a fully faithful functor
−⊗A e Ã : Mod A −→ Mod Ã
which is exact and maps projectives to projectives; also it preserves finite generation. This yields
the first assertion. The second assertion follows from Lemma 3 because we have M∗ ̸= 0 for every
finitely presented Ã-module M by Lemma 2. □
There is a somewhat more natural construction of a cover when the ring A is semilocal.
Recall that A is semilocal if the ring A/rad A is semisimple. In this case we have an idempotent
ε ∈ A/rad A and a Morita equivalence
A/rad A ∼ ε(A/rad A)ε ∼=
∏
S∈Σ(A)
EndA(S).
We set
Ã :=
[
A 0
A/rad A T (A/rad A)
]
and this is closely related to the cover Ã via the idempotent ε̃= [1 00 (ε,0)].
Lemma 5. For a semilocal ring A we have a Morita equivalence
Ã ∼ ε̃Ãε̃ ∼= Ã.
Proof. We use some general facts. Let Λ = [ A 0M B ] be a triangular matrix ring and e ∈ B an
idempotent such that B and eBe are Morita equivalent. Set ē = [1 00 e ]. Then Λ and ēΛē = [ A 0eM eBe ]
are Morita equivalent. Also, the trivial extensions T (B) and T (eBe) = (e,0)T (B)(e,0) are Morita
equivalent. □
We may identify Ã with Ã, as we are mostly interested in homological properties. In fact, Ã ∼= Ã
when A is semiperfect and basic. Note that the definition of Ã does not depend on any choices.
In particular, we have an identity Ãop = Ãop.
Next we discuss some ring theoretic properties which are preserved under the passage from A
to its cover Ã. Recall that a ring is semiprimary if it is semilocal and its radical is a nilpotent ideal.
Remark 6. If A is semilocal then Ã is semilocal. This follows from the isomorphism
top Ã = topP ⊕ topQ ∼−→ A/rad A⊕ (A/rad A)♮ = socQ
which is induced by the morphism Ã = P ⊕Q →Q ,→ Ã given by left multiplication with [0 01 (0,1)].
Moreover, in this case the inclusion A ,→ Ã yields the identity (rad A)n = (rad Ã)n for all n > 1.
Remark 7. If the ring A is left or right perfect then the same holds for Ã. This follows from
Remark 6, since A is right perfect if and only if A is semilocal and rad A is right T-nilpotent.
There is an analogue of Theorem 4 for the big finitistic dimension
Fin.dim A := sup{proj.dim M ∣∣M ∈ Mod A, proj.dim M <∞} .
We use the following fact which is a slight variation of [2, Theorem 6.3].
Proposition 8. For a ring A we have Fin.dim A = 0 if and only if A is right perfect and
fin.dim A = 0.
1452 Henning Krause
Proof. Suppose that A is right perfect and fin.dim A = 0. We need to show that every monomor-
phism φ : M → N between projective A-modules splits. This holds when M and N are finitely
generated since fin.dim A = 0. Because A is right perfect, any projective A-module decomposes
into a direct sum of finitely generated modules and can therefore be written as a filtered colimit
of finitely generated direct summands. Choose such a presentation M = colimi Mi . Then φ is a
filtered colimit of split monomorphisms φi : Mi → N , and therefore colimi Cokerφi ∼= Cokerφ is
a filtered colimit of projectives. Thus Cokerφ is projective and φ splits. For the other implication
we refer to [2]. □
Theorem 9. For a ring A we have
Fin.dim Ã ≥ Fin.dim A.
Moreover, Fin.dim Ãop = 0 if and only if A is left perfect.
Proof. The first assertion is easily checked as in the proof of Theorem 4. If Fin.dim Ãop = 0, then
Ã is left perfect by Proposition 8, and this implies that A is left perfect. For the converse suppose
that A is left perfect. Then Ã is left perfect by Remark 7. Thus Fin.dim Ãop = 0 by Theorem 4 and
Proposition 8. □
The preceding results demonstrate a failure of symmetry for the notion of ‘finite finitistic
dimension’, as pointed out in the recent work of Cummings [3]. In particular we have the following
examples.
Recall that a noetherian ring is regular if all its finitely generated modules have finite projective
dimension.
Corollary 10. Let A be a commutative noetherian ring that is regular of infinite Krull dimension.
Then
fin.dim Ã =∞ and fin.dim Ãop = 0.
Proof. The finitistic dimension fin.dim A is infinite by [1, Theorem 1.6 and Corollary 1.7]. Thus
the assertion follows from Theorem 4. □
Specific examples of regular rings of infinite Krull dimension have been constructed by Na-
gata; cf. [5, Example 7.2.20].
We continue with an example due to Kirkman and Kuzmanovich [4]. Let k be a field and
consider the quotient Λ= kQ/I of the path algebra kQ given by the quiver
Q : ◦ ◦ (i ∈N)
ai
bi
(with k-basis given by the paths in Q and multiplication induced by the composition of paths,
where for any pair of paths α,β we write βα for the composite when the terminal vertex of α
equals the initial vertex of β) modulo the ideal I that is generated by the elements
bt as br (r, s, t ∈N) bt as −at bt (t > s) ar br (r ∈N).
Note that Λ is a semiprimary ring with (radΛ)4 = 0.
Corollary 11. The ring Λ̃ is semiprimary satisfying
Fin.dimΛ̃= fin.dimΛ̃=∞ and Fin.dimΛ̃op = fin.dimΛ̃op = 0.
Proof. From Remark 6 it follows that the ring Λ̃ is semilocal with (radΛ̃)4 = 0. Thus Λ̃ is
semiprimary. In [4] it is shown that fin.dimΛ =∞. Then the assertion follows from Theorem 9,
using that Λ is left perfect. □
Henning Krause 1453
References
[1] M. Auslander, D. A. Buchsbaum, “Homological dimension in noetherian rings. II”, Trans. Am. Math. Soc. 88 (1958),
p. 194-206.
[2] H. Bass, “Finitistic dimension and a homological generalization of semi-primary rings”, Trans. Am. Math. Soc. 95
(1960), p. 466-488.
[3] C. Cummings, “Left-right symmetry of finite finitistic dimension”, 2022, https://arxiv.org/abs/2211.04394v1.
[4] E. Kirkman, J. Kuzmanovich, “Algebras with large homological dimensions”, Proc. Amer. Math. Soc. 109 (1990), no. 4,
p. 903-906.
[5] H. Krause, Homological theory of representations, Cambridge Studies in Advanced Mathematics, vol. 195, Cambridge
University Press, 2022.


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An entropic generalization of Caffarelli’s contraction theorem via covariance inequalities

Arpita Nayek

Pinakinath Saha

Torus quotient of the Grassmannian $G_{n,2n}$

Erik Burman

Guillaume Delay

Alexandre Ern

Lauri Oksanen

A stability estimate for data assimilation subject to the heat equation with initial datum

Michel Duprez

Vanessa Lleras

Alexei Lozinski

Killian Vuillemot

$\phi $-FEM for the heat equation: optimal convergence on unfitted meshes in space

Omar Anza Hafsa

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Remarks on homogenization and $3D$-$2D$ dimension reduction of unbounded energies on thin films

Hipolito Treffinger

Comptes Rendus
Mathématique
Hipolito Treffinger
The size of a stratifying system can be arbitrarily large
Volume 361 (2023), p. 15-19
https://doi.org/10.5802/crmath.385
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 15-19
https://doi.org/10.5802/crmath.385
Algebra, Representation theory / Algèbre, Théorie des représentations
The size of a stratifying system can be
arbitrarily large
Hipolito Treffinger∗, a
a Institut de Mathématiques Jussieu - Paris Rive Gauge, Université Paris Cité. Paris,
France
E-mail: treffinger@imj-prg.fr
In honour of Ibrahim Assem on the occasion of his retirement.
Abstract. In this short note we construct two families of examples of large stratifying systems in module
categories of algebras. The first examples consist on stratifying systems of infinite size in the module category
of an algebra A. In the second family of examples we show that the size of a finite stratifying system in the
module category of a finite dimensional algebra A can be arbitrarily large in comparison to the number
of isomorphism classes of simple A-modules. We note that both families of examples are built using well-
established results in higher homological algebra.
Funding. The author is supported by the European Union’s Horizon 2020 research and innovation pro-
gramme under the Marie Sklodowska–Curie grant agreement No 893654. He is also partially funded by the
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strat-
egy Programme – EXC-2047/1 – 390685813.
Manuscript received 28 January 2021, revised 15 September 2021 and 13 May 2022, accepted 25 May 2022.
1. Introduction
In this paper, A is a basic finite-dimensional algebra over an algebraically closed field K , mod A is
the category of finitely presented (right) A-modules and K0(A) denotes the Grothendieck group
of A.
The notion of exceptional sequences originated in algebraic geometry [1,8,9,22] and was later
introduced to representation theory in [5, 21], becoming an important subject of study in both
disciplines.
On the representation theory side, the definition of exceptional sequence can be stated in
the module category of any finite-dimensional algebra. Despite this, most of the articles on the
subject studied exceptional sequences in the module category of hereditary algebras. Outside the
hereditary case, the notion that have been mostly studied is the more general notion of stratifying
systems, firstly introduced in [7] (see also [17, 18]). If the algebra is hereditary, it has been proven
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
16 Hipolito Treffinger
in [3] that every stratifying system is an exceptional sequence. However, this is not true in general.
For instance, the stratifying systems that we build in this note are not exceptional sequences (see
Remarks 2 and 7). We now recall the definition of stratifying systems and exceptional sequences.
Definition. Let A be an algebra. A stratifying system of size t ∈ N∪ {∞} in the category mod A
of finitely generated A-modules is a pair (Θ,≤) where Θ := {θi : i ∈ [1, t ], i 6= ∞} is a family of
indecomposable objects in mod A and ≤ is a linear order on the set [1, t ] := {1, . . . , t } such that
HomA(θ j ,θi ) = 0 if i < j and Ext1A(θ j ,θi ) = 0 if i ≤ j . Moreover, we say that a stratifying system
(Θ,≤) is an exceptional sequence if EndA(θi ) ∼= K for all i ∈ [1, t ] and ExtnA(θ j ,θi ) = 0 for all i ≤ j
and n ∈N.
There are numerous works studying the consequences of the existence of a stratifying system
in the module category of an algebra, see for instance [3, 4, 6, 7, 17, 18]. However, the existence
of stratifying systems in module categories is a problem that has received less attention. To
our knowledge, the only works addressing the existence of exceptional sequences or stratifying
systems outside the hereditary case are [15] for canonical algebras, [12] for the Auslander algebra
of K [x]/x t , [20] for quotients of type A zig-zag algebras and [2, 16] for arbitrary algebras using
techniques from τ-tilting theory.
The classical examples of stratifying systems are the so-called canonical stratifying systems.
These are stratifying systems which are constructed using all indecomposable projective modules
and, as a consequence, the size t of every canonical stratifying system coincides with rk(K0(A)).
Also, it was shown in [3] that the size of every stratifying system in the module category of a
hereditary algebra H is bounded by rk(K0(H)).
This is not true for stratifying systems in the module category of an arbitrary algebra since
there are examples of stratifying systems whose size is bigger than the rank of the Grothendieck
group of the algebra, see for instance [7, § 3.2] and [17, Remark 2.7]. However, it was conjectured
the existence of an upper-bound for the size of a stratifying system in the module category A
which was a function of rk(K0(A)).
In Section 2, we show that this conjecture is false by proving the existence of a family of
algebras having a stratifying system of infinite size. Later, in Section 3, we construct a family of
finite stratifying systems whose size cannot be linearly bounded by the rank of the Grothendieck
group of their ambient module category.
2. stratifying systems of infinite size
The notion of d-representation infinite algebras was introduced and first studied by Herschend,
Iyama and Oppermann in [10]. In this section we use a particular family of d-representation
infinite algebras, the so-called Bellinson algebras [10, Example 2.15], to construct examples of
a stratifying systems of infinite size. We note that the choice of this particular family is made to
give an explicit example. However the same construction of stratifying system of infinite size can
be performed in any d-representation infinite algebra.
Fix a positive integer d > 1. Then the Beilinson algebra Bd is the path algebra of the quiver
1
a10
...
((
a1d
66 2
a20
...
((
a2d
66 3 · · · d
ad0
...
**
add
44 d +1
modulo the ideal of relations generated by the elements of the form aki a
k+1
j − akj ak+1i . Also, we
recall that the d-Auslander–Reiten translations in modBd are defined as
τd (−) := H 0 (νd (−)) : modBd → modBd
Hipolito Treffinger 17
τ−d (−) := H 0
(
ν−d (−)
)
: modBd → modBd
where νd is the autoequivalence of D
d (modBd ) defined as the composition νd := ν◦ [−d ] of the
Nakayama functor ν : Dd (modBd ) → Dd (modBd ) with the inverse [−d ] of the d th suspension
functor [d ] : Dd (modBd ) → Dd (modBd ).
It is clear that as a module over itself, Bd ∼=
⊕d+1
i=1 P (i ), where every P (i ) is indecomposable
projective in modBd and every indecomposable projective P is isomorphic to P (i ) for some
1 ≤ i ≤ d + 1. Moreover, since Bd is basic, we have that P (i ) is not isomorphic to P ( j ) if i is
different from j . Likewise, every indecomposable injective is isomorphic to one of the modules
I (i ), where DBd ∼=
⊕d+1
i=1 I (i ). Following this notation we define Pd (i ) := {τ
− j
d P (i ) : j ∈ N} and
Id (i ) := {τ jd I (i ) : j ∈ N} for every 1 ≤ i ≤ d + 1. We note that the natural order ≤ in N induces a
natural order in Pd (i ) and Id (i ) for every 1 ≤ i ≤ d +1.
Theorem 1. Let d be a positive integer greater than two and let ≤ be the natural order in N. With
the notation above, (Pd (i ),≤) is a stratifying system in modBd of infinite size for all 1 ≤ i ≤ d +1.
Similarly, (Id (i ),≤op ) is a stratifying system modBd of infinite size for all 1 ≤ i ≤ d +1.
Proof. We only prove the case of (Pd (i ),≤) for a given 1 ≤ i ≤ d + 1, since the other cases are
similar. First, [10, Proposition 4.10(a)] states that τ− jd P (i ) and τ
−l
d P (i ) are not isomorphic if j 6= l .
Hence Pd (i ) has infinitely many objects.
It follows from [10, Propositions 2.3 and 2.9] that HomBd (τ
− j
d P (i ),τ
−l
d P (i )) = 0 if l < j . Also,
[10, Proposition 4.10(f)] implies that
Ext1Bd
(
τ
− j
d P (i ),τ
−l
d P (i )
)
= 0
for all j , l ∈ N. In particular Ext1Bd (τ
− j
d P (i ),τ
−l
d P (i )) = 0 if l ≤ j . Then (Pd (i ),≤) is a stratifying
system of infinite size. 
Remark 2. We note that the stratifying system (Pd (i ),≤) is not an exceptional sequence, since
Extd
Bd
(τ−1d M , M) 6= 0 for all M ∈ Pd (i ). Likewise, (Id (i ),≤op ) is not an exceptional sequence since
Extd
Bd
(M ,τd M) 6= 0 for all M ∈ Id .
3. Stratifying systems for higher Auslander algebras
In the previous section we show the existence of stratifying systems of infinite size. In this section
we show the existence of a family of algebras having finite stratifying systems whose size grows
quicker than the rank of the Grothendieck group of the algebras.
Theorem 3. For every positive integer m, there exists an algebra Am and a stratifying system
(Θm ,≤) of size sm in mod Am such that sm > m.rk(K0(Am)).
Before proving our theorem, we need to recall some notions and results of higher homological
algebra that will be used in our proof. A subcategory M of mod A is said to be d-cluster tilting if
M =
{
X ∈ mod A : ExtiA(X , M) = 0 for all M ∈M and 1 ≤ i ≤ d −1
}
=
{
Y ∈ mod A : ExtiA(M ,Y ) = 0 for all M ∈M and 1 ≤ i ≤ d −1
}
.
An A-module M is said to be d-cluster tilting if add M is a d-cluster tilting subcategory of mod A.
In general, given an algebra A it is very difficult to know if A has a d-cluster tilting subcategory
(see [23] and the references therein). However, we are interested in the so-called higher Auslander
algebras introduced in [13]. These algebras are defined inductively as follows.
Definition/Theorem 4 ([13]). Let n be a positive integer and d be a non-negative integer. The d-
Aulander algebra Adn is the path algebra of the linearly oriented quiver of type An if d = 0, or is the
endomorphism algebra End
Ad−1n
(M) of the d-cluster tilting module M of Ad−1n for all d > 0.
18 Hipolito Treffinger
In fact, there are several combinatorial descriptions of the quiver and relations of Adn [11, 13,
14, 19]. In this paper we follow the notation appearing in [19].
Proposition 5 ([19, Theorem 3.6]). Let Adn be the d-Auslander algebra of An . Then there is a
bijection between the vertices of the quiver of Adn with the set
Vn,d =
{
(x0, x1, . . . , xd ) : 1 ≤ x0 < x1 < ·· · < xd ≤ n +d
}
.
Moreover, given two elements x, y ∈ Vn,d there is an arrow x → y if there exists 0 ≤ k ≤ d such that
yk = xk +1 and yi = xi for all i 6= k.
Remark 6. Note that it follows from the previous proposition that the number of vertices of the
quiver of Adn is equal to the number of integer lattice points inside the canonical d-simplex in
Rd+1 generated by the interval [0,n].
Proof of Theorem 3. Let A1n be the (1-)Auslander algebra of An . It follows from Theorem 4 that
A1n has a 2-cluster tilting module M =
⊕tn
i=1 Mi such that EndA1n M =A2n . We claim that there exists
of an order ≤ in the set [1, tn] := {1, . . . , tn} such that (Θn ,≤) := ({Mi : i ∈ [1, tn]},≤) is a stratifying
system.
We first note that it follows from Proposition 5 that the quiver of A2n is an acyclic quiver. As a
consequence, there exists a total order ≤ in [1, tn] such that HomA1n (Mi , M j ) = 0 if j < i . We also
have that Ext1
A1n
(Mi , M j ) = 0 for all i , j ∈ [1, tn] because M is a 2-cluster tilting object. In particular,
Ext1
A1n
(Mi , M j ) = 0 if i ≤ j . Then (Θn ,≤) is a stratifying system.
Now, it is easy to see that rk(K0(A1n)) = n(n+1)2 . Moreover the size tn of (Θn ,≤) is equal to the
tetrahedral number tn = n(n+1)(n+2)6 , see Remark 6. Hence the ratio R(n) between the size of the
stratifying system (Θn ,≤) and rk(K0(An)) is
R(n) = n(n +1)(n +2)/6
n(n +1)/2 =
n +2
3
.
In particular, for each m ∈ N, if we fix n = 3m − 1 we have that size t3m−1 of the stratifying
system (Θ3m−1,≤) inA13m−1 is greater than m.rk(K0(A13m−1)). Indeed,
t3m−1 = 3m −1+2
3
.rk
(
K0
(
A13m−1
))= (m + 1
3
)
.rk
(
K0
(
A13m−1
))> m.rk (K0 (A13m−1)) .
The result follows by taking Am =A13m−1 and sm = t3m−1. 
Remark 7. We note that the stratifying system (Θm ,≤) constructed in the proof of Theorem 3 is
not an exceptional sequence since we can find objects Mi , M j ∈Θm such that Ext2Am (Mi , M j ) 6= 0
for all m ∈N.
Remark 8. In the previous proof we fixed d = 1 to construct our example, but it is easy to see that
a similar argument is valid for every d . Indeed, by Propositon 5 we have that the quiver of Adn is
acyclic for every d . Hence we can construct a stratifying system inAdn for all n and d by means of
its (d +1)-cluster tilting module.
Moreover, the number of vertices of the quiver of Adn grows as n
d as n goes to infinity. Since
there is a stratifying system in modAdn of size K0(A
d+1
n ), we conclude that for every m ∈N there are
infinitely many algebras Am with a stratifying system (Θm ,≤) such that the size of Θm is greater
than m.rk(K0(Am)).
Remark 9. A stratifying system (Θ,≤) is said to be complete if there is no indecomposable
module N such that (Θ∪ {N },≤′) is a stratifying system such that i ≤′ j if i ≤ j . It follows from the
construction of (Θm ,≤) that there is no indecomposable Am-module M0 such that ({M0}∪Θm ,≤),
where ≤ is the natural order in [0, tm]. Likewise, there is no indecomposable Mtm+1 such that
(Θm ∪ {Mtm+1},≤) is a stratifying system where ≤ is the natural order in [1, tm +1]. However, is not
clear that (Θm ,≤) is a complete stratifying system in general.
Hipolito Treffinger 19
Acknowledgements
The author would like to thank Octavio Mendoza, Corina Saenz and Aran Tattar for their com-
ments and remarks. A special thanks goes to the anonymous referee for their careful reading and
helpful suggestions.The author is specially grateful of Gustavo Jasso for pointing out that the ar-
guments of Section 3 could be used to construct a stratifying system of infinite size, leading to the
inclusion of Section 2.
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(2019), p. 1139-1200.
[15] H. Meltzer, “Exceptional sequences for canonical algebras”, Arch. Math. 64 (1995), no. 4, p. 304-312.
[16] O. Mendoza Hernández, H. Treffinger, “Stratifying systems through τ-tilting theory”, Doc. Math. 25 (2020), p. 701-
720.
[17] E. do N. Marcos, O. Mendoza, C. Sáenz, “Stratifying systems via relative simple modules”, J. Algebra 280 (2004), no. 2,
p. 472-487.
[18] ——— , “Stratifying systems via relative projective modules”, Commun. Algebra 33 (2005), no. 5, p. 1559-1573.
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The size of a stratifying system can be arbitrarily large

Naoufel Battikh

Formes différentielles non commutatives et Algèbres de Gerstenhaber

Dandan Ma

Fan Wu

Comptes Rendus
Mathématique
Dandan Ma and Fan Wu
Shinbrot’s energy conservation criterion for the 3D
Navier–Stokes–Maxwell system
Volume 361 (2023), p. 91-96
https://doi.org/10.5802/crmath.379
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2023, Vol. 361, p. 91-96
https://doi.org/10.5802/crmath.379
Partial differential equations / Equations aux dérivées partielles
Shinbrot’s energy conservation criterion for
the 3D Navier–Stokes–Maxwell system
DandanMaa and FanWu∗, a
a College of Science, Nanchang Institute of Technology, Nanchang, Jiangxi 330099,
China
E-mails: 981423315@qq.com, wufan0319@yeah.net
Abstract. This paper concerns the energy conservation for the weak solutions to the Navier–Stokes–Maxwell
system. Although the Maxwell equation with hyperbolic nature, we still establish a Lq Lp type condition guar-
antee validity of the energy equality for the weak solutions. We mention that there no regularity assumption
on the electric field E .
2020 Mathematics Subject Classification. 76W05, 35Q30, 35Q61.
Manuscript received 24 January 2022, accepted 9 June 2022.
1. Introduction
In this paper, we study the incompressible Navier–Stoke–Maxwell system
∂t u +u ·∇u −µ∆u =−∇p + j ×B , divu = 0,
1
c ∂t E −∇×B =− j , j =σ(cE +u ×B),
1
c ∂t B +∇×E = 0, divB = 0,
(1)
with the initial data
u(0, x) = u0,E(0, x) = E0,B(0, x) = B0 (2)
for (t , x) ∈ R+ ×Ω, where Ω = T3 is a periodic domain in R3. Here c > 0 denotes the speed of
light, µ > 0 is the viscosity of the fluid, and σ > 0 is the electrical conductivity. In the above
system (1), u = (u1,u2,u3) = u(t , x) stands for the velocity field of the (incompressible) fluid,
while E = (E1,E2,E3) = E(t , x) and B = (B1,B2,B3) = B(t , x) are the electric and magnetic fields,
respectively. The scalar function p = p(t , x) is the pressure and is also an unknown. Observe,
though, that the electric current j = j (t , x) is not an unknown, for it is fully determined by (u,E ,B)
through Ohm’s law. For simplicity, we will take µ= c =σ= 1.
The Navier–Stokes–Maxwell system (1) describes the evolution of a plasma (i.e., a charged
fluid) subject to a self-induced electromagnetic Lorentz force j × B . Mathematically, (1) is a
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
92 Dandan Ma and Fan Wu
coupled system, constituted by the parabolic nature of the Navier–Stokes equations from fluid
dynamics and the hyperbolic of the Maxwell equation from electromagnetism. Moreover, it can
be derived from the Vlasov–Maxwell–Boltzmann system [2]. In the 2D case, Masmoudi [8] proved
global existence of regular solutions to the Maxwell–Navier–Stokes system (1), and also provide an
exponential growth estimate for the H s norm of the solution when the time goes to infinity. In the
3D case, Ibrahim–Keraani [10] showed the existence of global small mild solutions of system (1).
Recently, Arsenio–Gallagher [1] have made important and new progress in this direction, they
established that global existence of solutions to the 3D system (1) holds whenever the initial
datum (u0,E0,B0) is chosen in the natural energy space L2, while the electromagnetic field
(E0,B0) alone lies in Ḣ s , for some given s ∈ [ 12 , 32 ), and is sufficiently small when compared to
some non-linear function of the initial energy
E0 := 1
2
(‖u0‖2L2 +‖E0‖2L2 +‖B0‖2L2) .
Before discussing the contents of this paper let us recall some well-established facts regarding
the incompressible Navier–Stokes equations (corresponding to the case when (E ,B) = 0 in (1)), in
relation with this work. Pioneering works of Leray [5] and Hopf [3] showed the global existence of
a weak solution called Leray–Hopf solution with initial data u0 ∈ L2(Ω). As usual, a weak solution
u satisfies the energy inequality
‖u(t )‖2L2 +2
∫ T
0
∫
Ω
|∇u|2 d xd t ≤ ‖u0‖2L2 (3)
for any t ∈ (0,T ). On the other hand, regular solutions to the 3D Navier–Stokes equations satisfy
the energy equality:
‖u(t )‖2L2 +2
∫ T
0
∫
Ω
|∇u|2 d xd t = ‖u0‖2L2 . (4)
A natural question that still remains open is whether energy equality, which should be expected
from a physical point of view, is valid for weak solutions. Lions [6] and Ladyzhenskaya [4]
proved independently that such solutions satisfy the (global) energy equality (4) under the
additional assumption u ∈ L4L4. Shinbrot [9] generalized the Lions–Ladyzhenskaya condition to
u ∈ Lr (0,T ;Ls (Ω)) with 2r + 2s ≤ 1, s ≥ 4. Recently, Tan and Yin [12] proved that u ∈ L2,∞(0,T ;B MO)
implies energy equality (4).
For system (1), by results of Arsenio–Gallagher [1], it is know that for initial data (u0,E0,B0),
with divu0 = divB0 = 0, belongs to(
L2 × (H s)2)(R3) , 1
2
≤ s < 3
2
,
and
‖(E0,B0)‖Ḣ s C∗E
s− 12
0 e
C∗E0 ≤ 1,
then there is a global weak solution (u,E ,B) to the 3D Navier–Stokes–Maxwell system (1) such
that
u ∈ L∞ (R+;L2)∩L2 (R+; Ḣ 1) , (E ,B) ∈ L∞ (R+;L2) , j ∈ L2 (R+;L2) (5)
satisfying the energy inequality, for almost all t > 0,
1
2
(‖u(t )‖2L2 +‖E(t )‖2L2 +‖B(t )‖2L2)+∫ t
0
(‖∇u(τ)‖2L2 +‖ j (τ)‖2L2)dτ
≤1
2
(‖u0‖2L2 +‖E0‖2L2 +‖B0‖2L2) . (6)
Dandan Ma and Fan Wu 93
Moreover, the regular solutions to (1) satisfy the corresponding energy equality
1
2
(‖u(t )‖2L2 +‖E(t )‖2L2 +‖B(t )‖2L2)+∫ t
0
(‖∇u(τ)‖2L2 +‖ j (τ)‖2L2)dτ
=1
2
(‖u0‖2L2 +‖E0‖2L2 +‖B0‖2L2) . (7)
Although we can not prove that the weak solution is regular on time, due to the fact that the
Navier–Stoke equations is a sub-system of (1). An interesting question is whether or not the
energy equality (7) is holds for such weak solutions. By using the lemma introduced by Lions,
which has been used in the Yu’s article [13], we shall give an positive answer for this problem if
(u,B) ∈ Lq Lp .
We state our main result as follows:
Theorem 1. Let u ∈ L∞(0,T ;L2(Ω)) ∩ L2(0,T ; H 1(Ω)), (E ,B) ∈ L∞(0,T ;L2(Ω)) and j ∈ L2(0,T ;
L2(Ω)) be a weak solution to system (1). In addition, if u ∈ Lq (0,T ;Lp (Ω)) for any 1q + 1p
≤ 12 , p ≥ 4, and B ∈ Lr (0,T ;Ls (Ω)) for any 1r + 1s ≤ 12 , s ≥ 4, then∫
Ω
|u(t , x)|2 +|E(t , x)|2 +|B(t , x)|2 d x +2
∫ T
0
∫
Ω
|∇u|2 +| j |2d xd t
=
∫
Ω
|u0|2 +|E0|2 +|B0|2 d x
for any t ∈ [0,T ].
Remark 2. This result extends the well-known Shinbrot’s energy conservation criterion to the
Navier–Stokes–Maxwell system (1).
Remark 3. By means of arguement in [11], it is interesting to generalize Theorem 1 from periodic
domain to bounded Lipschitz domain.
2. Proof of Theorem 1.1
This section is devoted to the proof of Theorem 1. By using interpolation inequalities∥∥ f ∥∥L4(0,T ;L4(Ω)) ≤C ∥∥ f ∥∥θL∞(0,T ;L2(Ω)) ∥∥ f ∥∥1−θLn (0,T ;Lm (Ω)) ≤C
with 1n + 1m = 12 ,m > 4, we need only prove the case p = q = r = s = 4. For the sake of simplicity,
we will proceed as if the solution is differentiable in time. The extra arguments needed to mollify
in time are straightforward.
Let η : R3 → R be a standard mollifier, i.e. η(x) = C e
1
|x|2−1 for |x| < 1 and η(x) = 0 for |x| > 1,
where constant C > 0 selected such that ∫R3 η(x)dx = 1. For any ε > 0, we define the rescaled
mollifier ηε(x) = ε−3η
( x
ε
)
. For any function f ∈ L1loc(Ω), its mollified version is defined as
f ε(x) = ( f ∗ηε) (x) = ∫
Ω
ηε(x − y) f (y)dy.
If f ∈W 1,p (Ω), the following local approximation is well known
f ε(x) → f in W 1,pl oc (Ω) ∀ p ∈ [1,∞).
The crucial ingredient to prove Theorem 1 is the following important lemma proved by Lions [7]
Lemma 4. Let ∂ be a partial derivative in one direction. Let f ,∂ f ∈ Lp (R+ ×Ω), g ∈ Lq (R+ ×Ω)
with 1 ≤ p, q ≤∞, and 1p + 1q ≤ 1. Then, we have∥∥∂( f g )∗ηε−∂( f (g ∗ηε))∥∥Lr (R+×Ω) ≤C ∥∥∂ f ∥∥Lp (R+×Ω) ‖g‖Lq (R+×Ω)
for some constant C > 0 independent of ε, f and g , and with 1r = 1p + 1q . In addition,
94 Dandan Ma and Fan Wu
∂( f g )∗ηε−∂
(
f
(
g ∗ηε
))→ 0 in Lr (R+×Ω)
as ε→ 0, if r <∞.
With Lemma 4 in hand, we are ready to prove our main result. First, applying the mollifier to
the system (1) and multiplying all equations of the system by uε, Eε and Bε, respectively, one has
∫
Ωu
ε
(
∂t u +u ·∇u −∆u +∇p − j ×B
)εd x = 0,∫
ΩE
ε
(
∂t E −∇×B + j
)εd x = 0,∫
ΩB
ε (∂t B +∇×E)εd x = 0.
(8)
This yields
1
2
d
d t
∫
Ω
(∣∣uε∣∣2 + ∣∣Eε∣∣2 + ∣∣Bε∣∣2)d x +∫
Ω
∣∣∇uε∣∣2 d x
=−
∫
Ω
div(u ⊗u)ε ·uεd x +
∫
Ω
( j ×B)ε ·uεd x +
∫
Ω
(∇×B)ε ·Eεd x
−
∫
Ω
j ε ·Eεd x −
∫
Ω
(∇×E)ε ·Bεd x.
(9)
Clearly,∫
Ω
(∣∣uε∣∣2 + ∣∣Eε∣∣2 + ∣∣Bε∣∣2)d x −∫
Ω
(∣∣uε0∣∣2 + ∣∣Eε0∣∣2 + ∣∣Bε0 ∣∣2)d xd t +2∫ T
0
∫
Ω
∣∣∇uε∣∣2 d xd t
=−2
∫ T
0
∫
Ω
div(u ⊗u)ε ·uεd xd t +2
∫ T
0
∫
Ω
( j ×B)ε ·uεd xd t +2
∫ T
0
∫
Ω
(∇×B)ε ·Eεd xd t
−2
∫ T
0
∫
Ω
j ε ·Eεd xd t −2
∫ T
0
∫
Ω
(∇×E)ε ·Bεd xd t
=I1 + I2 + I3 + I4 + I5.
(10)
Notice that
div(u ⊗u)ε = [div(u ⊗u)ε−div(u ⊗uε)]+ [div(u ⊗uε)−div(uε⊗uε)]+div(uε⊗uε)
= I11 + I12 + I13,
thus
I1 =−2
∫ T
0
∫
Ω
(I11 + I12 + I13) ·uεd xd t . (11)
From Lemma 4, one obtains
‖I11‖
L
4
3
(
0,T ;L
4
3 (Ω)
) ≤C‖u‖L4(0,T ;L4(Ω))‖∇u‖L2(0,T ;L2(Ω))
and it converges to zero in L
4
3 (0,T ;L
4
3 (Ω)) as ε tends to zero. Thus, as ε goes to zero, it follows that∣∣∣∣−2∫ T
0
∫
Ω
I11u
εd xd t
∣∣∣∣= ∣∣∣∣∫ T
0
∫
Ω
[
div(u ⊗u)ε−div(u ⊗uε)] ·uεd xd t ∣∣∣∣
≤ ‖I11‖
L
4
3
(
0,T ;L
4
3 (Ω)
) ∥∥uε∥∥L4(0,T ;L4(Ω))
→ 0.
(12)
Dandan Ma and Fan Wu 95
Moreover, we have∣∣∣∣−2∫ T
0
∫
Ω
I12u
εd xd t
∣∣∣∣= ∣∣∣∣∫ T
0
∫
Ω
[
div
(
u ⊗uε)−div(uε⊗uε)] ·uεd xd t ∣∣∣∣
=
∣∣∣∣∫ T
0
∫
Ω
(
u ⊗uε−uε⊗uε) ·∇uεd xd t ∣∣∣∣
≤ ∥∥u −uε∥∥L4(0,T ;L4(Ω)) ∥∥uε∥∥L4(0,T ;L4(Ω)) ∥∥∇uε∥∥L2(0,T ;L2(Ω))
→ 0.
(13)
Since divuε = 0, one has
−2
∫ T
0
∫
Ω
I13 ·uεd x = 0. (14)
Combining (11)-(14), we know that
I1 → 0, as ε→ 0. (15)
For the term I2, we claim that
I2 = 2
∫ T
0
∫
Ω
( j ×B)ε ·uεd xd t → 2
∫ T
0
∫
Ω
( j ×B) ·ud xd t , as ε→ 0. (16)
Indeed,
2
∫ T
0
∫
Ω
( j ×B)ε ·uε− ( j ×B) ·ud xd t
=2
∫ T
0
∫
Ω
( j ×B)ε ·uε− ( j ×B) ·uε+ ( j ×B) ·uε− ( j ×B) ·ud xd t
=2
∫ T
0
∫
Ω
[
( j ×B)ε− ( j ×B)] ·uε+ ( j ×B) · (uε−u)d xd t
≤2∥∥( j ×B)ε− ( j ×B)∥∥
L
4
3
(
0,T ;L
4
3 (Ω)
) ‖uε‖L4(0,T ;L4(Ω))
+2∥∥ j ×B∥∥
L
4
3
(
0,T ;L
4
3 (Ω)
) ‖uε−u‖L4(0,T ;L4(Ω))
→ 0, as ε→ 0,
(17)
where used the fact that∥∥ j ×B∥∥
L
4
3
(
0,T ;L
4
3 (Ω)
) ≤ ∥∥ j∥∥L2(0,T ;L2(Ω)) ‖B‖L4(0,T ;L4(Ω)) ≤C .
By using the same trick, we find that
I4 =−2
∫ T
0
∫
Ω
j ε ·Eεd xd t →−2
∫ T
0
∫
Ω
j ·E d xd t , as ε→ 0. (18)
After integration by parts, the term I3 can be dominated as
I3 =2
∫ T
0
∫
Ω
(∇×B)ε ·Eεd xd t
=2
∫ T
0
∫
Ω
(
εi j k∂ j Bk
)ε ·Eεi d xd t
=−2
∫ T
0
∫
Ω
εi j k B
ε
k ·∂ j Eεi d xd t
=2
∫ T
0
∫
Ω
εk j i B
ε
k ·∂ j Eεi d xd t
=2
∫ T
0
∫
Ω
Bε · (∇×E)εd xd t ,
(19)
96 Dandan Ma and Fan Wu
So, it follows that
I3 + I5 = 0. (20)
Letting ε go to zero in (10), and using (15)-(20), what we have proved is that in the limit∫
Ω
(|u|2 +|E |2 +|B |2)d x +2∫ T
0
∫
Ω
|∇u|2d xd t
=
∫
Ω
(|u0|2 +|E0|2 +|B0|2)d xd t +2∫ T
0
∫
Ω
( j ×B) ·ud xd t −2
∫ T
0
∫
Ω
j ·E d xd t . (21)
Thus, it holds that∫
Ω
(|u|2 +|E |2 +|B |2)d x +2∫ T
0
∫
Ω
|∇u|2 +| j |2d xd t =
∫
Ω
(|u0|2 +|E0|2 +|B0|2)d xd t , (22)
where we have used the fact that
2
∫ T
0
∫
Ω
( j ×B) ·ud xd t −2
∫ T
0
∫
Ω
j ·E d xd t =−2
∫
Ω
(u ×B) · j dx −2
∫
Ω
E · j dx
and
j = E +u ×B.
The proof of Theorem 1 is completed.
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Shinbrot’s energy conservation criterion for the 3D Navier–Stokes–Maxwell system

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Shinbrot’s energy conservation criterion for the 3D Navier–Stokes–Maxwell system

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