Some dimensions of DG polynomial algebras
Xuefeng Mao, Maoyun Zhang

TL;DR
This paper investigates various homological dimensions of a specific class of differential graded polynomial algebras, providing explicit calculations for their DG Krull, global, ghost, and Rouquier dimensions.
Contribution
It explicitly determines multiple homological dimensions of cochain DG polynomial algebras with polynomial underlying graded algebra.
Findings
DG Krull dimension computed
Global dimension determined
Ghost and Rouquier dimensions calculated
Abstract
Assume that is a cochain DG polynomial algebra such that its underlying graded algebra \mathcal{A}^{#} is a polynomial algebra generated by degree elements. We determine the DG Krull dimension, the global dimension, the ghost dimension and the Rouquier dimension of .
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Taxonomy
TopicsAlgebraic structures and combinatorial models · Advanced Topics in Algebra · Nonlinear Waves and Solitons
Some dimensions of DG polynomial algebras
X.-F. Mao
Department of Mathematics, Shanghai University, Shanghai 200444, China
and
M.-Y.Zhang
Department of Mathematics, Shanghai University, Shanghai 200444, China
Abstract.
Assume that is a cochain DG polynomial algebra such that its underlying graded algebra is a polynomial algebra generated by degree elements. We determine the DG Krull dimension, the global dimension, the ghost dimension and the Rouquier dimension of .
Key words and phrases:
DG polynomial algebra, DG Krull dimension, global dimension, ghost dimension, Rouquier dimension
2010 Mathematics Subject Classification:
Primary 16E10, 16E45, 16E65, 16W20,16W50
1. introduction
Let is an algebraically closed field of characteristic [math]. This paper deals with DG polynomial -algebras, which are connected cochain DG algebras whose underlying graded algebras are polynomial algebras generated by degree one elements. In [MGYC], the differential structures and homological properties of DG polynomial algebras are systematically studied. However, we still know very little about various homological invariants of DG polynomial algebras.
In dimension theory for rings, it is well known that that the Krull dimension and the global dimension of the polynomial ring are both equal to . The DG Krull dimension and the global dimension for DG algebras have been introduced in [BSW] and [MW3], respectively. It is natural for one to be intensely curious about the DG Krull dimension and the global dimension of DG polynomial rings. In group theory, the terminology ‘class’ is used to measure the shortest length of a filtration with sub-quotients of certain type. Avramov-Buchweitz-Iyengar [ABI] introduced free class, projective class and flat class for differential modules over a commutative ring. Inspired from their work, the first author and Wu [MW3] introduced the notion of ‘DG free class’ for semi-free DG modules over a DG algebra . In brief, the DG free class of a semi-free DG -module is the shortest length of all strictly increasing semi-free filtrations. For a more general DG module , the first author and Wu [MW3] introduced the invariant ‘cone length’, which is the minimal DG free class of all its semi-free resolutions. The studies of these invariants for DG modules can be traced back to Carlsson’s work in 1980s. In [Car], Carlsson studied ‘free class’ of solvable free DG modules over a graded polynomial ring in variables of positive degree. By [Car, Theorem 16], one sees that the cone length of any totally finite DG -module satisfies the inequality . Note that the invariant ‘’ was denoted by ‘’ in [Car]. The invariant ‘cone length’ of a DG -module plays a similar role in DG homological algebra as the ‘projective dimension’ of a module over a ring does in classic homological ring theory (cf.[MW3]). The left (resp. right) global dimension of a connected DG algebra is defined to be the supremum of the set of cone lengths of all DG -modules (resp. -modules). The difficulty in the studies of the DG Krull dimension and the global dimension of DG polynomial rings comes from the fact that these two invariants are determined by a combination of the graded algebra structure and the differential system. Due to the classifications of DG polynomial algebras in [MGYC], we show the following theorem (see Theorem 4.2 and Theorem 4.4):
** Theorem A. ** Assume that is a cochain DG polynomial algebra such that with . Then and
[TABLE]
Christensen [Chr] introduces the invariant ‘ghost length’ for objects in triangulated categories. This concept can be applied to complexes and DG modules. The ghost length together with other invariants, including level, cone length and trivial category, of DG modules are studied in [Kur1, Kur2, Mao]. In [HL1, HL2], Hovey-Lockridge introduce the ghost dimension of a ring, which is defined as the maximum ghost lengthes of all perfect complexes. One can naturally extend this definition to DG algebras. Another important invariant of DG modules is called level. In the derived category of a DG algebra , the level of a DG -module counts the number of steps required to build out of via triangles. In [ABIM], some important and fundamental properties of the level of DG modules are investigated. To study topological spaces with categorical representation theory, Kuriayashi [Kur1] introduced the levels for space and gave a general method for computing the level of a space and studied the relationship between the level and other topological invariants such as Lusternik-Schnirelmann category. Later, Schmidt [Sch] used the properties of level of DG modules to study the structure of the Auslander-Reiten quiver of some important cochain DG algebra. In this paper, we show that , for any compact DG -module . This implies that . Especially, we prove the following theorem for DG polynomial algebras (see Theorem 6.2).
** Theorem B. ** Assume that is a cochain DG polynomial algebra such that with . Then and
[TABLE]
2. Notations and some invariants of dg algebras
We assume that the reader is familiar with basic definitions concerning DG homological algebra. If this is not the case, we refer to [FHT, MW1, MW2] for more details on them.
2.1. Notations and terminology
Recall that a cochain DG algebra is a graded -algebra together with a differential of degree such that
[TABLE]
And a cochain DG algebra is called connected if . In the section, will always be a connected cochain DG algebra over a field , if no special assumption is emphasized. We use
[TABLE]
to denote its cohomology graded algebra. Given a cocycle element , we write as the cohomology class in represented by . We denote as the opposite DG algebra of , whose product is defined as for all graded elements and . Right DG modules over can be identified with DG -modules. We write as the maximal DG ideal of . Via the canonical surjection , is both a DG -module and a DG -module. The underlying graded algebra of is written by .
Let be a DG -module. We write as its underlying graded -module, which is obtained by forgetting its differential. For any , the -th suspension of a DG -module is the DG -module defined by . If the corresponding element in is denoted by . The action of on is for all graded elements and . The differential of is defined by for all graded elements . We set .
The category of DG -modules is denoted by whose morphisms are DG morphisms. The homotopy category is the quotient category of , whose objects are the same as those of and whose morphisms are the homotopic equivalence classes of morphisms in . The derived category of DG -modules is denoted by , which is constructed from the category by inverting quasi-isomorphisms ([We],[KM]).
Let be a DG -module. We say is locally finite if each is a finite-dimensional -vector space. If is finite-dimensional as a -vector space, then we say is totally finite (cf.[Car, Definition 12]). One sees that is totally finite if and only if it is both locally finite and homologically bounded. So it makes sense to denote the full subcategory of consisting of totally finite DG -modules by . A DG -module is called compact if the functor preserves all coproducts in . By [MW1, Proposition 3.3], a DG -module is compact if and only if it admits a minimal semi-free resolution with a finite semi-basis. The full subcategory of consisting of compact DG -modules is denoted by .
2.2. DG Krull dimension
In classical ring theory, the Krull dimension of a ring is one of the most important invariants. In [BSW], this invariant was generalized to DG rings. Recall that a DG ideal is called prime if is a graded prime ideal of . Let denote the set of DG prime ideals of . According to [BSW, Definition 2.5], the DG Krull dimension of , denoted , is the supremum of lengths of chains of DG prime ideals of .
2.3. Global dimension
Let be a semi-free DG -module. The DG free class of is defined to be the number
[TABLE]
We denote it as . Let be a non-quasi-trivial DG -module. The cone length of is defined to be the number
[TABLE]
And we define if . These two invariants of DG modules were introduced and studied in [MW3]. One sees that is just the invariant ‘’ defined in [Car, Definition 9] when is a DG polynomial algebra with zero differential. According to [MW3, Definition 5.1], the left global dimension and the right global dimension of are respectively defined by
[TABLE]
Remark 2.1**.**
Note that may be . By the existence of Eilenberg-Moore resolution, we have and hence .
2.4. Ghost dimension
A DG morphism in is called a ghost morphism if . The concept of ‘ghost length’ was first introduced by Christensen in [Chr]. Later, Hovey-Lockridge [HL1] and Kuribayashi [Kur1] applied this invariant to complexes and DG modules, respectively. A DG -module is said to have ghost length , written by , if every composite
[TABLE]
of ghosts is [math] in , and there is a composite of ghosts from that is not [math] in . We set if is zero object in . In [HL1, HL2], Hovey-Lockridge introduced and studied ghost dimension for rings. In a similar way, we define the left ghost dimension of as
[TABLE]
Similarly, we can define the right global dimension of as
[TABLE]
2.5. Rouquier dimension
Let be a subcategory or simply a set of some objects of . We denote by the minimal strictly full subcategory which contains and is closed under taking (possible) direct summands. And we write (resp. ) as the intersection of all strict and full subcategories of that contain and are closed under direct sums (resp. finite direct sums) and all suspensions.
Let and be two strict and full subcategories of . We define as a full subcategory of , whose objects are described as follows: if and only if there is an exact triangle
[TABLE]
where and . For any strict and full subcategory of , one has (see [BBD] or [BVDB, 1.3.10]). Thus, the following notation is unambiguous:
[TABLE]
We refer to the objects of as -fold extensions of objects from . Define . Inductively, we define
[TABLE]
Similarly, we define . We have the associativity of and the formula
[TABLE]
(see [BVDB, Section 2]). Denote For an DG -module , its * -level* is defined to be
[TABLE]
This invariant is originally introduced by Avramov, Buchweitz, Iyengar and Miller in [ABIM]. The -level of counts the number of steps required to build out of via triangles in . It is a very important numerical invariant in the study of the derived category of compact DG -modules. According to the definition in [Rou], we define the Rouquier dimension of as the smallest such that . Actually, we have
[TABLE]
Remark 2.2**.**
By [Kur1, Proposition 7.5], we have for any . Hence
3. some facts on dg polynomial algebras
In this section, we list some useful facts on the structures and classifications of DG polynomial algebras.
Proposition 3.1**.**
[MGYC, Theorem 3.1]** Let be a DG polynomial algebra such that with , for any . Then there exist some such that is defined by
[TABLE]
for any .
It is reasonable to write the DG polynomial algebra in Proposition 3.1 by . The set of DG polynomial algebras in degree one variables is
[TABLE]
When it comes to the isomorphism classes of DG polynomial algebras, we have the following proposition.
Proposition 3.2**.**
[MGYC, Corollary 4.2]** In space , there are only two isomorphism classes and .
In general, the cohomology algebra of a DG algebra contains much important information on its properties (see [AT, DGI]). Proposition 3.2 indicates that the computations of the cohomology algebra of all non-trivial DG polynomial algebras in can be reduced to the calculation of . We have the following proposition (see [MGYC, Proposition 5.1, Theorem 5.2].
Proposition 3.3**.**
The DG polynomial algebra has formal property and its cohomology graded algebra is the polynomial algebra
[TABLE]
By Proposition 3.2 and Proposition 3.3, the following corollary is immediate.
Corollary 3.4**.**
Let be a DG polynomial algebra in . Then is a polynomial algebra and we have
[TABLE]
4. global dimension and dg krull dimension
In this section, we will compute the global dimension, the ghost dimension and the Rouquier dimension of a DG polynomial algebra in .
Lemma 4.1**.**
Let be a connected cochain DG algebra such that is finite. Then . Furthermore, if , then we have
[TABLE]
Proof.
In [MW3, Theorem 4.8], the author and Wu give a detailed proof for the same statement, when is an Adams connected DG algebra. It is easy for one to check that the proof there also carries through for a general connected cochain DG algebra. ∎
Theorem 4.2**.**
Let be a DG polynomial algebra in . Then
[TABLE]
Proof.
By Corollary 3.4, we have
[TABLE]
By Remark 2.1, we have . On the other hand, by Lemma 4.1. Then we get
[TABLE]
Therefore,
[TABLE]
By Remark 2.1 again, we have . This implies that
[TABLE]
∎
Remark 4.3**.**
For the DG polynomial algebra in Theorem 4.2, one can similarly show
[TABLE]
Theorem 4.4**.**
Let be a DG polynomial algebra in . Then , i.e., the DG Krull dimension of is independent of the choice of its differential.
Proof.
If , then any DG prime ideal of is just the graded prime ideal of the graded polynomial algebra . Hence
[TABLE]
in this case. If , then by Proposition 3.2. Therefore,
[TABLE]
and we only need to compute . Visibly, we have
[TABLE]
since for any . On the other hand, we have for any . So each is a DG prime ideal of when . Thus
[TABLE]
is a chain of prime ideals of length and . Hence .
∎
5. some basic lemmas
In this section, we assume that is a connected cochain DG algebra. We list some fundamental lemmas, which will be used in the studies of ghost dimension and Rouquier dimension for DG polynomial algebras.
The following lemma can be proved by the so-called Ghost lemma. One can also see it in [Sch, Lemma 6.7] and [Kur1, Proposition 7.5]. For a detailed proof, we refer the reader to [Mao, Proposition 4.10].
Lemma 5.1**.**
Suppose that is a DG -module. Then if and only if .
Remark 5.2**.**
By Lemma 5.1, one sees that if and only if and , i.e., .
Lemma 5.3**.**
For any DG -module , if and only if is an object in but not in .
Proof.
We only need to show that if and only if . If , then admits a semi-free resolution , which has a strictly increasing semi-free filtration
[TABLE]
of length . This yields a sequence of short exact sequences
[TABLE]
Since and are in , induction shows that is in . Hence is an object in .
Conversely, let be an object in for some . We will prove by induction on . For , the assertion is evident. For , there exists an exact triangle
[TABLE]
where and are in and respectively. Since is in , the DG morphism can be represented by a DG morphism , where is a direct sum of shifted copies of and is a semi-free resolution of with a strictly increasing semi-free filtration
[TABLE]
of length . Clearly, in . We have the following cone exact sequence
[TABLE]
Since is an injective DG morphism,
[TABLE]
is a strictly semi-free filtration of of length . Since in , we have . ∎
Remark 5.4**.**
For any , we have . So Lemma 5.1 and Lemma 5.3 imply . Then
[TABLE]
Lemma 5.5**.**
Let be a connected cochain DG algebra such that is a left Noetherian graded algebra. Then
[TABLE]
Proof.
If , then the inequality holds obviously. We only need to consider the case that . For any , we have . Set . The left graded -module is finitely generated since . Thus admits a finitely generated minimal free resolution
[TABLE]
where each is a finite-dimensional vector space. By [FHT, Proposition 20.11 ], the resolution (1) induces a semi-free resolution , which is called Eilenberg-Moore resolution, of . From the proof of [FHT, Proposition 20.11 ], one sees that
[TABLE]
and admits a semi-free filtration
[TABLE]
where each , . The semi-free filtration above yields a sequence of short exact sequences
[TABLE]
Since each , we prove by induction. Since , we show that and then . Therefore, .
∎
Lemma 5.6**.**
[Rou, Corollary 3.13]** Let be a subcategory or just a set of some objects of . Then .
Lemma 5.7**.**
For any , we have . And hence .
Proof.
We have
[TABLE]
where and are obtained by Remark 5.2 and Lemma 5.6, respectively. Then
[TABLE]
∎
6. ghost dimension and rouquier dimension
In this section, we determine the ghost dimension and the Rouquier dimension of DG polynomial algebras. To achieve this goal, we need the following lemma.
Lemma 6.1**.**
[ABIM, Theorem ]** Let be a DG algebra with zero differential and a DG -module. If the ring is a commutative, Noetherian algebra over a field, then , where is the annihilator of the -module .
Applying Lemma 6.1 and the listed lemmas in the previous section, we can prove the following theorem.
Theorem 6.2**.**
Let be a DG polynomial algebra in . Then and
[TABLE]
Proof.
By Theorem 4.2, we have
[TABLE]
By Remark 5.4 and Lemma 5.7, we have
[TABLE]
If , then and we have
[TABLE]
by Lemma 6.1. Therefore, .
If , then by Proposition 3.2. By Proposition 3.3, has formal property and is the polynomial algebra
[TABLE]
Therefore, is also formal and
[TABLE]
By the definition of formality, can be connected with the trivial DG algebra by a zig-zag of quasi-isomorphisms. This implies that
[TABLE]
On the other hand, we have
[TABLE]
by Lemma 6.1. So we also have if .
In both cases, we have
[TABLE]
where is obtained by Lemma 5.7. On the other hand,
[TABLE]
by Lemma 5.5. Then
[TABLE]
∎
Acknowledgments
The first author is supported by NSFC (Grant No. 11871326) and the Innovation Program of Shanghai Municipal Education Commission (Grant No. 12YZ031).
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