Hochschild cohomology related to graded down-up algebras with weights $(1,n)$
Ayako Itaba, Kenta Ueyama
TL;DR
This paper computes Hochschild cohomology for graded down-up algebras with weights (1,n) for n ≥ 2, revealing how the derived category structure varies with algebra parameters and weights.
Contribution
It extends Hochschild cohomology calculations to the case n ≥ 2 for graded down-up algebras, showing how derived categories differ based on algebraic conditions.
Findings
Hochschild cohomology groups are explicitly calculated for n ≥ 2.
Derived category structures depend on specific algebraic matrix conditions.
Differences in Grothendieck groups are observed between cases n=2 and n≥3.
Abstract
Let be a graded down-up algebra with and , and let be the Beilinson algebra of . If , then a description of the Hochschild cohomology group of is known. In this paper, we calculate the Hochschild cohomology group of for the case . As an application, we see that the structure of the bounded derived category of the noncommutative projective scheme of is different depending on whether is zero or not. Moreover, it turns out that there is a difference between the cases and in the context of Grothendieck groups.
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Hochschild cohomology related to graded down-up algebras with weights
Ayako Itaba
Department of Mathematics, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka, Shinjyuku, Tokyo, 162-8601, Japan
and
Kenta Ueyama
Department of Mathematics, Faculty of Education, Hirosaki University, 1 Bunkyocho, Hirosaki, Aomori, 036-8560, Japan
Abstract.
Let be a graded down-up algebra with and , and let be the Beilinson algebra of . If , then a description of the Hochschild cohomology group of is known. In this paper, we calculate the Hochschild cohomology group of for the case . As an application, we see that the structure of the bounded derived category of the noncommutative projective scheme of is different depending on whether is zero or not. Moreover, it turns out that there is a difference between the cases and in the context of Grothendieck groups.
Key words and phrases:
Hochschild cohomology, down-up algebra, Beilinson algebra, derived equivalence
2020 Mathematics Subject Classification:
16E40, 16S38, 16E05, 18G80
1. Introduction
Throughout let be an algebraically closed field of characteristic [math]. A graded algebra
[TABLE]
with parameters is called a graded down-up algebra. Down-up algebras were originally introduced by Benkart and Roby [4] in the study of the down and up operators on partially ordered sets. Since then, various aspects of these algebras have been investigated; for example, structures [5], [16], [23], representations [9], homological invariants [8], connections with enveloping algebras of Lie algebras [3], [4], invariant theory [14], [15], and so on. In particular, from the viewpoint of noncommutative projective geometry, the following property is of importance.
Theorem 1.1** ([16]).**
Let be a graded down-up algebra. Then is a noetherian AS-regular algebra of dimension if and only if .
A graded down-up algebra has played a key role as a test case for more complicated situations in noncommutative projective geometry.
Let be a graded down-up algebra with , so that is AS-regular. Let , so that is the Gorenstein parameter of . Then the Beilinson algebra of is defined by
[TABLE]
with the multiplication . For example, if , then is isomorphic to the quotient of the path algebra of the quiver
[TABLE]
modulo the ideal generated by the relations
[TABLE]
Let denote the quotient category of finitely generated graded right -modules by the Serre subcategory of finite dimensional modules, and the category of finitely generated right -modules. Note that is considered as the category of coherent sheaves on the noncommutative projective scheme associated to (see [1]). We write and for the bounded derived categories of and , respectively. The following is obtained as a special case of Minamoto-Mori’s theorem [20, Theorem 4.14].
Theorem 1.2**.**
If is a graded down-up algebra with , then is extremely Fano of global dimension , and there exists an equivalence of triangulated categories
[TABLE]
We remark that a Fano algebra was renamed as an -representation infinite algebra in [13] from the viewpoint of higher dimensional Auslander-Reiten theory. By Theorem 1.2, the Beilinson algebras of down-up algebras are important not only in noncommutative projective geometry but also in representation theory of finite dimensional algebras.
Our interest here is to study the Hochschild cohomology of the Beilinson algebra of a down-up algebra . It is known that the Hochschild cohomology of the Beilinson algebra of an AS-regular algebra is closely related to the Hochschild cohomology of and the infinitesimal deformation theory of (see [2], [18], [19]). In [2], Belmans computed the Hochschild cohomology of noncommutative planes and noncommutative quadrics, or in other words, the Hochschild cohomology of the Beilinson algebras of 3-dimensional quadratic or cubic AS-regular -algebras. It should be noted that, to describe the Hochschild cohomology, he used a geometric technique based on the classification of the point schemes of 3-dimensional AS-regular algebras. Since the point schemes of down-up algebras are divided into three cases (, a double curve of bidegree , or two curves of bidegree in general position), [2, Table 2] implies the following result.
Theorem 1.3**.**
Let be a graded down-up algebra with and . Then
- •
**
- •
**
- •
**
- •
* for .*
It is natural to ask what happens when is not generated in degree 1, i.e., how the structure of depends on the grading of . (If is a graded down-up algebra with such that , then the -th Veronese algebra is a graded down-up algebra with , and by [22, Theorem 9.1.8(1)], so it is enough to consider the case that .) In this paper, we devote to compute when is a graded down-up algebra with (so that and is a multiple of ). We will show the following theorem.
Theorem 1.4**.**
Let be a graded down-up algebra with , and . We define
[TABLE]
(e.g. ). Then
- •
**
- •
\operatorname{dim}_{k}\operatorname{HH}^{1}(\nabla A)=\begin{cases}4\quad\text{if}\ n\ \text{is odd and}\ \alpha=0\ (\text{in this case \delta_{n}=0}),\\ 3\quad\text{if}\ n\ \text{is odd},\alpha\neq 0,\text{and}\ \delta_{n}=0,\text{or if}\ n\ \text{is even and}\ \delta_{n}=0,\\ 2\quad\text{if}\ \alpha^{2}+4\beta=0\ (\text{in this case \delta_{n}\neq 0}),\\ 1\quad\text{if}\ \delta_{n}\neq 0\ \text{and}\ \alpha^{2}+4\beta\neq 0;\end{cases}**
- •
\operatorname{dim}_{k}\operatorname{HH}^{2}(\nabla A)=\begin{cases}8\quad\text{if}\ n=2\ \text{and}\ \delta_{2}=0,\\ 7\quad\text{if}\ n=2\ \text{and}\ \alpha^{2}+4\beta=0\ (\text{in this case \delta_{2}\neq 0}),\\ 6\quad\text{if}\ n=2,\delta_{2}\neq 0,\text{and}\ \alpha^{2}+4\beta\neq 0,\\ n+5\quad\text{if}\ n\ \text{is odd and}\ \alpha=0\;(\text{in this case \delta_{n}=0}),\\ n+4\quad\text{if}\ n\ \text{is odd},\alpha\neq 0,\text{and}\ \delta_{n}=0,\text{or if}\ n\geq 4\ \text{is even and}\ \delta_{n}=0,\\ n+3\quad\text{if}\ n\geq 3\ \text{and}\ \alpha^{2}+4\beta=0\ (\text{in this case \delta_{n}\neq 0}),\\ n+2\quad\text{if}\ n\geq 3,\delta_{n}\neq 0,\text{and}\ \alpha^{2}+4\beta\neq 0;\end{cases}**
- •
* for .*
Note that it is crucial for this result that is a multiple of ; see Remark 2.5. Since is not generated in degree 1, the geometric theory of point schemes does not work naively in our case, so our proof of Theorem 1.4 is purely algebraic.
It is known that Hochschild cohomology is invariant under derived equivalence. Using Theorem 1.2, we have the following consequence.
Corollary 1.5**.**
Let and be graded down-up algebras with , where . If
[TABLE]
then .
In the last section, we will apply our results to the study of Grothendieck groups. For a graded down-up algebra with , and , we can observe that if , then behaves a bit like a geometric object (a smooth projective surface), but if , then is not equivalent to the derived category of any smooth projective surface (Proposition 3.2).
2. Proof of Theorem 1.4
In this section, we present the proof of Theorem 1.4. Throughout this section, let be a graded down-up algebra with , and . Then is given as the quotient of the path algebra of the quiver
[TABLE]
modulo the ideal generated by the relations
[TABLE]
Note that the arrows in come from the generators of , respectively, and moreover, the relations come from the defining relations of , respectively.
Let be the enveloping algebra of . Then is a -bimodule, or equivalently, a right -module. For , the -th Hochschild cohomology group of is defined by
[TABLE]
It is known that coincides with the center of . Since is connected and has no oriented cycles, we have
[TABLE]
(by a similar argument as in the proof of [17, Lemma 4.4]).
To compute for , we first construct a minimal projective resolution of as a right -module by using Green-Snashall’s method [10, Section 2]. We define the sets for by
[TABLE]
We set
[TABLE]
where is the vertex at which starts (source) and is the vertex at which ends (target). Then each is a projective right -module.
We define to be the multiplication map, and to be the right -homomorphism determined by
[TABLE]
for all . We also define to be the right -homomorphism determined by
[TABLE]
for all .
Lemma 2.1**.**
With the above definitions, the sequence
[TABLE]
forms a minimal projective resolution of as a right -module.
Proof.
By construction, it follows from [10, Theorem 2.9] that forms part of a minimal projective resolution. Since by Theorem 1.2, we see by [11, Lemma 1.5], so the third term, , is [math] as required. ∎
By applying the functor to (2.2), we have the Hochschild complex
[TABLE]
We next calculate a -basis of . For , we define the right -homomorphism by
[TABLE]
for .
For , we define the right -homomorphisms by
[TABLE]
for .
For and , we define the right -homomorphisms
[TABLE]
by
[TABLE]
for .
When , for , we additionally define the right -homomorphism by
[TABLE]
for .
Lemma 2.2**.**
- (1)
* has a -basis , so .* 2. (2)
* has a -basis , so .* 3. (3)
If , then has a -basis , so . 4. (4)
If , then has a -basis , so .
Proof.
Since as -vector spaces, if , then is given as , and is described by a linear combination of basis elements of , so the result follows. ∎
We then compute a matrix presentation of . For , we define by
[TABLE]
Lemma 2.3**.**
For any , the equality
[TABLE]
holds in .
Proof.
We prove this by induction. The case is clear. Since
[TABLE]
it follows that and , so we get the result. ∎
Let denote the matrix
[TABLE]
and let denote the matrix
[TABLE]
where the blank entries represent zeros.
By Lemma 2.3, , so we see
[TABLE]
Lemma 2.4**.**
(1)* Assume that . Let be the ordered basis for , and let be the ordered basis for . Then the matrix representation of with respect to and is*
[TABLE]
(2)* Assume that . Let be the ordered basis for , and let be the ordered basis*
[TABLE]
for . Then the matrix representation of with respect to and is
[TABLE]
Proof.
We prove only the last two columns of (2.10); the others and (1) are similar. First, for , we have
[TABLE]
Since by Lemma 2.3,
[TABLE]
so we get the th column. Next, for , we have
[TABLE]
It follows from Lemma 2.3 that
[TABLE]
Thus we conclude
[TABLE]
This gives the last column. ∎
Remark 2.5**.**
If and is not a multiple of , then there are no paths of length in the quiver of , except for ’s, so we see that “the part” does not appear in .
Lemma 2.6**.**
Let .
- (1)
If is odd and , then . 2. (2)
If , then .
Proof.
(1) Clearly, . For ,
[TABLE]
by Lemma 2.3, so the result follows.
(2) If , then
[TABLE]
so it follows that
[TABLE]
Therefore, we have
[TABLE]
Since is nonzero, so is and hence so is . ∎
Lemma 2.7**.**
The rank of is if is odd and , and it is otherwise.
Proof.
Since , we have
[TABLE]
by elementary row operations, so we get
[TABLE]
Hence the assertion follows. ∎
Lemma 2.8**.**
Let . If , then . If and , then . If and , then .
Proof.
By elementary row operations, we have
[TABLE]
Since and , the result follows. ∎
We are now ready to prove Theorem 1.4.
Proof of Theorem 1.4.
By (2.1), we have , and so . It follows from Lemma 2.2(1) that . Lemmas 2.2(2) and 2.4 imply
[TABLE]
By Lemmas 2.6, 2.7, and 2.8, we obtain
[TABLE]
Moreover, since
[TABLE]
by Lemmas 2.2(3),(4), it follows that
[TABLE]
Clearly for , so the proof is completed. ∎
3. Discussion on Grothendieck groups
At the end of the paper, we give a discussion of our results in the context of Grothendieck groups based on [7, Section 3] and [3, Section 3.1]. Let be a triangulated category, and let be the Grothendieck group of (see [7, Section 3] for details). If admits a full strong exceptional sequence of length , then is , so . If has the Serre functor in the sense of Bondal and Kapranov [6], then induces an automorphism of .
Theorem 3.1**.**
Let be the bounded derived category of coherent sheaves on a smooth projective variety .
- (1)
([7, Lemma 3.1])* The action of on is unipotent.* 2. (2)
([2, Corollary 25])* If admits a full strong exceptional sequence, then*
[TABLE]
where .
Let be a graded down-up algebra with , and . Then has a full strong exceptional sequence of length by [20, Propositions 4.3, 4.4], so . Moreover has the Serre functor by [21, Appendix A]. Note that .
If , then acts unipotently on (see [2, comments after Remark 26]), and it follows from Theorem 1.3 that
[TABLE]
where , so an analogue of Theorem 3.1 holds. For the case , we have the following.
Proposition 3.2**.**
If , then acts unipotently on and
[TABLE]
If , then does not act unipotently on and
[TABLE]
Proof.
First, (3.1) and (3.2) follow from Theorem 1.4. If , then the Gram matrix of is given by
[TABLE]
so one can verify that
[TABLE]
is unipotent. We now consider the case . Suppose that acts unipotently on . Then is unipotent where is the Gram matrix. Since coincides with the Cartan matrix of , the Coxeter matrix of is obtained as (see [12, Section 1]), so we have
[TABLE]
Unipotency of implies , so it follows that . By Happel’s trace formula [12], we have , but this contradicts (3.2). Hence does not act unipotently. ∎
In respect of Proposition 3.2, when , behaves a bit like a geometric object (a smooth projective surface), but when , is not equivalent to the derived category of any smooth projective surface.
Acknowledgments
The authors are grateful to Teruyuki Yorioka for his support and helpful discussions. They also thank the referee for useful comments in improving the paper. The first author was supported by JSPS Grant-in-Aid for Early-Career Scientists 18K13397. The second author was supported by JSPS Grant-in-Aid for Early-Career Scientists 18K13381.
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