Combinatorial study of stable categories of graded Cohen--Macaulay modules over skew quadric hypersurfaces

Let $S$ be a graded ($\pm 1$)-skew polynomial algebra in $n$ variables of degree $1$ and $f =x_1^2 + \cdots +x_n^2 \in S$. In this paper, we prove that the stable category $\mathsf{\underline{CM}}^{\mathbb Z}(S/(f))$ of graded maximal Cohen--Macaulay module over $S/(f)$ can be completely computed using the four graphical operations. As a consequence, $\mathsf{\underline{CM}}^{\mathbb Z}(S/(f))$ is equivalent to the derived category $\mathsf{D^b}(\operatorname{\mathsf{mod}} k^{2^r})$, and this $r$ is obtained as the nullity of a certain matrix over ${\mathbb F}_2$. Using the properties of Stanley--Reisner ideals, we also show that the number of irreducible components of the point scheme of $S$ that are isomorphic to ${\mathbb P}^1$ is less than or equal to $\binom{r+1}{2}$.


Introduction
Triangulated categories play an increasingly important role in many areas of mathematics, including representation theory, (commutative and noncommutative) algebraic geometry, algebraic topology, and mathematical physics. In particular, there are two major classes of triangulated categories, namely, the (bounded) derived categories D b (A) of abelian categories A, and the stable categories C of Frobenius categories C. For example, the derived categories D b (coh X) of coherent sheaves on algebraic varieties X have been studied extensively in algebraic geometry, and the stable categories CM(A) of maximal Cohen-Macaulay modules over (not necessary commutative) Gorenstein algebras A have been studied extensively in representation theory of algebras. In this paper, we compute the stable categories CM Z (A) of graded maximal Cohen-Macaulay modules over certain noncommutative quadric hypersurface rings A (in the sense of Smith and Van den Bergh [6]) using combinatorial methods.
Throughout let k be an algebraically closed field of characteristic not 2. It is well-known that if A is the homogeneous coordinate ring of a smooth quadric hypersurface in P n−1 , then A ∼ = k[x 1 , . . . , x n ]/(x 2 1 + · · · + x 2 n ), so we have if n is odd, CM Z (k[x 1 , x 2 ]/(x 2 1 + x 2 2 )) ∼ = D b (mod k 2 ) if n is even.
(1.1) by Knörrer's periodicity theorem ([4, Theorem 3.1]). The main aim of this paper is to give a skew generalization of this equivalence. More precisely, we consider the following setting.
Notation 1.1. For a symmetric matrix ε := (ε ij ) ∈ M n (k) such that ε ii = 1 and ε ij = ε ji = ±1, we fix the following notations: (1) the standard graded algebra S ε := k x 1 , . . . , x n /(x i x j − ε ij x j x i ), called a (±1)-skew polynomial algebra in n variables, In [8], the second author gave a classification theorem for CM Z (A ε ) with n ≤ 5. After that, in [5], Mori and the second author introduced graphical methods to compute CM Z (A ε ). They presented the four operations, called mutation, relative mutation, Knörrer reduction, and two points reduction for G ε , and showed that CM Z (A ε ) can be completely computed up to n ≤ 6 by using these four graphical operations (see [5,Section 6.4]). We first extend this result to arbitrary n ∈ Z >0 . Theorem 1.2. Let A ε be as in Notation 1.1. By using mutation, relative mutation, Knörrer reduction, and two points reduction, we can completely compute CM Z (A ε ).
Thanks to this theorem, we obtain the following two consequences. Theorem 1.3. Let A ε and G ε be as in Notation 1.1. Then we have . . .
In particular, A ε has 2 r indecomposable non-projective graded maximal Cohen-Macaulay modules up to isomorphisms and degree shifts.
Theorem 1.4. Let A ε be as in Notation 1.1. Then A ε is a noncommutative graded isolated singularity (in the sense of [7]).
It is easy to see that Theorem 1.3 is a generalization of (1.1). Moreover, Theorem 1.4 tells us that A ε is a homogeneous coordinate ring of a noncommutative "smooth" quadric hupersurface (see [6], [5] for details).
Let A be a graded algebra finitely generated in degree 1. A graded A-module M is called point module if M is cyclic and has Hilbert series H M (t) = (1 − t) −1 . If A is commutative, these modules correspond to the closed points of the projective scheme Proj A. In [1], Artin, Tate, and Van den Bergh introduced a scheme E whose closed points parametrize the isomorphism classes of point modules over A; so it is called the point scheme of A. Since then, point schemes are an essential tool to study graded algebras in noncommutative algebraic geometry.
In [8,Conjecture 1.3], it was conjectured that the structure of CM Z (A ε ) is determined by the number of irreducible components of the point scheme E ε of S ε that are isomorphic to P 1 . This is true if n ≤ 6 (see [5,Theorem 6.20]), but unfortunately, it is known to fail for n = 7 (see [5,Remark 6.21]). Using a similar approach to the proof of Theorem 1.2 and the point of view of Stanley-Reisner ideals, we give a combinatorial proof of the following result. Theorem 1.5. Let A ε be as in Notation 1.1 so that there is some r such that CM Z (A ε ) ∼ = D b (mod k 2 r ). Then the number ℓ ε of irreducible components of E ε that are isomorphic to P 1 is less than or equal to r+1 2 . This theorem shows that the upper bound of ℓ ε which appears in [8, Conjecture 1.3] holds true for arbitrary n ∈ Z >0 .

Stable Categories of Graded Maximal
Cohen-Macaulay Modules. Throughout this paper, we continue to use Notation 1.1. It is well-known that S ε = k x 1 , . . . , x n /(x i x j − ε ij x j x i ) is a noetherian Koszul AS-regular algebra. Since f ε = x 2 1 + · · · + x 2 n is a regular central element of S ε of degree 2, it follows that A ε = S ε /(f ε ) is a noetherian Koszul AS-Gorenstein algebra. A finitely generated graded A ε -module M is called maximal Cohen-Macaulay if Ext i Aε (M, A ε ) = 0 for all i = 0. We denote by CM Z (A ε ) the category of (finitely generated) graded maximal Cohen-Macaulay A ε -modules with degree preserving A ε -module homomorphisms.
The stable category of graded maximal Cohen-Macaulay modules, denoted by CM Z (A ε ), has the same objects as CM Z (A ε ) and the morphism space is given by where P (M, N ) consists of degree preserving A ε -module homomorphisms factoring through a graded projective module. Since A ε is AS-Gorenstein, CM Z (A ε ) is a Frobenius category and CM Z (A ε ) is a triangulated category whose translation functor [1] is given by the cosyzygy functor Ω −1 (see [3,Section 4], [6, Theorem 3.1]).

Two Mutations and Two Reductions of Graphs.
A graph G consists of a set of vertices V (G) and a set of edges E(G) between two vertices. In this paper, we assume that there is no loop and multiple edge. An edge between two vertices v, w ∈ V (G) is written by First, we recall the notion of two mutations, which preserve the stable category of graded maximal Cohen-Macaulay modules. (1) Definition 2.4 (Relative Mutation, [5, Definition 6.6]). Let u, v ∈ V (G) be distinct vertices. Then the relative mutation µ v←u (G) of G at v with respect to u is the graph µ v←u (G) where V (µ v←u (G)) = V (G) and (1) Lemma 2.6 (Relative Mutation Lemma, [5,Lemma 6.7]). Suppose that G ε contains an isolated vertex u.
Next, we recall two ways to reduce the number of variables in computing the stable category of graded maximal Cohen-Macaulay modules over A ε .
Lemma 3.1. Let G be a graph and v its vertex. Then there exists a sequence of mutations Proof. We may apply the mutations at all u's in N G (v). See Example 2.2 (2).
Given two nonnegative integers a and b, let G(a, b) denote the graph on the set of vertices G(a, b) consists of a isolated edges and b isolated vertices.
Lemma 3.2. Let G be a graph with n vertices having at least one isolated vertex. Then there exists a sequence of relative mutations µ v 1 ←w 1 , . . . , µ v k ←w k such that where 2α + β = n and β > 0.
Proof. We prove the statement by the induction on n. The statement is trivial in the case n = 1 since G is already equal to G(0, 1).
Suppose that n > 1. Take an isolated vertex v 0 of G. We fix a vertex v in G with v = v 0 and we let G ′ = G\{v}. By the hypothesis of induction, there exists a sequence of relative mutations on G ′ such that G ′ can be transformed into G(α ′ , β ′ ) for some α ′ ∈ Z ≥0 and β ′ ∈ Z >0 . Let G be the graph after applying those relative mutations to G. Then G \ {v} = G(α ′ , β ′ ). Note that v 0 is still an isolated vertex in G. Let . . , α ′ } be the set of its edges. We let the set of edges of G as follows: , G eventually becomes the graph whose edge set is This is nothing but G(α ′ + 1, β ′ − 1) if r ≥ 1 (i.e. β ′ ≥ 2) and G(α ′ , β ′ + 1) if r = 0. See the figure below.
Proof of Theorem 1.2. By Lemma 3.1, we can transform G ε into a graph G ε ′ having at least one isolated vertex by using mutations several times. Moreover, by Lemma 3.2, it follows that G ε ′ can be transformed into G ε ′′ := G(α, β) with α ≥ 0, β > 0 by using relative mutations several times. It is easy to see that G ε ′′ can be reduced to the one-vertex graph by applying Knörrer's reductions α times and two points reductions (β − 1) times. Hence we see that every G ε can be reduced to the one-vertex graph using two mutations and two reductions. In addition, we have For a matrix M with its entry in F 2 , let rank F 2 (M ) (resp. null F 2 (M )) denote the rank (resp. the nullity) of M over F 2 .
For a graph G, let M (G) denote the adjacent matrix of G. We denote by X(G) the adjacent matrix of the graph whose vertex set is Note that X(G) looks like as follows: . . .
In what follows, we will regard each entry of M (G) and X(G) as an element of F 2 .
Lemma 3.3. Work with the same situation and notation as in Lemma 3.2. Then we have rank F 2 (X(G)) = 2α + 2.
Proof. By definition of mutation, we can observe the following: where E i,j is the matrix such that (i, j)-entry is 1 and the other entries are all 0, and E is the identity matrix. (Note that this holds true without assuming that G has an isolated vertex.) Since G has an isolated vertex, we may assume that n-th row (resp. n-th column) of X(G) is ). Then, by definition of relative mutation, we can observe the following: By Lemmas 3.1 and 3.2 together with the above observation, we see that there exists a sequence of regular matrices P 1 , . . . , P N , Q 1 , . . . , Q N such that where A = 0 1 1 0 and O denotes the zero matrix of size n − 2α. We can easily see that rank F 2 (X(G)) = rank F 2 (Q N · · · Q 1 X(G)P 1 · · · P N ) = 2α + 2, as required.
Now we are ready to prove Theorems 1.3 and 1.4.

Proof of Theorem 1.5
This section is devoted to the proof of Theorem 1.5. For a graph G, let Iso(G) be the set of isolated vertices of G and let i(G) = # Iso(G). For two graphs G and G ′ , we write G ∼ G ′ if G and G ′ can be transformed by finitely many times of mutations and relative mutations. By Lemmas 3.1 and 3.2, we know that G ∼ G(α, β) for some α ∈ Z ≥0 and β ∈ Z >0 . Moreover, by Lemma 3.3, we also see that G(α, β) ∼ G(α ′ , β ′ ) if β = β ′ and β, β ′ > 0. (Note that this is not true if β = 0 or β ′ = 0. For example, G(2, 0) ∼ G (1, 2).) Here, we see the following: Proof. If i(G ′ ) = 0, then the result is clear, so assume that i(G ′ ) ≥ 1. Notice that any relative mutation µ v←w keeps the isolated vertices unchanged when v is not an isolated vertex. Thus, by Lemma 3.2, there is a sequence of relative mutations which transforms We recall the following useful lemma.
Let ℓ ε denote the number of irreducible components of E ε that are isomorphic to P 1 . For the investigation of ℓ ε , we will recall some fundamental facts on the Stanley-Reisner ideals of simplicial complexes. Consult, e.g., [2,Section 5].
Let ∆ be a simplicial complex on the vertex set V = {x 1 , . . . , x n }, i.e., ∆ ⊂ 2 V satisfies "F ∈ ∆, F ′ ⊂ F =⇒ F ′ ∈ ∆". A maximal F ∈ ∆ is said to be a facet of ∆. We define the Stanley-Reisner ideal I ∆ ⊂ k[x 1 , . . . , x n ] of ∆ as follows: Clearly, I ∆ is a squarefree monomial ideal. Conversely, every squarefree monomial ideal can be realized as a Stanley-Reisner ideal I ∆ for some ∆.
For a facet F of ∆, let I F be the prime ideal generated by all variables x i with x i ∈ F , i.e., For the purpose of this section, for a graph G on the vertex set V = {x 1 , . . . , x n } with its edge set E, we consider the squarefree monomial ideal I G defined as follows: In fact, we can easily see that V(I Gε ) = E ε . Let ∆ G be the associated simplicial complex, i.e., I ∆ G = I G . We write ℓ(G) for the number of facets of ∆ G with cardinality 2. By the primary decomposition (4.1), we see that so we focus on the calculation of ℓ(G). By definition of the Stanley-Reisner ideal, we have the following: Assume that G contains an isolated vertex x. In this case, x i x ∈ E and x j x ∈ E hold for any On the other hand, the condition Hence, in the case G contains an isolated vertex, we see that Notice that N G (u) = ∅ is equivalent to what u is an isolated vertex. Therefore, in the case G contains an isolated vertex, we conclude that , so it follows that ℓ ε = ℓ(G ε ) = 2 + 2 2 = 3. In fact, we can verify that and so ℓ ε = 3.
Thus i(G ′ ) = i(G)+m−1. Moreover, we also see that J(G ′ ) = J(G)\{{u i , u j } | 1 ≤ i < j ≤ m}. In fact, since only the adjacencies of u i 's and the adjacencies of the vertices in N G (u 1 ) change after applying the above relative mutations, we may observe the vertices of N G (u 1 ), but we can easily see that {w, w ′ } ∈ J(G) if and only if {w, w ′ } ∈ J(G ′ ) for any w, w ′ ∈ N G (u 1 ).
Proof of Theorem 1.5. By [5, Lemma 6.5], mutation does not change the point scheme, so we may assume that G ε is a graph containing an isolated vertex by Lemma 3.1. It follows from Lemma 3.2 that G ε ∼ G(α, β) for some α ∈ Z ≥0 and β ∈ Z >0 . By the proof of Theorem 1.2, we see that r = β − 1. Thus our goal here is to prove that ℓ ε ≤ β 2 . Applying Lemma 4.4 repeatedly, we can obtain a graph G ′ such that G ′ ∼ G ε , j(G ′ ) = 0, and ℓ(G ′ ) ≥ ℓ(G ε ). Since i(G ′ ) ≤ β by Lemma 4.1, we conclude by Lemma 4.4 that as desired.