Annihilators and decompositions of singularity categories

Given any commutative Noetherian ring $R$ and an element $x$ in $R$, we consider the full subcategory $\C(x)$ of its singularity category consisting of objects for which the morphism that is given by the multiplication by $x$ is zero. Our main observation is that we can establish a relation between $\C(x), \C(y)$ and $\C(xy)$ for any two ring elements $x$ and $y$. Utilizing this observation, we obtain a decomposition of the singularity category and consequently an upper bound on the dimension of the singularity category.


Introduction
Hilbert's syzygy theorem is one of the most beautiful theorems in commutative algebra and algebraic geometry.It is considered to be the introduction of homological methods into this area.In today's language, Hilbert's theorem states that over a polynomial ring with n variables over a field, any finitely generated module is quasi-isomorphic to a perfect complex of length at most n.During the first half of the twentieth century, the goal was to answer this question for other rings: over which rings can we say that any finitely generated module is quasi-isomorphic to a perfect complex?And during the second half of the century, with the language and the rising importance of derived categories, the question became: over which rings can we say that any bounded complex of finitely generated modules is quasi-isomorphic to a perfect complex?This question motivates the definition of the singularity category (or the stabilized derived category) of a Noetherian ring which is defined as the Verdier quotient of the bounded derived category of finitely generated modules by the (full) subcategory of perfect complexes.Note that this quotient vanishes if and only if the answer to the above question is affirmative.The name singularity category suggests that there is a geometric connection.Indeed, due to the work of Auslander, Buchsbaum and Serre, we know that the singularity category D sg (R) of a commutative Noetherian ring R vanishes if and only if R is regular.Thus, it provides a measure of the singularities of the affine space Spec(R).
The goal of this paper is to further our understanding of the structure of the singularity category of a commutative Noetherian ring R. Our approach uses the annihilators of the singularity category and our main observation is the following: for any x, y ∈ R, we have C(xy) = smd(C(x) * C(y)).
(1.0.1)Let us explain the notation.For each r ∈ R we denote by C(r) the subcategory of D sg (R) consisting of objects X such that the morphism X r − → X in D sg (R) given by the multiplication by r is zero.The subcategory C(x) * C(y) consists of all objects E that fits into an exact triangle X → E → Y → X [1] in D sg (R) with X ∈ C(x) and Y ∈ C(y).Finally, by smd(C(x) * C(y)), we denote the subcategory consisting of all direct summands of objects belonging to C(x) * C(y).This observation is motivated by a result of Dugas and Leuschke [7] who proved the special case C(z m+n ) = smd(C(z m ) * C(z n )) where R = S/(f + z k ) is the k-fold branched cover of a hypersurface singularity S/(f ).We note that if C(xy) = D sg (R), then (1.0.1) gives a decomposition of the singularity category.This is the first part of our main theorem.In fact, using the associativity of the * operator, we have the following.
Theorem A. Let x 1 , . . ., x n be elements of a commutative Noetherian ring R such that the product The second part of our main theorem concerns the dimension of the singularity category of a commutative Noetherian ring.Recall that the (Rouquier) dimension dim T of a triangulated category T is an invariant which measures how much it costs to build it from a single object using the "free" operations of finite direct sums, direct summands and shift and the cone operation which costs "1 unit" each time it is applied.Theorem B. Let x 1 , . . ., x n be elements of a commutative Noetherian ring R such that the product x 1 • • • x n belongs to ann D sg (R).If no x i is a unit or a zerodivisor, then there is an inequality In these theorems, ann D sg (R) is the annihilator of the singularity category which is defined as the ideal of R consisting of elements r such that C(r) = D sg (R).We prove these theorems in Section 2 after providing the necessary background and preliminaries.It is worth comparing them with similar results shown in [9,14].
In Section 3, we consider the case of a sequence elements.After proving a weaker version of Theorem A, we prove the following.
In particular, dim We dedicate Section 4 to examples and applications of these theorems.We compute several examples and discuss the case of isolated singularities: When dim D sg (R) is finite, the vanishing locus of ann D sg (R) is the singular locus of R. In particular, if (R, m) is a local ring with an isolated singularity for which dim D sg (R) < ∞, then ann D sg (R) is m-primary.In this case, for any x ∈ m, we have x n ∈ ann D sg (R) for some positive integer n.If we define α(x) = min{n | x n ∈ ann D sg (R)}, then we have the following corollary.
Corollary D. Let (R, m) be a commutative Noetherian local ring with an isolated singularity.
where x runs over the nonzerodivisors in m.In particular, if dim D sg (R/xR) = 0 for some nonzerodivisor x ∈ m, then dim D sg (R) ≤ α(x) − 1.

Proof of the main theorem
In this section, we introduce and prove preliminaries about the subcategories which appear in our main observation 1.0.1.Let us start with the conventions and the definitions that we will use.We assume that all R-modules are finitely generated and all subcategories are strictly full.We may omit subscripts/superscripts unless there is a danger of confusion.
Definition 2.2.(1) For an element x ∈ R, we denote by C(x) the subcategory of D sg (R) consisting of objects X such that the morphism X x − → X in D sg (R) given by the multiplication by x is zero.Note that C(x) is closed under finite direct sums, direct summands, and shifts.
(2) Let A be an additive category.For a subcategory X of A, we denote by smd A X the subcategory of A consisting of direct summands of objects in X .
(3) Let T be a triangulated category.For subcategories X , Y of T , we denote by X * Y the subcategory of T consisting of objects E ∈ T such that there exists an exact triangle (4) For a sequence x = x 1 , . . ., x n of elements of R, we denote by K(x) the Koszul complex of x over R.
In general, taking derived tensor product T ⊗ L R − does not preserve an isomorphism in D sg (R), but it does if T is isomorphic to a perfect complex.
To be precise, let P be a perfect complex over R. Let (X ) is isomorphic to a perfect complex.One sees that the functor holds for any objects X, Y of D sg (R) and any sequence x = x 1 , . . ., x n in R.
The first part of the following lemma shows us the role that the Koszul complex plays in the rest of the section. Proof.
(1) The chain map K(x) x − → K(x) given by multiplication by x is null-homotopic by [2, Proposition 1.6.5(a)],which shows the inclusion (⊇).To show the inclusion (⊆), pick an object Y ∈ C(x).There is an exact triangle (2) The argument given in [9, Remark 2.6] shows the assertion.
Before proving our main observation, we need one more technical lemma.
Lemma 2.5.Let T be a triangulated category.Let X , Y be subcategories of T .Then there are equalities smd((smd Proof.Since smd X contains X , we have that smd((smd X ) * Y) contains smd(X * Y).To show the opposite inclusion, pick an object A ∈ smd((smd X ) * Y).Then there exists an object B ∈ T such that A ⊕ B ∈ (smd X ) * Y, which gives an exact triangle (2.5.1) in T with C ∈ smd X and Y ∈ Y.There exists an object D ∈ T such that C ⊕ D ∈ X .
Taking the direct sum of (2.5.1) with the exact triangle and it follows that A belongs to smd(X * Y).The first equality of the lemma is now obtained, and the second equality is shown similarly.Now we are ready to prove our main observation.We note once again that this observation works in the generality of all commutative Noetherian rings.Proposition 2.6.Let x 1 , . . ., x n ∈ R. Then there is an equality Proof.In view of Lemma 2.5, it suffices to show that C(xy) = smd(C(x) * C(y)) for x, y ∈ R. Indeed, if we have done it and if C( ), then we will have Let us show the inclusion C(xy) ⊆ smd(C(x) * C(y)).Pick an object A ∈ C(xy).Applying the octahedral axiom, we get a commutative diagram , and so is its direct summand A.
We are interested in the situation where the left hand side of the equality in the preceding proposition is the entire singularity category.That is, we would like to consider the case where for any object in the bounded derived category, multiplication by x 1 • • • x n factors through a perfect complex.Let us make this more precise and introduce notation with the following definition.
Definition 2.7.We introduce two kinds of annihilators as follows.
(1) The annihilator of an object X ∈ D sg (R) is defined by: ann (2) The annihilator ann D sg (R) of the category D sg (R) is defined by: ann Dsg(R) X.
Note that both of the annihilators ann Dsg(R) X and ann D sg (R) are ideals of R.
Now we recall the definition of the dimension of a triangulated category.
Definition 2.8.Let T be a triangulated category.
(1) Let X be a subcategory of T .We denote by X the additive closure in T , that is, the smallest subcategory of T containing X and closed under finite direct sums, direct summands and shifts.We set X T 0 = 0 and define X T n = X T n−1 * X by induction for any n ≥ 1.Note that X 1 = X .When X consists of a single object T , we simply write T n instead of X n .
(2) The (Rouquier) dimension of T is defined by We are now ready to give the main application of our main observation.Theorem 2.9.Let x 1 , . . ., x n be elements of R such that the product If no x i is a unit or a zerodivisor, then there is an inequality . Since x i is a non-zerodivisor, we have (2.9.1) .
As a perfect R/x i R-complex is quasi-isomorphic to a perfect R-complex, the natural surjection R → R/x i R induces an exact functor D sg (R/x i R) → D sg (R).Applying this functor to (2.9.1), we observe that there is an inclusion C(x i ) ⊆ G i Dsg(R) Hence the equality We should compare the above theorem with [9, Theorem 1.1 and Corollary 2.12], which have similar flavors.

The case of a sequence of elements
In the previous section, we considered the full subcategory of objects in the singularity category of a commutative Noetherian ring that are annihilated by a given ring element.
In this section, we are going to further investigate such subcategories by considering their intersections.That is, we will consider the full subcategory of objects that are annihilated by a given sequence of ring elements.
(1) For a sequence x = x 1 , . . ., x n of elements of R, set C(x) = n i=1 C(x i ).Namely, C(x) is the subcategory of D sg (R) consisting of all objects X with (X (2) For an ideal I of R, we set C(I) = a∈I C(a).This is the subcategory of D sg (R) consisting of those objects X which satisfy (X a − → X) = 0 in D sg (R) for all a ∈ I.Note that for any ideals I, J of R, if I ⊆ J, then C(I) ⊇ C(J).
The second assertion of the following proposition is a generalization of Lemma 2.4(1).(2) Let x = x 1 , . . ., x n be a sequence of elements of R. Then every object C ∈ C(x) satisfies To show the opposite inclusion, let X ∈ C(x).Take any a ∈ I. Then a = n i=1 a i x i for some a 1 , . . ., a n ∈ R. Note that x 1 , . . ., x n are in ann Dsg(R) X.Since ann Dsg(R) X is an ideal of R, the element a is also in ann Dsg(R) X. Hence X belongs to C(I).
(2) The last assertion follows from the first assertion and the fact by Lemma 2.4(1) that for each 1 ≤ i ≤ n.To prove the first assertion, we use induction on n.The case n = 0 is obvious.Let n > 0. The induction hypothesis gives rise to an isomorphism K(x 1 , . . ., x n−1 )⊗ , where the fourth isomorphism follows from (2.4.2), and the last isomorphism holds since The following proposition is a weaker version of Proposition 2.6 for this setting.
The following proposition is an immediate consequence of Proposition 3.3.
Lemma 3.4.Let x 1 , . . ., x n ∈ R and assume that m 1 , . . ., m n are nonnegative integers.Then, we have and the assertion obviously holds.Assume that m 1 , . . ., m n are all positive.By induction on m n , it follows from Proposition 3.3 and Lemma 2.5 that we should have Then, the proof is finished by induction on n.
As an application of Lemma 3.4, we get the following.Lemma 3.5.Let x 1 , . . ., x n ∈ R and assume that m 1 , . . ., m n are such that where a i is a nonnegative integer and Proof.Note that for any 1 ≤ j ≤ n, we have C(x i+1 , . . ., x mn n ).Then, the result follows from Lemma 3.4.
Combining our results, we have the following proposition.Proposition 3.6.Let x 1 , . . ., x n ∈ R and assume that a 1 , . . ., a n are nonnegative integers.Let ω 1 , . . ., ω n be as in Lemma 3.5.Then, we have We are now ready to state our main theorem in this section.We should compare it with Theorem 2.9.
Theorem 3.7.Let x = x 1 , . . ., x n a regular sequence on R such that x m 1 1 , . . ., x mn n ∈ ann D sg (R) for some positive integers m 1 , . . ., m n .Assume that a

R). (1)
There is an inequality There is an inequality Let (S, n) be a regular local ring, and let R = S/(f ), where 0 = f ∈ n 2 .We say that R is a simple singularity if there exist only finitely many ideals I of R such that f ∈ I 2 .When R = k[[x 1 , . . ., x n ]]/(f 1 , . . ., f m ) with k a field, we denote by jac R the Jacobian ideal of R, which is defined as the ideal of R generated by the h-minors of the Jacobian matrix ( ∂f i ∂x j ), where h is the height of the ideal (f 1 , . . ., f m ) of the formal power series ring k[[x 1 , . . ., x n ]].Take the parameter ideal Q = (x a−1 ) of R contained in J.We claim that J a = QJ a−1 .In fact, we have J a = (x a−1 , y b−1 ) a = ({(x a−1 ) i (y b−1 ) a−i } a i=1 ) + (y (b−1)a ), and As a ≤ b, we have y b−1 ∈ (y a−1 ).There are equalities Therefore, we get This claim says that the parameter ideal Q is a reduction of J, and we obtain e As a, b are at least 2, both of the integers a + 2b − 5 and (a − 1)(b − 1) are positive.Therefore, the upper bound (4.4.1) for the dimension of the triangulated category MCM(R) produced by Corollary 4.3 is better than the upper bounds produced by [1,6].Furthermore, we should notice that when a = 3 and b ≥ 6, the ring R is not of finite CM-representation type by [15,Chapter 9], so that (4.4.1) and [13, Theorem 1.2] imply dim MCM(R) = 1.This also says that the inequality (4.4.1) is the best possible.
We should also compare Corollaries 4.2, 4. Our results on the dimension of singularity categories require the condition that the product of some nonzerodivisors belong to ann D sg (R).Hence, it is reasonable to ask whether this condition is too strong or when it happens.Now we recall a recent result on the annihilation of singularity categories and formulate our results from this point of view in the case of isolated singularities.Therefore, the annihilator of the singularity is not only a homological invariant, but also plays a role geometrically.For instance, it tells us that when the dimension of the singularity category of a commutative Noetherian ring is finite, then the singular locus is a closed subset.
We start with the finiteness condition on dim D sg (R) in Liu's theorem.To this end, let us consider the following three conditions on a commutative Noetherian local ring (R, m): A: There exists a nonzerodivisor x ∈ m such that dim D sg (R/xR) < ∞ and x i ∈ ann D sg (R) for some i > 0.
C: For any x ∈ m, there is a positive integer ℓ x such that x ℓx ∈ ann D sg (R).
Then, Theorem 3.7 immediately tells us that A implies B. If, moreover, R has only isolated singularities, then the above theorem due to Liu tells us that ann D sg (R) is m-primary, which tells us that B implies C. Note that the ℓ x are bounded above by the Loewy length ℓℓ(R/ ann D sg (R)) of R/ ann D sg (R).
For any nonzerodivisor x ∈ m, let ℓ x be as above and put d x = dim D sg (R/xR).Then, we have the following corollary of Theorem 3.7.

Convention 2 . 1 .
Throughout the rest of this section, let R be a commutative Noetherian ring.Denote by mod R the category of finitely generated R-modules.Denote by D b (R) the bounded derived category of mod R. Let D sg (R) stand for the singularity category of R, which is by definition the Verdier quotient of D b (R) by the perfect complexes.

Proposition 3 . 2 . ( 1 )
Let I be an ideal of R. Let x = x 1 , . . ., x n be a system of generators of I. Then C(I) = C(x).In particular, the implication (a) = (b) =⇒ C(a) = C(b) holds for sequences of elements a = a 1 , . . ., a r and b = b 1 , . . ., b s in R.