On non-commutative formal deformations of coherent sheaves on an algebraic variety

We review the theory of non-commutative deformations of sheaves and describe a versal deformation by using an A-infinity algebra and the change of differentials of an injective resolution. We give some explicit non-trivial examples.


Introduction
We consider non-commutative deformations of sheaves on an algebraic variety in this paper. We consider also multi-pointed deformations, and give some non-trivial examples. The point is that such deformation theory is more natural than the commutative ones as long as we consider infinitesimal deformations.
Let F be a coherent sheaf on an algebraic variety X defined over a field k such that the support of F is proper. We can consider a moduli space M which parametrizes flat deformations of F . The infinitesimal study of M is to investigate the completed local ringÔ M,[F ] at a point corresponding to F . The tangent space of M at [F ] is isomorphic to Ext 1 (F, F ), and the singularitiy at [F ] is described by using the obstruction space Ext 2 (F, F ). Thus we can writeÔ where * denotes the dual vector space, k[[Ext 1 (F, F ) * ]] is the completed symmetric tensor algebra of Ext 1 (F, F ) * and the denominator is a certain ideal determined by Ext 2 (F, F ) * , an ideal generated by power series on a basis of Ext 1 (F, F ) * corresponding to the members of a basis of Ext 2 (F, F ) * .
But it is more natural to consider the completed (non-symmetric) tensor algebra. We obtain the non-commutative (NC) deformation algebra, the parameter algebra of a versal NC deformation where k Ext 1 (F, F ) * is the completed tensor algebrâ and the denominator is a certain two sided ideal determined by Ext 2 (F, F ) * .
The abstract existence of a versal (formal) NC deformation is proved in the same way as in the case of commutative deformations ( [11], [10]).
We can describe a versal deformation, as well as proving its existence, by using A ∞ -algebra formalism. Such a description is apparently well known to experts, e.g., [12] §4. But we use injective resolutions instead of locally free resolutions. This has advantage that our argument works not only for non-smooth non-projective varieties X but also for objects in a klinear abelian category with enough injectives. We also put emphasis on the non-commutativity of the parameter algebras. We treat only formal deformations, but there are results on the convergence (cf. Remarks 7.11 and 7.12).
The abstract description of the versal deformation using an A ∞ -algebra does not necessarily give solutions to explicit deformation problems because it involves injective resolutions etc. So we consider simple but non-trivial examples where the versal deformations are explicitly calculated. We prove that the versal deformation of a structure sheaf of a subvariety is described by a left ideal (Lemma 7.6). We apply this for lines in a projective space and prove that the relation ideal is generated by quadratic NC polynomials. We also calculate the relation NC polynomials for deformations of conics and prove that they have degree 3.
The content of this paper is as follows. In §2, we give a definition of noncommutative deformations of a coherent sheaf, and express NC deformations as a change of differentials in an injective resolution. We describe them by using Maurer-Cartan equation in a differential graded associative algebra.
We review the theory of A ∞ -algebras in §3 in order to use it in later sections. In §4, we describe a versal deformation and the deformation algebra, the parameter algebra of a versal deformation, by using a minimal model A ∞ -algebra of the DG-algebra considered in §2. The advantage of 2 the minimal model A ∞ formulation is that the vector spaces are finite dimensional for fixed degrees, while the DG algebra is infinite dimensional in each degree. In order to achieve this, we need to introduce infinitely many multi-linear maps. We prove the versality of the deformation constructed by using the injective resolution (Theorem 4.6). We extend the whole theory to its refined version of multi-pointed NC deformations (Theorem 5.2) in §5.
We make some remarks on the relationship of NC deformations and iterated self extensions in §6.
We consider some explicit non-trivial examples in §7. In Example 7.8 on lines in a projective space, we prove that the higher multiplications m i for i ≥ 3 vanish, while in Example 7.9 on conics in P 4 , we prove that m 3 = 0 but m i = 0 for i ≥ 4.
The author would like to thank Professors Keiji Oguiso, Spela Spenko and Michel Van den Bergh for the information on the refernces [3] and [13] in Remark 7.10. The author would also like to thank Professors Yukinobu Toda and Zheng Hua for useful discussions (cf. Remarks 7.11 and 7.12).

Non-commutative deformations and DG algebra
We consider 1-pointed non-commutative (NC) deformations of a coherent sheaf in this section. The extension to multi-pointed case is treated in a later section.
Let X be an algebraic variety defined over a field k, and let F be a coherent sheaf on X with proper support.
Let (Art k ) be the category of associative k-algebras R with a maximal two-sided ideal M such that R is a finite dimensional k-module, R/M ∼ = k, and that M n+1 = 0 for some n. It follows that R/M is the only simple R-module and any finitely generated R-module is obtained as a successive extension of R/M . Definition 2.1. Let X, F be as above, R ∈ (Art k ), let F R be a left R⊗ k O Xmodule which is coherent as an O X -module, and let φ : R/M ⊗ R F R ∼ = F be an isomorphism. Then a pair (F R , φ) is said to be a non-commutative deformation of F over R, if F R is flat as a left R-module.
Unless M = 0 and R = k, we can define a two-sided ideal J = M n for the maximal integer n such that M n = 0. Then we have M J = 0. If we put R ′ = R/J, then we have dim k R ′ < dim k R. We use this fact for the purpose of inductive argument on dim k R We will describe NC deformation by using injective resolutions.
Lemma 2.2. Let F be a coherent sheaf on X. Then there is an injective O X -homomorphism i : F → I to an injective O X -module which satisfies the following condition: for any deformations F R of F over any R ∈ (Art k ), Proof. For any point x ∈ X, we define stalks of I by I x = Hom k (O X,x , F x ).
Then I x has an O X,x -module structure given by af given by Since the stalk F R,x has an R ⊗ k O X,x -module structure, the k-module By scalar extension, we obtain an R ⊗ k O X -homomorphism R ⊗ k I → I R , which is bijective due to the flatness of F R over R. Therefore the lemma is proved.
The above lemma is non-trivial in some sense because R ⊗ k I appears in the middle of the flow of arrows in the following diagram: as O X -modules such that, for any deformation which is reduced to the first exact sequence when the functor R/M ⊗ R is applied.
Proof. We apply the lemma to the cokernels.
We describe NC deformations by using differential graded (DG) associative algebras. Let F → I • be an injective resolution as above, and let is the i-th graded piece, and the differential of A is given by , where d I denotes the differential of I.
(2) In this case, H p (R ⊗ k I • , d R,I + y) = 0 for p > 0 and F R : (2) We proceed by induction on dim k R. We take a two-sided ideal J such that M J = 0, and let R ′ = R/J. Then we have an exact sequence of complexes 0 → J ⊗ I • → R ⊗ I • → R ′ ⊗ I • → 0. The associated long exact sequence yields the result.
The existence of a versal deformation, or a hull, for NC deformations is proved in the same way as in the case of commutative deformations ( [11], [10]). One can describe a versal deformation using the formalism of A ∞algebras as explained in subsequent sections.

Review on A ∞ -algebra
We recall the definition of A ∞ -algebras (cf. [9]).
consists of k-linear maps f i : A ⊗i → B of degree 1 − i for i ≥ 1 satisfying the following relations: For example, A DG (differential graded) associative algebra is a special case of an A ∞algebra where m 1 is the differential, m 2 is the associative algebra multiplication, and m i = 0 for i ≥ 3.
Let A be a DG algebra. Then its cohomology group H(A) = i H i (A) is a graded k-module.
Theorem 3.2 (Kadeishvili [8]). Let A be a DG associative algebra. Then there is an A ∞ -algebra structure on the cohomology group H(A) such that m 1 = 0, m 2 is induced from the algebra multiplication m A 2 of A, and that there is a morphism of A ∞ -algebras f : H(A) → A such that f 1 lifts the identity of H(A).
sketch of proof. We define k-linear maps m n : H(A) ⊗n → H(A) of degree 2 − n and f n : H(A) ⊗n → A of degree 1 − n by induction on n ≥ 1, which satisfy the following relations: where m A 1 = d A and m A 2 is the associative multiplication. First we set m 1 = 0. Let us choose f 1 : H(A) → A to be any k-linear map which sends cohomology classes to their representatives. 6 Now assume that m i and f i are already defined for i < n. Let U n : where we need to be careful on the sign changes.
We define m n = [U n ], where [ ] denotes the cohomology class in H(A). (2) is satisfied. Then we can check the relation (1) by a complicated calculation again.
The composition of A ∞ -morphisms f : A → B and g : B → C is defined as follows: The identity morphism f = 1 : A → A is defined by f 1 = 1 and f n = 0 for n ≥ 2. Proof. We will define the g n inductively. The conditions are r,t≥0,r+1+t=n , If the g i for i < n are already determined, then g n is chosen such that it has given values on f 1 (A) ⊗n and the k-subspace V of A ⊗n generated by elements of the form x 1 ⊗ · · · ⊗ x r ⊗ dx r+1 ⊗ x r+2 ⊗ · · · ⊗ x n . Such a g n exists because f 1 (A) ⊗n ∩ V = 0. 7

Description using A ∞ -structure
Let F be a coherent sheaf on an algebraic variety X, and let A = Hom • (I • , I • ) be the DG algebra considered in §2. We know that H p (A) = Ext p (F, F ). The cohomology space H(A) has an A ∞ -structure, and there are A ∞ -morphisms f : H(A) → A and g : A → H(A). We will describe versal NC deformation of F using these A ∞ -algebras and morphisms.
In general, for R ∈ (Art k ), we define m R,n : and so on by the extensions of scalars.
We consider the Maurer-Cartan equation in A ∞ -algebras using the following proposition: We note that the sums are finite because M is nilpotent.
where we dropped the subscripts R for simplicity.
In the above argument, we followed the Koszul rule of the signs: Lemma 4.2. Let A be a DG algebra, and let f : Now we construct a versal deformation over its parameter algebra, called the deformation ring.
be the tautological element corresponding to the identity of H 1 (A). Then we can write , and the product of total degree more than n is set to be zero. There are natural surjective ring homomorphisms Proof. Let {w j } be a basis of H 2 (A), and let {w * i } be the dual basis of We have By Lemma 2.4, we define an NC deformation The following theorem is apparently well-known to experts (cf. [12]): Theorem 4.6. LetF = lim ← − F n be the inverse limit. Then the formal defor-mationF overR is a versal non-commutative deformation of F Proof. We have to prove the following statement: "Let R be a quotient algebra of n i=0 T i H 1 (A) * such that R n is its quotient algebra. Assume that there is an element y ∈ R ⊗ A 1 which satisfies the Maurer-Cartan equation and induces y n on R n ⊗ A 1 . Then R = R n ".
We will derive a contradiction assuming that R = R n . We may assume that the images of R and R n to quotient algebras of n−1 We note that y satisfies the MC equation over R but y R does not, because neither does x over R.
We have y ⊗i = (y R +z) ⊗i = y ⊗i R for i ≥ 2, and we have The parameter algebraR of the versal deformation is called a deformation algebra of F . 10

1-pointed versus r-pointed deformations
Now we consider r-pointed deformations for a positive integer r ≥ 1. If r = 1, then they are NC deformations in the previous sections. It is a refined version in the case where the coherent sheaf F has a direct sum decomposition to r ordered factors F = r i=1 F i . We consider the base ring k r , the product ring of r copies of k, instead of k. F has a structure of a left k r -module, where the orthogonal idempotents Let (Art r k ) be the category of pairs (R, M ) such that R is an associative k ralgebra with an augmentation R → k r and M is a two-sided ideal satisfying the conditions that R is a finite dimensional k-module, R/M ∼ = k r , and that M n+1 = 0 for some n. We have R/M ∼ = r i=1 R/M i for maximal two-sided ideals M i . It follows that the R/M i are the only simple R-modules and any finitely generated R-module is obtained as a successive extension of the R/M i (cf. [6]).
Definition 5.1. Let F = r i=1 F i be a direct sum of coherent sheaves with proper supports on an algebraic variety X and R ∈ (Art r k ). Let F R be a left R ⊗ k O X -module which is coherent as an O X -module. Then a pair (F R , φ) is said to be an r-pointed non-commutative deformation of F over R, if F R is flat as a left R-module and φ : k r ⊗ R F R → F is an isomorphism.
The injective resolution F → I • are k r -equivariant in the sense that The graded ring A = Hom • (I • , I • ) has a structure of k r -bimodules; we have a direct sum decomposition Hom The constructions of the deformation ring and the versal deformation are generalized from the 1-pointed case to the r-pointed case in the following way. The cohomology groups H p (A) = Ext p (F, F ) have also k r -bimodule structures. H(A) = i H i (A) has an A ∞ -structure with a k r -bimodule structure. If n > 0, then there is an injective homomorphism from a direct summand T n k r H 1 (A) → T n k H 1 (A) between tensor products. For example, The A ∞ -multiplications m r n : T n k r H 1 (A) → H 2 (A) for n ≥ 2 are induced from the 1-pointed case m n = m 1 n . We define In order to defineF r , we take the tautological element n of 1-pointed and r-pointed deformations. Their Zariski cotangent spaces are the same H 1 (A) * = (Ext 1 (F, F )) * . The truncated deformation ring R r n of r-pointed deformations is a quotient algebra of the tensor algebra over k r : where the tensor products are taken over the base ring k r . There is a split surjective ring homomorphism k r ×T The degree 0 part of T • k r H 1 (A) * is k r , which is larger than k, but positive degree parts are quotients of the usual tensor products T i k H 1 (A) * . Therefore the r-pointed deformation ringR r is not exactly a quotient of the 1-pointed deformation ringR 1 , but almost is. In particular, r-pointed deformations are derived from a special case of 1-pointed deformations.
We note that the deformation F R r n over R r n is different from the one induced from the deformation F R 1 n by the natural ring homomorphism R 1 n → R r n . For example, F R r n is flat over R r n and k r ⊗ R r n F R r n = F , but k ⊗ R 1 n F R 1 n = F . We have F R r n = (R r n ⊗ R 1 n F R 1 n )/N 12 where N is a submodule consisting of irrelevant factors of F R 1 n that are attached in the extension process; we have to attach F i 's instead of F (cf. Example 7.3).
The following theorem is a consequence of Theorem 4.6: Theorem 5.2. The formal deformationF r of F overR r is a versal rpointed non-commutative deformation of F in the following sense. If (F r R , φ r 0 ) is any r-pointed non-commutative deformation over (R, M ) ∈ (Art r k ) such that φ r 0 : R/M ⊗ k r F R → F is an isomorphism, then there exist an integer n and a k r -algebra homomorphism ψ r : R r n → R such that there is an isomorphism φ r : R ⊗ R r n F r n → F R which induces φ r 0 over R/M .

Remark on universal extensions
We consider iterated self extensions of F = r i=1 F i in this section. NC deformations of F are iterated self extensions of F . Conversely, any iterated self extensions of F are expected to be expressed as NC deformations of F , and the versal deformation is given by a tower of universal extensions. Indeed if F is a simple collection, i.e., if End(F ) ∼ = k r , then it is the case ([6] Theorem 4.8). The point is that the parameter algebra in this case is naturally given as the endomorphism ring of the iterated non-trivial self extensions.
We define inductively a tower of universal extensions by E r 0 = F and (6.1) 0 → Ext 1 (E r n , F ) * ⊗ k r F → E r n+1 → E r n → 0 for n ≥ 0, or equivalently where we note that E r n = r i=1 E r n,i is a left k r -module and Ext 1 (E r n , F ) is a k r -bimodule. The above exact sequence corresponds to a natural morphism On the other hand, in the notation of the previous sections, from an exact sequence 0 → (M r n+1 ) n+1 → R r n+1 → R r n → 0 we obtain an exact sequence We expect that (6.1) and (6.2) are isomorphic as exact sequences of O Xmodules. For example, we have M 1 ∼ = M r 1 ∼ = Ext 1 (F, F ) * , and this is the case for n = 0.
In the case n = 1, from an exact sequence Thus each corresponding terms in (6.1) and (6.2) coincide for n = 1.
We compare universal extensions corresponding to 1-pointed and r-pointed deformations: Lemma 6.1. There are natural split surjective homomorphisms E 1 n → E r n . Proof. For n = 0, we have E 1 0 = E r 0 = F . Assume that there is a split surjective homomorphism E 1 n → E r n . Then there is an induced split surjective homomorphism we have a split surjective homomorphism

Examples
We consider some examples of versal NC deformations in this section. We start with a trivial example: Example 7.1. Let F = O x be the structure sheaf of a point x ∈ X. We claim that the versal deformationF of F is isomorphic to the deformation algebraR, which is commutative and isomorphic to the formal completion of the local ringÔ X,x .
F is a simple collection with r = 1, i.e., a simple sheaf in this case, hence the versal deformation is given by the tower of universal extensions ( [6] Theorem 4.8). Therefore it is sufficient to prove that any NC deformation F R of 14 F over some R ∈ (Art k ) obtained by successive non-trivial extensions is of the form F R ∼ = O X /J for an ideal J such that Supp(F R ) = {x}. We proceed by induction on dim R. Let and dimR 2 = 4. On the other hand, the 1-pointed deformation ring is: y]]/(xy) = k x, y /(xy, yx) and dimR 1 = ∞. The corresponding deformations are as follows. There are non-trivial 1,x (resp. F 2 1,y ) are invertible sheaves on X whose degrees on the irreducible components L x = {y = 0} and L y = {x = 0} of X are (0, 1) (resp. (1, 0)). We haveF On the other hand, we have Example 7.4. Let X = {x 2 + y 2 + y 3 = 0} ⊂ P 2 be a rational curve with one node, and let F be the structure sheaf of the normalization of X.
X has a singularity which is analytically isomorphic to the singularity of the variety considered in the previous example. We have againR = k[[x, y]]/(xy), andF becomes an invertible sheaf on an infinite chain of rational curves.
We have End X (F ) = k but End Dsg (F ) = k[t]/(t 2 + 1) ( [7]). Lemma 7.6. Let X be a proper variety and let Y be a closed subvariety. Proof. Since F is a simple sheaf, a versal NC deformation is obtained by a sequence of universal extensions. We prove that a deformation F R over be a non-trivial extension of NC deformations over an extension 0 of O X -modules. Since the vertical arrows at both ends are surjective, so is the middle vertical arrow. Since is an isomorphism. Using this isomorphism, we define a left R-module structure on O ⊕i X . Then the middle vertical arrow becomes a homomorphism of left R ⊗ O X -modules, and we have We prove that the generating sections of J ⊗ O X (1) extend to generating sections of J R ⊗O X (1) by induction again. From an exact sequence of kernels 1)) is surjective, hence the global sections are liftable. By Nakayama's lemma, they are generating.
The following lemma says that the NC deformations of a Cartier divisor is not interesting: Therefore we consider NC deformations of higher codimensional subvarieties: Example 7.8. Let X = P n be a projective space with homogeneous coordinates [x 1 , . . . , x n+1 ], and let F = O L = k[x 1 , . . . , x n+1 ]/(x 1 , . . . , x n−1 )b e the structure sheaf of a line L, where˜denotes a coherent sheaf on X associated to a graded module.
We claim that the deformation algebra is given bŷ and the versal deformation is given as a quotient by a left ideal: . . , x n−1 + a n−1 x n + b n−1 x n+1 )w here˜denotes a coherentR ⊗ O X -module associated to a graded module. We use Lemma 7.6. The sheaf J ⊗ O X (1) for the ideal sheaf J of L ⊂ X is generated by global sections x 1 , . . . , x n−1 and H 1 (X, J ⊗ O X (1)) = 0. Hence F R should be of the form (R ⊗ O X )/J R for an ideal sheaf J R such that J R ⊗ O X (1) is generated by the following global sections which are linear forms on the x i : x 1 + a 1 x n + b 1 x n+1 , . . . , x n−1 + a n−1 x n + b n−1 x n+1 where we note that elements of the form 1 + r with r ∈ M are invertible, so that the coefficients can be reduced to the above form.
LetR ab be the maximal abelian quotient ofR. Then it is the completed local ring of a Grassmann variety G(2, n + 1) at a point. SinceR andR ab have the same Zariski cotangent spaces, the variables ofR are the a i , b i as in the above expression ofF . SinceR ab is a smooth commutative ring, the relations forR are contained in the commutator ideal of the variables.
In order to determine the quadratic terms in the relations, we calculate explicitly. We have where N L/X is the normal bundle of L in X. Let t 1 , t 2 be the homogeneous coordinates on L, and let ν 1 , . . . , ν n−1 be the normal directions of L. Then . Therefore m 2 is surjective and its kernel has a basis where 1 ≤ j ≤ n − 1 for the first 4 terms, and 1 ≤ j < k ≤ n − 1 for the rest. The dual basis of Im(m * 2 ) = Ker(m 2 ) ⊥ ⊂ (Ext 1 (F, F ) * ) ⊗2 is given by They are the leading terms of the relations forR. Now we prove that there are no higher order terms in the relations, i.e., we prove that there is no higher Massey products. We use the fact that the variables x 1 , . . . , x n+1 inF are commutative. We have inF , inF . If there were higher order terms in the relations ofR on top of the quadratic relations above, then there were more relations of order ≥ 3, a contradiction to the fact that the relations are given by m * Ext 2 (F, F ), and their number is 3(n − 1)(n − 2)/2. Thus the claim is proved.
In particular, if n ≥ 3, then the NC deformations of F are obstructed, because there are non-trivial relations forR, but there are more NC deformations than commutative deformations. 18 For example, if n = 3, then lines on P 3 are parametrized by G(2, 4) under commutative deformations, but the deformation ring for NC deformations isR = k a, b, c, d /(ab − ba, cd − dc, ad − da − bc + cb). We note that this kind of examples are not artificial. For example, if we consider a Calabi-Yau manifold Y such that L ⊂ Y ⊂ P n , then the deformation ring of O L on Y , which is an important invariant of an analytic neighborhood of L in Y , is a quotient ring ofR (cf. [2]). In this sense, it is interesting to calculate versal deformations of rational normal curves of higher degrees. Example 7.9. Let X = P 4 with homogeneous coordinates [x, y, z, w], and let F = O L = k[x, y, z, w, t]/(x, y, zt − w 2 )˜for a conic L in X.
We claim that the deformation ringR of F is given bŷ and the versal deformationF is given bŷ We note that there are order 3 terms in the relations ofR, i.e., m 3 = 0, but m i = 0 for i ≥ 4. In order to prove the claim, we argue similarly to the previous example. We have N L/P 4 ∼ = O(2) 2 ⊕ O(4), and F is written in the above form by Lemma 7.6. We will determine the relations among variables a i , b j , c k inR.
The quadratic terms of the relations are determined by the multiplication We take the dual basis , and a basis , so that the multiplication map satisfies the following: Therefore the image of the map m * 2 : is spanned by the following: These terms give the relations in degree 2.
Remark 7.10. Let C be a smooth rational curve on a Calabi-Yau 3-fold. If C is contractible to a point by a bimeromorphic morphism X →X whose exceptional locus coincides with C, then the NC deformation ring of O C is finite dimensional. It is interesting to know whether the converse is true.
By [1], there is an example where C is not contractible but the abelianization of the deformation ring is finite dimensional. In this example, the 22 normal bundle of C is isomorphic to O(2) ⊕ O(−4) (hence not contractible). The deformation ring is a quotient of a non-commutative formal power series ring with 3 variables by an ideal generated by 3 relations. By [3] and [13], it is known that such a ring is finite dimensional if the 3 relations are generic quadratic forms (this information, opposite to author's naive expectation, was given to the author by Professor Spela Spenko through Professors Michel Van den Bergh and Keiji Oguiso).
Remark 7.11. By [12] Lemma 4.1, the versal formal NC deformation is convergent in the sense that m n < C n and f n < C n for suitable norms and a constant C > 0 which is independent of n.
Remark 7.12. Zheng Hua informed the author that, if the bounded derived category of coherent sheaves D b (coh(X)) has a strong exceptional collection consisting of line bundles, e.g., X ∼ = P n , then the NC deformation algebra of any coherent sheaf F on X is algebraic in the following sense; the A ∞algebra Ext * (F, F ) is quasi-isomorphic to a finite dimensional A ∞ -algebra B such that m B n = 0 for all n ≥ n 0 with a fixed n 0 (cf. [4] Theorem 4.4, [5]). We note that B is not necessarily minimal, i.e., m B 1 may not vanish.