Moduli of non-commutative polarized schemes

We construct, using geometric invariant theory, a quasi-projective Deligne–Mumford stack of stable graded algebras. We also construct a derived enhancement, which classifies twisted bundles of stable graded A∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A_\infty $$\end{document}-algebras. The tangent complex of the derived scheme is given by graded Hochschild cohomology, which we relate to ordinary Hochschild cohomology. We obtain a version of Hilbert stability for non-commutative projective schemes.


Introduction
All our graded algebras will be unital and associative, with finite dimensional graded pieces.
We study derived moduli of graded algebras. In the first part of this paper, we construct a differential graded stack X , classifying graded algebras of a fixed dimension d = (d 1 , d 2 , . . .). The construction is as a stack quotient of a vector bundle of curved differential graded Lie algebras over a linear space, divided by an algebraic 'gauge' group. The construction is infinite-dimensional, but equal to the projective limit of its finite dimensional truncations.
For a graded algebra A, representing the point P of X , the tangent complex of X at P has (shifted) Hochschild cohomology of A, computed with homogeneous cochains of degree 0, for cohomology groups: H i (T X | P ) = HH i+1 gr (A). In other words, the derived deformation theory of a graded algebra is given by its (shifted) graded Hochschild cohomology.
In the second section, we study graded Hochschild cohomology in some detail, and relate it to more familiar invariants. We do this for algebras A 'coming from geometry', by which we mean that there exists a C-linear Grothendieck category C , an object O of C , and an autoequivalence s of C , satisfying suitable hypotheses, such that A = n≥0 Hom C (O, s n O). In the commutative case, this essentially means that A = n≥0 X, O(n) , for a projective scheme X, O(1) .
Our results can be understood as supporting the idea that (under certain hypotheses), the derived deformation theory of the graded algebra A coincides with the derived deformation theory of the triple (C , O, s).
In the last part of the paper we define a notion of stability for graded algebras and construct a (derived) separated Deligne-Mumford stack X s classifying stable graded algebras of fixed dimension vector. Our notion of stability comes from geometric invariant theory for the finite-dimensional truncations X ≤q of X . We expect that for many interesting dimension vectors the stack X s (or at least an interesting substack) will be of finite type.
In the commutative case, our notion of stability coincides with the classical notion of Hilbert stability. Thus our stack X s extends the classical stack of Hilbert stable projective varieties into the non-commutative world.
We speculate that the stack X s (or a suitable open substack) is a moduli stack of non-commutative projective schemes in the sense of Artin-Zhang [1].

Notation and conventions
We work over a field of characteristic zero, which we shall denote by C. Unless specified otherwise, a graded vector space will refer to a Z-graded vector space. A graded vector space is locally finite, if each graded piece is finite dimensional over C.
All our graded algebras will be unital and associative, locally finite and graded in non-negative degrees. The component in degree zero will be assumed to be equal to the ground field C. Often we will replace such an algebra A by its graded ideal A >0 of elements of positive degree ( A can be recovered from A >0 in a canonical way).
Our algebraic stacks will have affine diagonal, but we do not require the diagonal to be of finite type, in general.
We follow the Bourbaki convention that set inclusion (proper or not) is denoted by '⊂'.

The derived stack of graded algebras 2.1 Gerstenhaber bracket
Let V = n>0 V n be a locally finite positively graded vector space. For p ≥ 0, let be the vector space of ( p + 1)-ary multilinear operations on V , which preserve degree. We have which is a product of finite dimensional vector spaces. Therefore, it is an affine Cscheme. If V is finite dimensional, it is an affine C-scheme of finite type.
On L = p≥0 L p we introduce a (non-associative) product • : L p ⊗ L q → L p+q by the formula We antisymmetrize and obtain the Gerstenhaber bracket The pair (L , [ , ]) is a graded Lie algebra (see [5]). It is finite dimensional, if V is finite dimensional.
Augmentation. Sometimes it will be convenient to augment L by putting a copy of C in degree −1, i.e., setting L −1 = C, and defining the differential L −1 → L 0 to be the map C → Hom gr (V, V ) given by the tautological graded endomorphism γ of V , which is multiplication by the degree, so γ (μ) = deg(μ)μ, for homogeneous elements μ ∈ V . Define the bracket by [L −1 , L] = 0. The fact that the augmented object is a differential graded Lie algebra follows from the fact that the tautological endomorphism γ is central in L. (This kind of construction would not work with the identity in place of γ , as the identity is not central.) We denote by L the graded Lie algebra obtained by dividing L in degree 0 by the ideal Cγ . Note that L is quasiisomorphic to the augmented L.

Maurer-Cartan equation
The Maurer-Cartan equation for L is Thus a Maurer-Cartan element μ is a degree preserving binary operation μ : V ⊗V → V , satisfying the equation Equivalently, i.e., μ is associative. Thus the Maurer-Cartan locus MC(L) of L is the scheme of all associative graded products on V . It is a closed subscheme of the affine scheme L 1 .

Gauge group: moduli stack
The gauge group of L is G = n G L(V n ). It is an affine group scheme over C, and it is algebraic, if V is finite dimensional. Its Lie algebra is L 0 = Hom gr (V, V ). The gauge group acts on L by conjugation, preserving the Gerstenhaber bracket, and hence the Maurer-Cartan locus. The moduli stack of L is the stack quotient It classifies graded associative products on V up to change of basis in V . In other words, X classifies graded associative algebras (without unit), whose underlying graded vector space is isomorphic to V . The stack X is an algebraic stack with affine diagonal, although the diagonal is not of finite type, unless V is finite dimensional. Let The groupoid X (T ), for a scheme T , is the category of bundles of graded algebras of rank d, parametrized by T . Such a bundle of algebras is given by a graded vector bundle V = n>0 V n over T , where rank V n = d n , endowed with O T -bilinear operations V i ⊗ V j → V i+ j , satisfying associativity. (We can always add to such a bundle of graded algebras a copy of O T in degree 0, and make it into a bundle of unital algebras, in a canonical way.) Definition 2.1 For a graded algebra, we call the automorphisms φ λ , for λ ∈ G m , given by φ λ (a) = λ deg a a on homogeneous elements, tautological. The tautological automorphisms define the tautological one-parameter group of automorphisms. This leads to a modified moduli problem:s Denote by the central one-parameter subgroup of G which acts with weight n on V n , for all n. Let G be G/ . The Lie algebra of G is L 0 . The group G acts by conjugation on L, and we call G the gauge group of L. It is an affine group scheme over C. Consider the quotient stack which is again an algebraic stack, the moduli stack of L.
We have a morphism of stacks X → X , which is a G m -gerbe. The moduli problem solved by X is the following: for a scheme T , the groupoid X (T ) is the groupoid of pairs (X, V ), where X is a G m -gerbe over T , and V = n>0 V n is an X-twisted vector bundle on T , where V n is n-twisted, and rank(V n ) = d n , for all n > 0. Moreover, V is endowed with the structure of graded algebra. We call such pairs (X, V ) twisted bundles of graded algebras.
For a review of twisted sheaves, see [9]. Our terminology is as follows: a quasicoherent X-twisted sheaf F on T is a quasi-coherent sheaf on X. Such a sheaf decomposes naturally into a direct sum F = n∈Z F n , where on F n the natural inertia action is equal to the n-th power of the linear action given by the O X -module structure on F . If F = F n , we refer to F as n-twisted. If all F n are vector bundles, we call F a twisted vector bundle.
If the components of the dimension vector d of V are strongly coprime, by which we mean that there exists a k such that (d 1 , 2d 2 , . . . , kd k ) = 1, then the gerbe X → X is trivial. In this case, the universal twisted bundle of algebras can be represented by a bundle of algebras. It can be constructed by twisting the given action of G on V by the character χ : G → G m , defined by χ(g 1 , g 2 , . . .) = det(g 1 ) r 1 det(g 2 ) r 2 . . ., where the r i are such that ir i d i = 1. The point is that this twist does not affect the action on L, but it changes the action on V in such a way that it factors through G.

Derived moduli stack of algebras
One of the simplest kinds of derived moduli stacks is given by a bundle of curved differential graded Lie algebras on a smooth algebraic stack (see [2], for the definitions). In the present case, the construction is as follows.
We start with the affine scheme L 1 , and construct over it a bundle of curved differential graded Lie algebras: the underlying graded vector bundle L = p≥2 L p is the trivial bundle over L 1 , with fibre L p in degree p, for p ≥ 2. The curvature map L 1 → L 2 , given by x → x • x, gives rise to a global section f of L 2 , the curvature of our bundle of curved differential graded Lie algebras. The twisted differential d μ : L i → L i+1 is given by d μ = [μ, · ], in the fibre over μ ∈ L 1 . The Lie bracket on L is constant over L 1 , induced from the Gerstenhaber bracket in each fibre.
Then we notice that the gauge group action on L 1 lifts to an action on all of L , preserving the structure of bundle of curved differential graded Lie algebras. Thus, this structure descends to the quotient stack M = [L 1 /G], giving rise to a bundle of curved differential graded Lie algebras over M. From now on, let us reserve the notation (L , f, d μ , [ , ]) for the descendant bundle on M. (If V is finite dimensional, each L p is a bundle of finite rank.) Our moduli stack X is now realized as the closed substack X ⊂ M, cut out by the vanishing of the curvature f of L .
In [2], it is explained how a bundle of curved differential graded Lie algebras (M, L ) gives rise to a differential graded stack, which we shall denote by (M, R M ). In fact, the curved differential graded Lie algebra structure on L defines a differential graded co-algebra structure on Sym L [1], which dualizes to a differential graded algebra structure on R M = (Sym L [1]) ∨ .
We also get a functor on differential graded schemes: if (T, R T ) is a differential graded scheme, we associate to it the set of pairs (V , μ), where V is a graded vector bundle of dimension d over T , and μ is a global Maurer-Cartan element of the sheaf of differential graded Lie algebras This is the same thing as the structure of a graded We also have a bundle of curved differential graded Lie algebras over M = [L 1 / G], giving rise to a differential graded stack ( M, R M ), whose underlying classical stack is X . This gives rise to the derived stack of twisted bundles of graded A ∞ -algebras.

Hochschild cohomology: deformation theory
Let us consider a point of X , represented by the Maurer-Cartan element μ ∈ L 1 . The derived stack (M, L ) gives rise to a complex of vector bundles on X , the tangent complex, which governs deformations and obstructions of morphisms from square zero extensions of differential graded schemes. (For details, see [2]). At the point μ, this complex is our original graded Lie algebra L, endowed with the twisted differential d μ = [μ, · ]. This differential is the Hochschild differential of the associative algebra (V, μ). It makes (L , d μ , [ , ]) into a differential graded Lie algebra.
Explicitly, for α ∈ L p , α : where we have written μ as concatenation.
The cohomology spaces are the graded Hochschild cohomology spaces of the graded associative algebra (V, μ), computed with Hochschild cochains which are homogeneous of degree zero. Graded deformations/obstructions of the graded algebra (V, μ) are given by The corresponding infinitesimal deformation of (V, μ) is given by V = V ⊕ V with multiplication * , which is determined on V ⊂ V by Associativity of * follows from the cocycle condition (1).

Truncation
Let V → V ≤q be the truncation of V into degrees less than or equal to q, for a positive integer q. We will always consider V ≤q as a quotient of V . Repeating the above constructions with V replaced by V ≤q , we obtain a finite dimensional graded Lie algebra L ≤q = L p ≤q , together with an epimorphism of graded Lie algebras L → L ≤q . Let X ≤q and X ≤q denote the corresponding moduli stacks, which are algebraic stacks of finite type, whose diagonal is affine of finite type. We have Let us write M = [L 1 /G] and M = [L 1 / G], etc. Then we have also τ q : M → M ≤q , and a morphism of bundles of curved differential graded Lie algebras (and a similar fact with tildes), as bundles of curved differential graded Lie algebras.
To state the compatibility with truncations on the level of deformation theory, let A = (V, μ) be an algebra giving rise to a point of X . Then we have as a direct consequence of (3). Thus, we also have Remark 2.2 The projective system L q is a projective system of C-vector spaces, and all transition maps are obviously surjective. The reason to insist that we think of V → V ≤q as a quotient map is only to prove that the Hochschild boundary commutes with the maps of the projective system. A simple argument with lim ← − 1 proves that taking cohomology commutes with the projective limit.

Graded Hochschild cohomology
In this section we will relate graded Hochschild cohomology to more familiar invariants. We will do this for certain graded rings S which 'come from geometry'. By this we mean that S is the homogeneous coordinate ring of a 'sufficiently amply polarized' non-commutative projective scheme (C , A, s) in the sense of [1]. Since our hypotheses are going to diverge slightly from [1], we will call our triples (C , A, s) polarized Grothendieck categories, rather than non-commutative projective schemes.
We will define the concept of reduced Hochschild cohomology for such a triple. We apologize for this abuse of established terminology.

Preliminaries
We summarize a result from [11], which allows us to write down a relatively small complex which computes the Hochschild cohomology of a Grothendieck category.
Let C be a C-linear Grothendieck category, and A : u → C a C-linear functor from a C-linear category u. This situation gives rise to the Yoneda functor C → Mod(u), where Mod(u) is the category of right u-modules, i.e., the category of C-linear functors u op → (C-vector spaces).
We will need to assume that C → Mod(u) is fully faithful and has an exact left adjoint. By the Gabriel-Popescu theorem, for this it suffices that {A(u)} u∈ob u is a generating family for C , and that A : u → C is fully faithful. By Theorem 1.2 of [10], this latter condition can be weakened to There exist (Section 1.10 in [7]) functorial injective resolutions for the objects of u. This means we have a 2-commutative diagram where C • (C ) denotes the differential graded category of complexes in C . For every u ∈ u, the homomorphism of complexes A(u) → E(u) (given by the natural transformation '⇒' in the diagram) is an injective resolution.
Denote by E the u-bimodule defined by the functor E : We shall consider the Hochschild cochain complex C • (u, E), see [11, (2.4)]. It is the product total complex of the double complex whose p-th column is given by The complex C • (u, E) computes the Hochschild cohomology of C as abelian category, and therefore governs the deformation theory of C as abelian category.
We will apply this result in the situation where {A(−n)} n∈N is a family of objects of C , such that for every N , the family {A(−n)} n<N generates C . We let u be the category whose objects are the negative integers, and whose morphisms are given by By construction, u comes with a faithful (although not necessarily full) functor A : u → C , which satisfies Condition 3.1.1, above. The Hochschild complex C • (u, E) is given by

Polarized Grothendieck categories
where s is an auto-equivalence of C , and A is an object of C , such that where we have written s n A = A(n).
In addition, we will make the assumption that Hom It satisfies the additional assumption, if X is connected.
For another example, if (C , A, s) is a finite-dimensional non-commutative projective scheme in the sense of [1], by which we mean that it satisfies the conditions (H1), (H2), (H3), (H4), and (H5) of [1], and has finite cohomological dimension, then (s, A) is a polarization of C , if we replace s by a sufficiently large power.
As explained in [1, Proposition 4.2], we may, and shall, assume that s is an automorphism of C , rather than an autoequivalence.
We choose functorial injective resolutions for the objects A(−n), n ∈ N, and use the complex C • (u, E), defined as above (5), to compute the Hochschild cohomology of C .

Reduced Hochschild cohomology
Let E * be the same u-bimodule as E, except that we set E(−1, −n) equal to zero: By the definition of u, we have that E * is a bi-submodule of E. Let E be the quotient bimodule Again, by the definition of u, we have for all p that the p-th column of We call the cohomology of C • (u, E * ) the reduced Hochschild cohomology of C , with respect to the base object A, notation HH There is a short exact sequence of complexes which gives rise to a long exact sequence in cohomology Remark 3.2 As HH * (C ) governs deformations of the abelian category C , and Ext * C (A, A) governs deformations of the object A within C (see [14]), the sequence (7) suggests that HH * (C , A) governs the deformations of the pair (C , A). This motivates our terminology. We apologize for the somewhat ad-hoc definition, which is motivated by its convenience for what follows.

Graded Hochschild cohomology
Define the unital graded C-algebra Hom C A, A(n) , and the graded differential graded S-bimodule The grading coming from the autoequivalence s will be called the projective grading and will be denoted using lower indices, in contrast to the cohomological grading, which is indicated with superscripts. We have the Hochschild complex of S with values in M • and the subcomplex of projective degree 0 cochains. These are the cochains which preserve the projective degree.

Lemma 3.3 We have a short exact sequence of complexes of C-vector spaces
Proof During this proof we will disregard the vertical degree (the coefficient degree) and consider only the horizontal degree (the Hochschild degree). Thus, C p (u, E) will denote the p-th column (5) is a multilinear map. We associate to χ the p-cochain ψ ∈ C p (u, E) given by the family (ψ n 0 ,...,n p ) n 0 ≥···≥n p ≥1 , where is the multilinear operation given by Sending χ to ψ defines the injective arrow in (8).
The functor s −1 restricts to a fully faithful functor s −1 : u → u, and defines an endomorphism of the diagram (4), and so induces an endomorphism s −1 of C • (u, E). Given a p-cochain ψ ∈ C p (u, E), the p-cochain s −1 ψ ∈ C p (u, E) is given by where, for i = 0, . . . , p, we have used the abbreviation n i = n + j>i j , for an arbitrary n ≥ 1. This proves that (8) is exact in the middle.
Sequences (6) and (8) , giving rise to the commutative diagram of complexes with exact rows and columns Lemma 3.4 Both α and β are quasi-isomorphisms.
Proof In fact, the two claims are equivalent, so let us prove the one for α. Let us start by noting that in C Consider the monomorphism of S-bimodules S >0 → M • ≥0 . By the second condition that we require of polarizations, the quotient of M • ≥0 modulo S >0 is quasi-isomorphic to the bimodule M 0 = Hom • C (E, E), which exists entirely in projective degree 0. It follows that Q is quasi-isomorphic to C • gr (S, M 0 ). But for every p, we have  A, A). The same is true for C • (u, E). We have used the fact that graded Hochschild cohomology of S is invariant under quasi-isomorphisms of the coefficient bimodule. This can be reduced to the case of Hochschild cohomology of the category u via Lemma 3.3, which applies to any ubimodule.

Corollary 3.5 There is a distinguished triangle of complexes of C-vector spaces
and hence a long exact sequence in cohomology Proof This is where we use the connectedness assumption that Hom C (A, A) = C. By this assumption, the normalized graded Hochschild complex of S with values in S >0 is C • (S >0 , S >0 ).
Thus Diagram (9) gives rise to a diagram of long exact sequences in cohomology:

Heuristic remarks
Unfortunately, this result about , is only about the tangent complex of our derived stack as a complex, disregarding the structure of deformation functor, i.e. the L ∞ -structure. We would like to make a few heuristic remarks, which may explain the provenance of Diagram (11).
There is an octahedron of deformation functors Then there is an isomorphism Def(C ) = Def C (s) [1], so we can rewrite this as and as Def C (A) [1] Def C (A) [1] We believe that this latter diagram is, in fact, (11), and this justifies our suspicion that C • gr (S >0 , S >0 )[+1] governs the deformation theory of the triple (C , A, s). From Sect. 2, we know that C • gr (S >0 , S >0 )[+1] governs the deformation theory of (nonunital) graded algebras. This is consistent with the Artin-Zhang philosophy that graded algebras are just triples (C , A, s) in disguise.

Relative Hochschild cohomology (commutative case)
In the commutative case, we can interpret graded Hochschild cohomology of a graded ring as reduced equivariant Hochschild cohomology. We will introduce this concept, and prove results analogous to the non-commutative case.

Relative Hochschild cohomology for schemes
Let X be a separated scheme and X → Y a separated morphism of algebraic stacks. Consider the diagonal morphism which is a closed immersion of schemes. As for any closed immersion of schemes, the derived category object L * * O X splits off H 0 (L * * O X ) = * * O X = O X , and we write (L * * O X ) red for the complement τ <0 (L * * O X ).
For a sheaf of O X -modules F , we define the relative Hochschild cohomology of X over Y with values in F to be We also call the reduced Hochschild cohomology of X over Y with values in F . For F = O X , we use the usual abbreviations We have

Equivariant Hochschild cohomology
If G is a reductive algebraic group, π : P → X is a principal G-bundle, and X → BG the associated classifying morphism, then we write HH * G for HH * BG , and HH * G for HH * BG , and speak of equivariant (reduced) Hochschild cohomology.

Proposition 3.6
For any quasi-coherent sheaf of O X -modules F , There is a natural G-action on HH * (P, π * F ), and we have canonical isomorphisms In particular, Proof Consider the cartesian diagram P π P × P π X X × BG X and write A = π * O P , so that P is the relative spectrum of the O X -algebra A over X . By flat base change, we have and therefore and by adjunction We have a G-action on A , and the invariants are A G = O X . We get an induced action on A ⊗ O X F with invariants F , and an induced action on HH * F ). This proves the claim for usual Hochschild cohomology. The proof goes through also in the reduced case.

Relation to ordinary Hochschild cohomology
We specialize to the case G = G m . Proposition 3.7 Let X be a separated scheme and X → BG m a morphism. Denote the diagonal X → X × X by , and the diagonal X → X × BG m X by . Then in D(O X ) there are distinguished triangles and Proof Let X be a scheme, and a central extension of groupoids over X . The example which will concern us is given by R = X × BG m X , and R = X × X . Denote the identity sections of R and R by and , respectively, and assume that is a closed immersion. Then ι is a closed immersion, as it is a pullback of . Let us denote the identity of G m × X by e, and let t be the standard coordinate on G m .
We have a short exact sequence of sheaves of O-modules on G m × X Applying ι * we get the short exact sequence We have ι * O G m ×X = π * * O X , and ι * e * O X = * O X , by the cartesian diagram So we can rewrite our exact sequence as Now we apply L * to this exact sequence of O-modules on R, to obtain the distinguished triangle (12).

A lemma on Hochschild cohomology of quasi-affine schemes
If X is quasi-affine, say X ⊂ V = Spec A, we can apply the usual tilde construction to the Hochschild complex C • (A) of A. We obtain a complex of quasi-coherent sheaves Removing the degree 0 part from C • (A) gives the reduced Hochschild complex C • (A), and the associated complex of quasi-coherent sheaves C • (A) ∼ | X , which is obtained from C • (A) ∼ | X by removing the component in degree 0.

Lemma 3.9 In the derived category of X , the complex C • (A) ∼ | X represents the object
Proof This follows from [13], where it is proved that on a quasi-projective scheme the complex of non-quasi-coherent sheaves C X • , which sheafifies the Hochschild complex, represents the derived category object L * * O X .
Then we have a canonical quasi-isomorphism because Hochschild homology commutes with localization.

Now suppose F = M| X , for an A-module M.
Lemma 3. 10 We have spectral sequences Proof By the previous lemma, the derived category object R H om(L * * O X , F ) can be represented by the complex C • (A, F ) of sheaves on X , whose degree p component is i.e., an infinite product of copies of F . It follows that Hochschild cohomology of X with values in F is equal to hypercohomology This hypercohomology can be computed using a finite affineČech cover U of X , because an infinite product of quasi-coherent sheaves is acyclic over an affine scheme (even though not quasi-coherent in itself). Thus We now consider the double complex. Computing cohomology in theČech direction gives us C • A, H q (X, F ) , because infinite products are exact in the category of A-modules. Next, computing cohomology in the Hochschild direction gives us HH p A, H q (X, F ) , by definition. Thus the desired spectral sequence is the standard E 2 spectral sequence of our double complex. The proof is the same for the reduced case.

Graded Hochschild cohomology
Let A be a locally finite commutative graded C-algebra, such that (i) A is graded in non-negative degrees: A = A ≥0 , (ii) A is connected: A 0 = C, (iii) A is generated in degree 1.
Let V = Spec A, and Y = V \0, where 0 ∈ V is the vertex defined by the homogeneous maximal ideal A >0 . Moreover, let X = Y/G m = Proj A, and denote the quotient map by π : Y → X . Assume further that (iv) for all n > 0, the homomorphism A n → X, O X (n) is bijective, (v) for all q > 0 and n > 0, we have H q X, O X (n) = 0.

Let us remark that
For example, if X is a connected projective scheme, and O X (1) is a sufficiently ample line bundle, then A = i X, O X (i) satisfies our assumptions. We can then use Lemma 3.10 to determine HH * gr (Y ). In fact, G m acts on the relevant spectral sequence, and we get an induced spectral sequence of invariants To deal with the E 2 -term, notice that, passing to the normalized complex, we have This implies also For q > 0 and p > 0, we have because there are no graded cochains in the relevant degrees (and taking invariants commutes with computing Hochschild cohomology). So the E 2 -term of the spectral sequence (13) is entirely contained in the row q = 0. We deduce that We have and we conclude that HH * gr (Y ) = HH * gr (A >0 , A >0 ).

Remark 3.12
This argument would fail for non-reduced Hochschild cohomology, because the corresponding spectral sequence would also contain the non-vanishing n = 0 column. This is the reason for working with reduced Hochschild cohomology. In fact, for Hochschild cohomology, we have

Corollary 3.13
There is a long exact cohomology sequence This sequence is also the sequence (10), for S = A.

Hochschild-Kostant-Rosenberg
We return to the case of a separated scheme X , with a separated morphism X → Y to an algebraic stack Y , and assume that X → Y is smooth, of relative dimension d. The usual proof of the Hochschild-Kostant-Rosenberg theorem goes through and gives

Corollary 3.14 For relative Hochschild cohomology, we have
In particular, consider the case Y = BG m , and X smooth. The bundles j T X/BG m can be related to the j T X by considering the short exact sequence of vector bundles on X (the Euler sequence), which induces, for every j > 0, another short exact sequence of vector bundles If A is a graded ring as in Theorem 3.11, and X = Proj A is smooth of dimension d, then for q > 0 we have

Further considerations
We consider the case that X, O X (1) is a smooth projective connected scheme of dimension d. The polarization O X (1) gives rise to the morphism X → BG m . Assume 0 The infinitesimal non-commutative polarized deformations of X are given by HH 2 G m (X ). This splits up into two direct summands There is the classical, commutative part H 1 (X, T ). This is an extension of the kernel of H 1 (X, T X ) → H 2 (X, O X ), i.e., the infinitesimal deformations of X lifting to the pair X, O X (1) , by the cokernel of H 0 (X, T X ) → H 1 (X, O X ), i.e., the infinitesimal deformations of O X (1), modulo those that come from infinitesimal automorphisms of X .
Then there is the non-commutative part H 0 (X, 2 T ). This is an extension of the kernel of H 0 (X, 2 T X ) → H 1 (X, T X ) by H 0 (X, T X ). The subspace H 0 (X, T X ) corresponds to non-commutative deformations of the graded sheaf of algebras n O(n) coming from automorphisms of X , via the 'twisted coordinate ring construction'. The quotient space consists of non-commutative deformations of the structure sheaf (given by H 0 (X, 2 T X ), which map to zero in H 1 (X, T X ).
The infinitesimal non-commutative polarized obstructions of X are given by HH 3 G m (X ), and split up into three parts. The classical, commutative part H 2 (X, T ), and two non-classical parts H 1 (X, 2 T ) and H 0 (X, 3 T ). In particular, they contain H 0 (X, 2 T X ) as a subspace.

Remark 3.15
For the obstruction theory to be perfect at X , i.e., for the higher obstructions to vanish, we could require For X a curve this is always true. This leads to the speculation that there may be interesting moduli spaces of non-commutative polarized curves, which admit virtual fundamental classes.
For surfaces, this would give the three conditions

Stability for graded algebras
In this section we study the geometric invariant theory quotient associated to the action of G on L 1 (notation from Sect. 2). Because of (2) we restrict to the case of truncated algebras. Then both L 1 and G are of finite type, and we are in a classical geometric invariant theory context.

The GIT problem
Here we construct quasi-projective moduli schemes of finite graded stable algebras. We start by setting up a Geometric Invariant Theory problem.
Let q be a positive integer, d = (d 1 , . . . , d q ) a vector of positive integers, and We write elements of V as x = (x 1 , . . . , x q ) and elements of G as g = (g 1 , . . . , g q ).
Let R = Hom gr (V ⊗2 , V ), with elements written as μ = (μ i j ) i j , where μ i j : Consider the left action of G on R by conjugation. More precisely, for g ∈ G and μ ∈ R, we have Remark 4.1 This is not a space of quiver representations. So we cannot directly quote results for moduli of quiver representations. Although similar techniques do apply.
There are two canonical one-parameter subgroups of G. The anti-diagonal −1 : G m → G acts by scalar multiplication (i.e., by weight 1) on R, and hence destabilizes every point of R. The other, : G m → G given by (t) = (t, t 2 , . . . , t q ) acts trivially on R, prompting us to pass from G to G = G/ .

Definition 4.2 We call a vector of integers
Any stability parameter defines a character χ θ of G by The second condition on θ says that θ factors through G → G, and the first condition implies that χ, −1 > 0.
We then linearize the action of G on R by taking the trivial line bundle on R, and lifting the action to R × C by the formula g * (μ, t) = (g * μ, χ(g) −1 t). Then the GIT quotient of R by G is are the twisted invariants. Note that the condition χ, −1 > 0 implies that n (R) G χ n is non-negatively graded. The GIT quotient is a projective scheme, because the affine quotient Spec (R) G is reduced to a point.
Let R s ⊂ R ss ⊂ R be the open subsets of stable and semi-stable points in R, respectively. Then [R s / G] is a separated Deligne-Mumford stack with quasi-projective coarse moduli space R s G. Moreover, R ss G = R G is a projective scheme, containing R s G as an open subscheme. If R s = R ss , then [R s / G] is a proper Deligne-Mumford stack with projective coarse moduli space R G.
When we need to specify the stability parameter, we call points of R s (R ss ) θ -(semi)-stable.

Proposition 4.3 [Hilbert-Mumford numerical criterion]
The point μ ∈ R is (semi)stable (with respect to the linearization given by χ ) if and only if for every non-trivial one-parameter subgroup λ of G, such that lim t→0 λ(t) * μ exists in R, we have χ, λ > 0 (≥ 0).  In each equivalence class there is a unique upper grading such that no weights are negative, but there exists a non-zero space V m i with m < i. Let us call such an upper grading standard. Now let μ ∈ R be given. Let λ be a one-parameter subgroup of G, corresponding to the double grading V = ⊕V m i on V . Then lim t→0 λ(t) * μ exists in R, if and only if none of the λ-weights of μ are negative. This is equivalent to μ preserving the descending filtration given by V ≥n = m≥n V m , by which we mean that μ(V ≥m , V ≥n ) ⊂ V ≥m+n . Note that this condition is preserved under equivalence of upper gradings, even though the upper filtration itself changes in the equivalence class. Now suppose that μ ∈ R preserves the filtration V ≥n , given by λ. Then Note that for standard upper gradings, we have V ⊂ V ≥0 , and hence Thus we conclude that μ ∈ R is stable if and only if for every descending filtration V = V (0) ⊃ V (1) ⊃ . . . (compatible with the lower grading), satisfying

Reformulation using test configurations
Suppose now that A is an associative and unital graded algebra, which is locally finite and connected, with A 0 = C.

Definition 4.5 A test configuration for
A is a bundle of graded unital algebras B (as defined in Sect. 2) over the affine line A 1 , together with a G m -action on the bundle B, lifting the natural action on A 1 , such that the restriction of B to G m ⊂ A 1 is G m -equivariantly isomorphic to the constant bundle with fibre A.
Two test configurations for A are equivalent, if one can be obtained from the other by multiplying the G m -action by a suitable power of the tautological action. A test configuration for A is trivial, if it is equivalent to a G m -equivariantly constant test configuration.
The special fibre B| 0 of a test configuration is a graded algebra with the same Hilbert function as A, endowed with a G m -action. The weight of the G m -action on the graded piece of degree i of B| 0 is denoted w i , and the function The test configuration is trivial, if and only if there exists an ∈ Z, such that Let us suppose B is an admissible test configuration written in this way. We have , for all k, > 0, (iv) for every i > 0, there exists an > 0, such that B Definition 4. 8 We call a sequence of two-sided ideals I (k) k>0 in A satisfying these conditions an admissible family of ideals in A.

An admissible family of ideals I (k)
k>0 defines a test configuration by where we set I (k) = A, for all k ≤ 0. The special fibre of this test configuration is

Remark 4.9
If A is finitely generated, then every test configuration for A is admissible.

Standard admissible test configurations
If a test configuration is admissible, there is a unique equivalent test configuration with the properties

Such a test configuration is called standard admissible.
A test configuration is standard admissible if and only if the corresponding admissible family of ideals does not contain the tautological admissible family given by I (k) = A ≥k . Such an admissible family of ideals is called standard admissible.

Stability
Now let us return to the setup of 4.1. Suppose that μ ∈ R = L 1 is a Maurer-Cartan element, so that A = (V, μ) is a graded algebra with A i = 0, for i > q. This proposition motivates the following definition.

Definition 4.11 A finite graded algebra
for every non-trivial test configuration for A, the weights satisfy

Standard stability parameters
We fix a dimension vector (d 1 , . . . , d q ) and a stability parameter θ , as above, and study θ -stability of finite graded algebras A with dim A i = d i .
Let us remark that there is no a priori reason to expect complete moduli of stable algebras: Remark 4. 13 We can eliminate θ 1 from the stability condition. The stability parameter condition becomes and as stability condition we obtain We see that no matter the choice of stability parameter θ , an admissible sequence of ideals with constant Futaki function will always violate stability. The Futaki function being constant means that There is no a priori reason why d 1 should not divide w 1 , and so there is no divisibility condition on the dimension vector (d 1 , d 2 , . . .) which would exclude the possibility of strictly semi-stable objects. Therefore, there is no such condition that would ensure a projective coarse moduli space of stable algebras.
For certain stability parameters, stability implies generated in degree 1: Proposition 4.14 Suppose that θ 1 < 0 and θ i ≥ 0, for all i > 1. Then θ -stable algebras are generated in degree 1. If, in addition, θ i > 0, for all i > 1, then θ -semistable algebras are generated in degree 1.
Proof Write I = A ≥1 , and consider the admissible sequence of ideals of powers of I , given by I (k) = I k , for all k ≥ 1. Assume that not I k = A ≥k , for all k. Then (I k ) is properly contained in the tautological filtration, and hence does not dominate it. Thus This is a contradiction, because w i ≤ id i , for all i. Thus I k = A ≥k , for all k, which implies that A is generated in degree 1.
To prove the additional claim, assume that V is θ -semi-stable. Then we can still conclude that i>1 iθ i d i ≤ i>1 θ i w i . Thus, from w i ≤ id i , and the fact that none of the θ i vanish, we conclude that w i = id i , for all i > 1. Again, we reach a contradiction, proving that A is generated in degree 1.

Remark 4.15
We have, in both cases, proved that any admissible sequence of ideals which is contained in the tautological one, and satisfies I (1) 1 = A 1 , is necessarily the tautological sequence. Proposition 4.16 Suppose that we have θ i ≤ 0, for all i < q. Then every θ -stable algebra has no non-zero ideal I , which vanishes in degree q. If θ i < 0, for all i < q, we can reach the same conclusion for θ -semi-stable algebras.
Proof In fact, if we assume that I (k) is an admissible sequence of ideals which vanishes in degree q, we can conclude that I (k) = 0, for all k ≥ 1, under either of the two assumptions.

Remark 4.17
If θ q = 0, there are no stable algebras.

Remark 4.18
If we want the assumptions of both Propositions 4.14 and 4.16 to hold, we need to have θ 1 < 0, and θ q > 0, as well as θ i = 0, for all 1 < i < q. For the conclusions to hold, we need to assume stability, not just semi-stability.

Definition 4.19
The stability parameter θ is standard, if θ 1 and θ q are the only nonzero θ i .
For a standard stability parameter θ , the stability condition reads This is equivalent to which is independent of the sizes of θ 1 and θ q .
When not specified otherwise, we always work with a standard stability condition, and make the following definition.

Corollary 4.21
Suppose A is stable. Then A is generated in degree 1, and has no non-trivial two-sided ideals which vanish in degree q.

Moduli
Consider the dimension vector d = (d 1 , . . . , d q ), and the associated stack of twisted bundles of graded algebras of dimension d, which we called X ≤q in Sect. 2. Let X s ≤q be the open substack of stable algebras. It is a closed substack of the quotient stack [R s / G], and it is a separated Deligne-Mumford stack with quasi-projective coarse moduli space, which is a closed subscheme of R s G. The C-points of this coarse moduli space correspond in a one-to-one fashion to isomorphism classes of stable algebras of dimension d.

Unbounded algebras
For simplicity, we will only consider stability, not semi-stability. In view of Corollary 4.21, we will only consider algebras generated in degree 1.

Proposition 4.22
Fix an integer q > 1, and let A be a graded algebra, finitely generated in degree 1. Proof The fact that (i) implies (ii), follows because if a test configuration B for A is generated in degrees ≤ q, and is non-trivial, then also its truncation B ≤q is non-trivial.
Obviously, (ii) implies (iii). Next we claim that (iii) implies (iv). Here we will use that A is generated in degree 1. The admissible sequence of ideals generated by the filtration {V (k) } is the smallest admissible sequence of ideals {I (k) }, with I (k) 1 , for all k > 0. The corresponding test configuration B is generated as C[t]-algebra by k≥0 V (k) t −k ⊂ A 1 [t, t −1 ] inside A[t, t −1 ], if we set V (0) = A 1 . It is generated in degree 1. Thus, (iii) implies (iv). Now let us assume that (iv) is satisfied. To prove (v), i.e., that A ≤q is stable, it suffices to check all standard admissible test configurations for A ≤q . Among these, it suffices to check those that are generated in degree 1, because adding generators in higher degree can only increase F(q), without affecting F(1). But non-trivial standard admissible test configurations generated in degree 1 are all generated by a filtration {V (k) } as in (iv).
Finally, the fact that (v) implies (i) is, again, trivial.

Remark 4.23
For a given dimension d 1 of A 1 , in Condition (iv), we can further reduce to considering only flags whose dimensions (dim V (1) , dim V (2) , . . .) come from a finite list of integer sequences, but as we currently have no use for this fact, we will not prove it here.

Definition 4.24
A connected graded algebra, finitely generated in degree 1, is called q-stable, if any of the equivalent conditions in Proposition 4.22 is satisfied. It is called stable, if there exists and N > 0, such that it is q-stable for all q ≥ N .

Commutative case
Suppose that (Y, O Y (1)) is a connected projective C-scheme, such that H i (Y, O(1)) = 0, for all i > 0. Let A be the homogeneous coordinate ring of (Y, O Y (1)). This is the image of Sym (Y, O(1)) → i≥0 (Y, O(n)), and is a connected graded algebra, generated in degree 1.

Proposition 4.25 The polarized scheme (Y, O Y (1)) is Hilbert stable if and only if A is stable in the sense of Definition 4.24.
Proof For the definition of Hilbert stability (more precisely, Hilbert stability with respect to r = 1), see [12]. By definition, the Hilbert stability of (Y, O Y (1)) is tested against all filtrations of A 1 = (Y, O Y (1)), exactly as in Proposition 4.22 (iv). This immediately implies the result.

Moduli
Return to the moduli stack We have now defined open substacks X s ≤q ⊂ X ≤q of stable algebras. We let X s q be the preimage of X s ≤q in X . This is the substack of q-stable algebras. Hence we have in X a family of open substacks X s q , parametrized by q ∈ N. A point in X represents a stable algebra if and only if it is in almost all open substacks X s q ⊂ X . The locus of stable algebras in X is We see no obvious reason why X s should be an open substack of X .

Discussion
We and we find it reasonable, that there should exist dimension vectors d and integers N , for which all arrows in (15) are isomorphisms, so that X s = X s ≤N , and X s is a finite type, separated, (in fact quasi-projective) Deligne-Mumford stack.
In the commutative case (where d is a numerical polynomial), the category of graded algebras generated in degree 1, with fixed Hilbert polynomial, is bounded. This is due to results of Macaulay-Gotzmann-Gasharov (see [4,6]) on persistence and the Macaulay bound. Furthermore, it can be shown that stability is also a bounded condition, namely it can be checked in the N -truncation, for a for sufficiently large N (depending on d).
From this it follows that, in the commutative case, the corresponding claim X s = X s ≤N , holds. This, in particular, implies that the commutative analogue of the stack X s can be realized as a moduli stack of projective polarized schemes (by taking Proj).
Lack of suitable persistence theorems and flattening stratifications currently keep us from generalizing this result to the non-commutative case. But it stands to reason, by analogy with the commutative case, that for certain dimension vectors d, the stack X s , or an open substack, is indeed a moduli stack for stable non-commutative projective schemes, where we propose the following definition for the latter. Further evidence for this expectation is provided by the deformation theory arguments from Sect. 2, which indicate that the derived deformation theory of a noncommutative projective scheme coincides with that of its algebra of homogeneous coordinates.