Introduction to graded geometry

This paper aims at setting out the basics of $\mathbb{Z}$-graded manifolds theory. We introduce $\mathbb{Z}$-graded manifolds from local models and give some of their properties. The requirement to work with a completed graded symmetric algebra to define functions is made clear. Moreover, we define vector fields and exhibit their graded local basis. The paper also reviews some correspondences between differential Z-graded manifolds and algebraic structures.


Introduction
In the 70s, supermanifolds were introduced and studied to provide a geometric background to the developing theory of supersymmetry. In the Berezin-Leites [2] and Kostant [11] approach, they are defined as Z 2 -graded locally ringed spaces. More precisely, this means that a supermanifold is a pair (|N |, O N ) such that |N | is a topological space and O N is a sheaf of Z 2 -graded algebras which satisfies, for all sufficiently small open subsets U , where R m is endowed with its canonical Z 2 -grading. Accordingly, a local coordinate system on a supermanifold splits into n smooth coordinates on U and m elements of a basis on R m . Both types of coordinates are distinguished via the parity function p with values in Z 2 = {0, 1}. Namely, the first n coordinates are said to be even (p = 0), while the others are odd (p = 1). From the late 90s, the introduction of an integer grading was necessary in some topics related to Poisson geometry, Lie algebroids and Courant algebroids. These structures could carry the integer grading only, as it appeared in the work of Kontsevitch [9], Roytenberg [17] and Severa [19], or could be endowed with both Z 2 -and Z-gradings, as it was introduced by Voronov [22]. The first approach, that we follow, led to the definition of a Z-graded manifold given by Mehta [14]. Similarly to a supermanifold, a Z-graded manifold is a graded locally ringed space. Its structure sheaf O N is a sheaf of Z-graded algebras which is given locally by O N (U ) ≃ C ∞ R n (U ) ⊗ SW , where W = ⊕ i W i is a real Z-graded vector space whose component of degree zero satisfies W 0 = {0}. Here, SW denotes the algebra of formal power series of supercommutative products of elements in W. Therefore, the local coordinate system of the Z-graded manifold can be described by n smooth coordinates and a graded basis of W. In particular, it might admit generators of even degree but not of smooth type. This implies that such coordinates do not square to zero and can generate polynomials of arbitrary orders. Their existence requires the introduction of formal power series to obtain the locality of every stalk of the sheaf. This condition, which has not always been accurately considered in past works, also occurs with Z n 2grading for n 2, as discussed in [6].
Complete references on supermanifolds can be found (e.g. [4,7,11,21]), but mathematicians have not yet prepared monographs on graded manifolds. The aim of this paper is to show what would appear at the beginning of such a book and it aspires to be a comprehensive introduction to Z-graded manifolds theory for both mathematicians and theoretical physicists.
The layout of this article is inspired by [4,21] and the reader needs a little knowledge of sheaf theory from § 2.2 onward. We first introduce graded vector spaces, graded rings and graded algebras in § 2. These objects allow us to define graded locally ringed spaces and their morphisms. After that, we introduce graded domains and we show that their stalks are local. In particular, we notice that this property follows from the introduction of formal power series to define sections. The graded domains are the local models of graded manifolds, which are studied in § 3. Specifically, we give the elementary properties of graded manifolds and define their vector fields. In particular, we prove that there exists a local graded basis of the vector fields which is related to the local coordinate system of the graded manifold. Finally, we illustrate in § 4 the theory of Z-graded manifolds with a few theorems stating the correspondence between differential graded manifolds and algebraic structures. These important examples are usually referred to as Q-manifolds. Other applications can be found in [5,16].
Conventions. In this paper, N and Z denote the set of nonnegative integers and the set of integers, respectively. We write N × and Z × when we consider these sets deprived of zero. Notice that some authors use the expression graded manifolds to talk about supermanifolds with an additional Z-grading (see [22]), but we only use this expression in the present paper to refer to Z-graded manifolds.
If a nonzero element v ∈ V belongs to one of the V i , one says that it is homogeneous of degree i. We write | · | for the application which assigns its degree to a homogeneous element. Moreover, we only consider Z-graded vector spaces of finite type, which means that A graded basis of V is a sequence (v α ) α of homogeneous elements of V such that the subsequence of all elements v α of degree i is a basis of the vector Given two Z-graded vector spaces V and W, their direct sum V ⊕ W can be defined with the grading (V ⊕W) i = V i ⊕W i , as well as the tensor product Both constructions are associative. A morphism of Z-graded vector spaces T : V → W is a linear map which preserves the degree : T (V i ) ⊆ W i for all i ∈ Z. We write Hom(V, W) for the set of all morphisms between V and W.
The category of Z-graded vector spaces is a symmetric monoidal category. It is equipped with the nontrivial commutativity isomorphism c V,W : V ⊗ W → W ⊗V which, to any homogeneous elements v ∈ V and w ∈ W, assigns the homogeneous elements Finally, duals of Z-graded vector spaces can be defined. Given a Z-graded vector space V, Hom(V, R) denotes the Z-graded vector space which contains all the linear maps from V to R. Its grading is defined by setting (Hom(V, R)) i to be the set of linear maps f such that It is the dual of V and we write V * =Hom(V, R). One can show that the latter satisfies ( Remark 2.1. From now on, we will refer to Z-graded objects simply as graded objects. However, we will keep the complete writing in the definitions or when it is needed, to keep it clear that the grading is taken over Z. Remark 2.2. The different notions introduced for Z-graded vector spaces extend to Z-graded R-modules over a ring R. In particular, taking R = Z, these constructions apply to any Z-graded abelian group (which is a direct sum of abelian groups) seen as a direct sum of Z-modules.

Graded ring.
A Z-graded ring R is a Z-graded abelian group R = i R i with a morphism R⊗R → R called the multiplication. By definition, it satisfies R i R j ⊆ R i+j .
A graded ring R is unital if it admits an element 1 such that 1 r = r = r 1 for any r ∈ R. In that case, the element 1 satisfies 1 ∈ R 0 . The graded ring R is associative if (ab)c = a(bc) for every a, b, c ∈ R. The graded ring R is supercommutative when ab = (−1) |a||b| ba for any homogeneous elements a, b ∈ R. This means that the multiplication is invariant under the commutativity isomorphism c.
We can introduce the definitions of left ideal, right ideal and two-sided ideal (which is referred to as ideal) of a graded ring in the same manner as in the non-graded case. It is easy to see that, in a supercommutative associative unital graded ring, a left (or right) ideal is an ideal. A homogeneous ideal of R is an ideal I such that I = k I k , for I k = I ∩ R k . Equivalently, a homogeneous ideal I is an ideal generated by a set of homogeneous elements H ⊆ ∪ k R k . In that case, we write I = H when we want to emphasize the generating set of I. An homogeneous ideal I R is said to be maximal if, when I ⊆ J with J another homogeneous ideal, then either J = I or J = R. A local graded ring R is a graded ring which admits a unique maximal homogeneous ideal.
Define J R as the ideal generated by the elements of R with nonzero degree. We can consider the quotient R/J R and the projection map π : R → R/J R . We say that R is a π-local graded ring if it admits a unique maximal homogeneous ideal m such that π(m) is the unique maximal ideal of R/J R . Note that a local graded ring is always π-local, but the converse is not true (see Remark 2.6 below).
Remark 2.3. The different notions introduced for Z-graded rings extend to any object with an underlying Z-graded ring structure.
2.1.3. Graded algebra. A Z-graded algebra is a Z-graded vector space A endowed with a morphism A ⊗ A → A called multiplication. Alternatively, it is a graded ring with a structure of R-module.
Consider a Z-graded vector space W. The free Z-graded associative algebra generated by W is W = k∈N ⊗ k W. The product on W is the concatenation, while its degree is induced by the degree on W. Then, the symmetric Z-graded associative algebra generated by W is given by SW This object gathers the non-graded exterior and symmetric algebras. Indeed, split W as W = W even ⊕ W odd with W even = ⊕ i W 2i and W odd = ⊕ i W 2i+1 . Then we obtain that SW ≃ SW even ⊗ W odd as Z 2 -graded algebras (with the two operations on the right considered on non-graded vector spaces).
Remark 2.4. The graded algebra SW can be seen as the set of polynomials in a graded basis of the graded vector space W. We write SW to indicate its completion by allowing formal power series. Assume that (w α ) α is a graded basis of W. Then, any element s ∈ SW admits a unique decomposition with respect to the (w α ) α : where s 0 and all the s α 1 ...α K are real numbers. We use the multiplicative notation w α 1 . . . w α K to denote w α 1 ⊗ . . . ⊗ w α K modulo the supercommutativity in SW.
Example 2.1. Any locally ringed space is a graded locally ringed space whose sheaf has only elements of degree zero.
where |φ| : |M | → |N | is a morphism of topological spaces and φ * : G → φ * F is a morphism of sheaves of associative unital supercommutative Z-graded rings, which means that it is a collection of A morphism of Z-graded locally ringed spaces φ : (|M |, F) → (|N |, G) is a morphism of Z-graded ringed spaces such that, for all x ∈ |M |, the induced morphism on the stalk φ x : Let (|M |, F) and (|N |, G) be two graded locally ringed spaces. Assume that for all x ∈ |M |, there exists an open set V ⊆ |M | containing x and an open set V ⊆ |N | such that there exists an isomorphism of graded locally ringed spaces φ V : Then, we say that the graded locally ringed space (|M |, F) is locally isomorphic to (|N |, G).
. This can be rephrased as a local isomorphism of graded locally ringed spaces whose sheaves have only elements of degree zero.

Graded domain.
Definition 2.3. Let (p j ) j∈Z be a sequence of non-negative integers and A global coordinate system on U (p j ) is given by a global coordinate system (t 1 , . . . , t p 0 ) on U , and by a graded basis (w α ) α of W. We write such a system on a graded domain as (t i , w α ) i,α . From Equation (2.1), we see that any section f ∈ O U (V ), with V ⊆ U , admits a unique decomposition in the global coordinate system : where f 0 and all the f α 1 ...α K are smooth functions on V . Let J (V ) be the ideal generated by all sections with nonzero degree on the open subset V . The map , then π 1 implies the existence of an isomorphism between C ∞ U (V ) and O U (V )/J (V ) as shown in the following diagram : The map π 1 can be used for evaluating a section at a point x ∈ V . It is the value at x.
The ideal of sections with null value at x ∈ V , written as I x (V ), is given by Then, for all k ∈ N, there exists a polynomial P k,x of degree k in the vari- . . , t p 0 ), one can take its Taylor series of order r ∈ N at x, written as T r x (g; t). It is a polynomial of order r that satisfies Since f 0 ∈ C ∞ (V ) and f α 1 ...α K ∈ C ∞ (V ) for all indices, we can develop them in Taylor series at x in the variables t = (t 1 , . . . , t p 0 ). If, for all k ∈ N, we define P k,x as then P k,x is a polynomial of order k. In particular, we obtain that As w α ∈ I x (V ) for all α and S r x (g; t) ∈ I r+1 x (V ) for any function g, we deduce that f − P k,x ∈ I k+1 x (V ).
Proof. We show that, if f, g ∈ O U (V ) are two sections such that for all k ∈ N, f − g ∈ I k+1 x (V ) for all x ∈ V , then f = g.
Consider a global coordinate system (t i , w α ) i,α on U (p j ) . Set h = f − g and decompose this section as in Equation (2.2) : Since this is true for all x ∈ V , we have that h 0 = 0 and h α 1 ...α K = 0 on V for all indices with K k. As k is arbitrary, we conclude that h = 0 on V .
Consider a section f ∈ I x (V ). Decomposing f as in Equation (2.2), we have From the definition of A, we only need to show that f 0 ∈ A. By assumption, f ∈ I x (V ), so that its value at x is zero. Therefore, considering the Taylor series of f 0 of order 1 gives that At a point x of a graded domain U (p j ) , we write the elements of the stalk, called germs, as [f ] x ∈ O U,x . The evaluation is defined on germs in the same manner as it is done for sections. We denote the ideal generated by all germs with nonzero degree by J x , and the ideal of germs with null value by I x . These two ideals can be obtained by inducing the ideals J (V ) and I x (V ) on the stalk.
namely we do not consider formal power series in the graded coordinates but only polynomials. Then, in general, the ideal I x of germs with null value is not the unique maximal homogeneous ideal of O U,x . For example, the homogeneous ideal 1 − [w] x (where w is a a non-nilpotent element of degree 0 which is a product of local coordinates of nonzero degree) is contained in a maximal homogeneous ideal different from I x , as the germ 1 − [w] x has value 1. Thus, the stalks of this alternative graded domain are not local. Nevertheless, they are π-local: I x is the only maximal homogeneous ideal of O U,x = C ∞ U,x ⊗ SW which projects onto the maximal ideal of C ∞ U,x . Though this definition can be chosen, it limits the study of Z-graded manifolds since a local expression of the form f (t + w) would not always admit a finite power series in w, which is an obstacle to introduce differentiability.

Graded manifold
3.1. Definition. Moreover, let (p j ) j∈Z be a sequence of non-negative integers. We say that M is a Z-graded manifold of dimension (p j ) j∈Z , if there exists a local isomorphism of Z-graded locally ringed spaces φ between M and R (p j ) . We say that an open subset V ⊆ |M | is a trivialising open set if the restriction of φ to V , written φ V , is an isomorphism of graded locally ringed spaces between . If a sequence (t i , w α ) i,α is a global coordinate system on the Z-graded domain R (p j ) , its pullbacks on trivialising open sets form a local coordinate system on M . By abuse of notation, one says that (t i , w α ) i,α is a local coordinate system on M . Morphisms of graded manifolds are defined as morphisms of graded locally ringed spaces.
Remark 3.1. Alternatively, one can define a graded manifold as a graded πlocally ringed space (i.e. stalks are π-local graded rings), locally isomorphic to the alternative graded domains defined in Remark 2.6. Full development of such a definition is left to the interested reader.
Example 3.1. Let M be a smooth manifold of dimension n and write |M | for its underlying topological space. If E → M is a vector bundle of rank m and k is a nonzero integer, we define the sheaf The graded manifold in Example 3.1 is usually denoted by E[k]. We prefer to use a different notation in these notes so that such a graded manifold obtained from a shift is written differently than a lifted graded vector space.
Remark 3.2. One can define N-graded manifolds as Z-graded manifolds whose dimension is indexed by N. Therefore, any homogeneous section has nonnegative degree. This leads to an interesting property: on a N-graded manifold, there does not exist a section of degree zero which can be obtained as a product of sections of nonzero degree. This means that the locality and the π-locality of the stalks are equivalent conditions. Hence, N-graded manifolds do not require the introduction of formal series to be studied, unlike Z-graded manifolds (see Remarks 2.6 and 3.1). Another difference between N-and Z-graded manifolds is that there exists a Batchelor-type theorem on N-graded manifolds [3]. It is not known if an analogous result holds for Z-graded manifolds.  Although it is not used for defining the value on a graded manifold, the ideal generated by all sections with nonzero degree exists. We write J (V ) to denote this ideal on an open subset V of a graded manifold M . If V is a trivialising open set, for any section f ∈ O M (V ) we setf to indicate the element π 1 (f ) ∈ C ∞ (V ) (see § 2.3). Under this notation, the following partition of unity holds true : The proof of this proposition is exactly the same that the one for the partition of unity on a supermanifold [4].
Let R p 0 and we get that (|M |, O M /J ) is a smooth manifold. Thus, one can assume from the beginning that a graded manifold has an underlying structure of smooth manifold. This justifies the frequent definition of graded manifold found in the literature, which considers graded manifolds as smooth manifolds with an additional glued structure. Remark 3.3. In recent developments (see [1,15]) formal polynomial functions on graded manifolds have been considered with respect to a Z-graded vector space that has a non-trivial part of degree zero. This means that, in Definition 3.1, we could choose W such that dim(W 0 ) 1. In that case, the underlying structure of smooth manifold still exists, but one has to obtain it from the morphism π 1 instead of π (see § 2.3) as their images are no longer isomorphic.   for the dual graded basis of (w α ) α in W * . These objects satisfy, for all indices i, j, α, β, Moreover, we can extend them as R-linear derivations on the whole sheaf by Leibniz's rule (3.1). These are graded vector fields and the degree of ∂ ∂t i is 0 while the degree of ∂ ∂wα is −|w α |.
If (t i , w α ) i,α is a global coordinate system on U (p j ) , the vector field X admits the following splitting : where X 0,i = X(t i ) ∈ O U (U ) and X 1,α = X(w α ) ∈ O U (U ) for all i, α.
Proof. Let X ∈ Vec U (p j ) be a vector field. For all open subsets V ⊆ U and every f ∈ O U (V ), f X is a derivation on O U (V ). Therefore Vec U (p j ) is a sheaf of O U -modules. The fact that it is free and Equation (3.4) are consequences of the existence of a graded basis, which is obtained below. Assume that the global coordinates on U (p j ) are (t i , w α ) i,α and let X ∈ Vec U (p j ) be a vector field. We definê where X(z) is the section in O U (U ) given by applying X to the section z.
SinceX is a derivation, D = X −X is also one and it vanishes on the global coordinates (t i , w α ) i,α . By Leibniz's rule (3.1), D is equal to zero on all polynomials generated by the global coordinates. Our aim is to show that D is null on all sections. Take an open subset V ⊆ U and a section f ∈ O U (V ). From Lemma 2.1 we have that, for all n 0, there exists a polynomial P n,x of degree n in the global coordinates such that f − P n,x ∈ I n+1 x (V ). We set h = f − P n,x . Notice that for any vector field Y we have Y (I n x (V )) ⊂ I n−1 x (V ) for n 1. Since D vanishes on every polynomial in (t i ) i and (w α ) α , we have by linearity that D(f ) = D(h) ∈ I n x (V ). Since this is true for all n 1 and any point x ∈ V , Proposition 2.2 gives that D(f ) = 0 on U . As f is any section of O U (V ) and V is any open set, we have proved that D = 0.
It remains to show that the decomposition is unique. Assume thatX is another decomposition of X such thatX = The relations (X −X)(t i ) = 0 and (X −X)(w α ) = 0 implies that f i = X(t i ) and g α = X(w α ).
The following result is a consequence of Definition 3.1 and Lemma 3.2 : If (t i , w α ) i,α is a local coordinate system on M , the vector field X admits the following splitting on a trivialising open set V :  Remark 3.5. It is usually not possible to recover a vector field from the tangent vectors that it defines at all points. Indeed, assume that a graded manifold M admits (t 1 , w 1 ) as local coordinates of respective degrees 0 and 1. Consider the vector field X given locally by X = ∂ ∂t 1 + w 1  A Z-graded manifold M endowed with a homological vector field Q is said to be a differential graded manifold or a Q-manifold. We write it (M, Q) and we refer to it as a dg-manifold.
Example 4.1. Let M be a smooth manifold of dimension n and let Ω be the sheaf of its differential forms. The graded manifold G T M [1] = (|M |, Ω) has dimension (p j ) j∈Z , with p j = n if j = 0, 1 and p j = 0 otherwise. If (x i ) n i=1 are local coordinates on M , then (x i , dx i ) n i=1 is a local coordinate system on G T M [1] whose degrees are respectively 0 and 1. The exterior differential d is a vector field on G T M [1] , since its restriction on any open set V ⊆ |M | is a derivation of Ω(V ). It is given locally by d = i dx i ∂ ∂x i . One can compute that |d| = 1 and [d, d] = 2 d 2 = 0. Therefore, (G T M [1] , d) is a dg-manifold.
Several geometrico-algebraic structures can be encoded in terms of dgmanifolds. The easiest example is certainly the Lie algebra structures on a vector space. ). If V 0 is a Lie algebra, one can show that its Chevalley-Eilenberg differential is a vector field of degree 1 which squares to zero. It endows G V 0 [1] with a homological vector field. Conversely, one can construct a Lie bracket, known as the derived bracket, from the homological vector field. This construction is detailed in [10].
In fact, Theorem 4.1 can be seen as a corollary of two more general correspondences. If we consider an arbitrary Z-graded vector space, we have Our definition of V being a strongly homotopy Lie algebra is that V[−1] admits a sh-Lie structure as stated by Lada and Stasheff in [13]. The proof for Theorem 4.1 strictly follows the correspondence of such structures with degree +1 differentials on the free graded algebra S (ΠV * ). It was proved in [13] that each strongly homotopy Lie algebra implies the existence of a differential, while the converse can be found in [12]. We refer to [18, §6.1] for additional details on this relation. Note that Voronov has provided an analogous definition in Z 2 -settings, which gives an equivalent result for supermanifolds [23,24]. An alternative generalization of Theorem 4.1 is to allow a non-trivial underlying topological space : This theorem is due to Vaȋntrob [20] in the framework of supermanifolds. The proof is exactly the same in the Z-graded case, since there are no local coordinates of even nonzero degree on G E [1] (see e.g. [8]).
To conclude, the interested reader is invited to find in [3] another general equivalence. It links Lie n-algebroids to N-graded manifolds (with a homological vector field and generators of degree at most n), for all n ∈ N × .