A Comparison of the functors of points of Supermanifolds
aa r X i v : . [ m a t h . R A ] S e p A Comparison of the functorsof points of Supermanifolds
L. Balduzzi ♮ , C. Carmeli ♯ , R. Fioresi ♭♮ Dipartimento di Fisica, Universit`a di Genova, and INFN, sezione di GenovaVia Dodecaneso 33, 16146 Genova, Italy e-mail: [email protected] ♯ DIME, Universit`a di Genova, and INFN, sezione di GenovaVia Cadorna 2, 17100 Savona, Italy e-mail: [email protected] ♭ Dipartimento di Matematica, Universit`a di BolognaPiazza di Porta San Donato 5, 40127 Bologna, Italy e-mail: fi[email protected]
Abstract
We study the functor of points and different local functors of pointsfor smooth and holomorphic supermanifolds, providing characteriza-tion theorems and fully discussing the representability issues. In theend we examine applications to differential calculus including the tran-sitivity theorems.
Contents A -points 12 A -points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.3 Natural transformations between functors of A -points . . . . . 221 The Weil–Berezin functor and the Shvarts embedding 26 A -smooth structure and its consequences . . . . . . . . . . . 274.2 Representability of the Weil–Berezin functor . . . . . . . . . . 354.3 The functors of Λ-points . . . . . . . . . . . . . . . . . . . . . 374.4 Batchelor’s approach . . . . . . . . . . . . . . . . . . . . . . . 38 A -points . . . . . . . . . . . 425.2 Transitivity theorem and applications . . . . . . . . . . . . . . 44 References 49
This paper is devoted to understand the approach to supergeometry via dif-ferent local functors of points for both the differential and the holomorphiccategory.In most of the treatments of supergeometry (see among many others[3, 5, 15, 4, 17, 19, 9, 24]) supermanifolds are understood as classical manifoldswith extra anticommuting coordinates, introduced as odd elements in thestructure sheaf.Only later in [19, 9], the functorial language starts to be used systemati-cally and the functor of points approach becomes a powerfull device allowing,among other things to give a rigorous meaning to otherwise just formal ex-pressions and to recover some geometric intuition otherwise lost. In this ap-proach, a supermanifold M is fully recovered by the knowledge of its functorof points, S M ( S ) := Hom( S, M ), which assigns to each supermanifold S , the set of the S -points of M , M ( S ). This is in essence the content ofYoneda’s Lemma (see, for example, [18]).Whereas in the sheaf theoretic approach the nilpotent coordinates areintroduced by enlarging the sheaf of a classical manifold without modifyingthe underling topological space, another possible approach to supermanifoldstheory consists in introducing new local models for the underlying set itself([2, 20, 7, 23]). In this setting, supermanifolds are obtained by gluing do-mains of the form Λ p × Λ q , where Λ and Λ are the even and odd part of2ome Grassmann algebra Λ. This idea is actually the original physicists ap-proach to supergeometry and only later its mathematical foundations weredeveloped from different perspectives. In particular, in [2] M. Batchelor, fol-lowing B. DeWitt, considers supermanifolds which are locally isomorphic to(Λ L ) p × (Λ L ) q where Λ L is an arbitrary fixed Grassmann algebra with L > q generators; the topology is non Hausdorff and the smooth functions over asupermanifold are defined in an ad hoc way in order to obtain the same classof morphisms as in Kostant’s original approach [15].From another point of view, A. S. Shvarts and A. A. Voronov (see [23, 26])consider simultaneously every finite dimensional Grassmann algebra Λ andlocal models that vary functorially with Λ; the topology is the classical Haus-dorff one and morphisms between superdomains are defined as appropriatenatural transformations between them. Both approaches turn out to beequivalent to Kostant’s original formulation, as it is proved in [2] and [26].Other possible frameworks involving a wider class of supermanifold mor-phisms have been considered in the literature (see, for example, [7] and [20]).In particular in DeWitt’s approach, supermanifolds have a non Hausdorfftopology and are modeled on a Grassmann algebra with countable infinitelymany generators. For a detailed review of some of these approaches we referthe reader to [3, 21].This paper represents the generalization to the holomorphic setting ofthe paper [1]; moreover it contains more observations, examples and proofsof the results, for both the differential and the holomorphic setting, thatoverall make the whole material more accessible. We have also made an ef-fort to compare our treatment with Batchelor’s one and also with Shvartsand Voronov point of view, which is directly inspiring our definition of Weil-Berezin functor of points. As far as we know, there is no organic treatmentof such local functors of points even in the differential setting, though we areaware that it is somehow common knowledge. It is our hope that our workwill fill such a gap, providing in addition a treatment which is able to accomo-date also the category of holomorphic supermanifolds, which is substantiallydifferent from the differential one. In fact, as far as the holomorphic categoryis concerned, we believe that our approach is novel and shows that the localfunctor of points can be employed easily beyond what it was originally de-veloped for. Furthermore we prove a representability theorem which enablesto single out among a certain class of functors, those which are the localfunctor of points of supermanifolds. To our knowledge, this appears for the3rst time in this work. Despite the abstraction, the importance of such a the-orem should not be overlooked. Very often physicists resort to the functor ofpoints to define supergeometric objects, hence the representability issue be-comes essential to show that there is a supergeometric object correspondingto the functor in exam.We have made an effort to make our work self-contained as much as pos-sible, though we are going to rely for the main results of supergeometry, likethe Chart’s theorem or Hadamard’s lemma to the many available referenceson the subject, providing punctual references whenever we need them.Our paper is organized as follows.In section 2 we review some basic definitions of supergeometry like thedefinition of superspace, supermanifold and its associated functor of points asit is discussed in J. Bernstein’s notes by P. Deligne and J. W. Morgan [9]. Webriefly recall the representability problem and we state the representabilitytheorem for supermanifolds.In section 3 we introduce super Weil algebras with their basic properties.The basic observation is that super Weil algebras are, by definition, localalgebras. At a heuristic level it is clear they are well suited to study thelocal properties of a supermanifold, or, in other words, the properties of thestalks at the various points of a supermanifold. Once we define the functor A M A from the category of super Weil algebras to the category of sets,it is only natural to look for an analogue of Yoneda’s lemma, in other wordsfor a result that allows to retrieve the supermanifold from its local functorof points. It turns out (see subsection 3.3) that, as it is stated, this resultis not true; in order for the local functor of points to be able to characterizethe supermanifold, its category of arrival (sets, as we defined it) needs to besuitably specialized by giving to each set M A an extra structure.In section 4, we discuss the modifications we need to introduce in or-der to obtain a bijective correspondence between supermanifold morphismsand natural transformations between the local functors of points. Follow-ing closely what is proved in [23, 26], it turns out that it is necessary toendow the set M A with the structure of an A -smooth manifold (see defini-tion 4.1). We call the functor A → M A , with M A an A -smooth manifold,the Weil–Berezin functor of M . The main result is that, in such a context,the analogue of Yoneda’s lemma holds (see theorem 4.5), and as a conse-quence supermanifolds embed in a full and faithful way into the category of4eil–Berezin functors ( Shvarts embedding ).In analogy with the classical theory, it is only natural to ask underwhich conditions a generic local functor is representable, meaning it is theWeil–Berezin functor of points of a supermanifold (strictly speaking we areabusing the word “representable”). We prove a representability theorem forgeneric functors from the category of super Weil algebras to the category of A -smooth manifolds.We end the section by giving an account of the functor of Λ-points origi-nally described by Shvarts, which is a restriction of the Weil–Berezin functorto Grassmann algebras which form a full subcategory of super Weil algebras.We also describe Batchelor’s approach, providing a comparison between thesetwo approaches and the Weil-Berezin functor discussed previously.In section 5 we examine some aspects of super differential calculus onsupermanifolds in the language of the Weil–Berezin functor. We describethe finite support distributions over the supermanifold M and their relationswith the A -points of M , establishing a connection between our treatmentand Kostant’s seminal approach to supergeometry.We also prove the super version of the Weil transitivity theorem, whichis a key tool for the study of the infinitesimal aspects of supermanifolds, andwe apply it to define the “tangent functor” of A M A . Acknowledgements.
We want to thank prof. G. Cassinelli, prof. A. Cat-taneo, prof. M. Duflo, prof. P. Michor, and prof. V. S. Varadarajan for helpfuldiscussions. We also wish to thank the Referee, whose comments have helpedus to improve our manuscript.
In this section we recall few basic definitions in supergeometry. Our mainreferences are [15, 19, 9, 24].
Let the ground field K be R or C .A super vector space is a Z -graded vector space, i. e. V = V ⊕ V ; theelements in V are called even , those in V odd . An element v = 0 in V ∪ V
5s said homogeneous and p ( v ) denotes its parity: p ( v ) = 0 if v ∈ V , p ( v ) = 1if v ∈ V . K p | q denotes the super vector space K p ⊕ K q .A superalgebra A is an algebra that is also a super vector space, A = A ⊕ A , and such that A i A j ⊆ A i + j (mod 2) . A is an algebra, while A is an A -module. A is said to be commutative if for any two homogeneous elements x and y xy = ( − p ( x ) p ( y ) yx .The category of commutative superalgebras is denoted by SAlg K or simplyby SAlg when no confusion is possible. From now on all superalgebras areassumed to be commutative unless otherwise specified.
Definition 2.1. A superspace S = ( | S | , O S ) is a topological space | S | , en-dowed with a sheaf of superalgebras O S such that the stalk at each point x ∈ | S | , denoted by O S,x , is a local superalgebra.
Definition 2.2. A morphism ϕ : S → T of superspaces is a pair ( | ϕ | , ϕ ∗ ),where | ϕ | : | S | → | T | is a continuous map of topological spaces and ϕ ∗ : O T →| ϕ | ∗ O S , called pullback , is such that ϕ ∗ x ( M | ϕ | ( x ) ) ⊆ M x where M | ϕ | ( x ) and M x denote the maximal ideals in the stalks O T, | ϕ | ( x ) and O S,x respectively.
Example 2.3 (The smooth local model).
The superspace R p | q is thetopological space R p endowed with the following sheaf of superalgebras. Forany open set U ⊆ R p define O R p | q ( U ) := C ∞ R p ( U ) ⊗ Λ R ( ϑ , . . . , ϑ q )where Λ R ( ϑ , . . . , ϑ q ) is the real exterior algebra (or Grassmann algebra )generated by the q variables ϑ , . . . , ϑ q and C ∞ R p denotes the C ∞ sheaf on R p . Example 2.4 (The holomorphic local model).
The superspace C p | q isthe topological space C p endowed with the following sheaf of superalgebras.For any open set U ⊆ C p define O C p | q ( U ) := H C p ( U ) ⊗ Λ C ( ϑ , . . . , ϑ q )where H C p denotes the holomorphic sheaf on C p and Λ C ( ϑ , . . . , ϑ q ) is nowthe complex exterior algebra.By a common abuse of notation, K m | n denotes both the super vectorspace K m ⊕ K n and the superspace defined above.6 efinition 2.5. A smooth (resp. holomorphic) supermanifold of dimension p | q is a superspace M = ( | M | , O M ) which is locally isomorphic to R p | q (resp. C p | q ), i. e. for all x ∈ | M | there exist open sets x ∈ V x ⊆ | M | and U ⊆ R p (resp. C p ) such that: O M | V x ∼ = O R p | q | U (resp. O M | V x ∼ = O C p | q | U ).The sheaf O M is called the structure sheaf of the supermanifold M . A morphism of supermanifolds is simply a morphism of superspaces. SMan K (or simply SMan ) denotes the category of supermanifolds. In particularsupermanifolds of the form ( U, O K p | q | U ) are called superdomains . If | U | ⊆ C n is a domain of holomorphy (see, for example, [12]), then we say that( | U | , O C p | q | U ) is a Stein superdomain .If U is open in | M | , ( U, O M | U ) is also a supermanifold and it is calledthe open supermanifold associated to U . We shall often refer to it just by U ,whenever no confusion is possible. Remark 2.6. [12, Corollary 2.5.6] gives many examples of domains of holo-morphy in C m and hence many examples of Stein superdomains. In partic-ular C m | n and the open supermanifolds associated with polydiscs are Steinsuperdomains. We recall that, given ( z , . . . , z m ) ∈ C m and m positive realnumbers ( r , . . . , r m ) the polydisc with center ( z , . . . , z m ) and polyradius( r , . . . , r m ) is P ( z , . . . , z n ; r , . . . , r n ) := { ( z , . . . , z n ) ∈ C n | | z i − z i | < r i ∀ ≤ i ≤ n } Since the Stein property is stable under biholomorphic morphisms, itis easy to see that every complex supermanifold admits an atlas of Steinsuperdomains.In order to avoid duplications and heavy notations, we will simply refer tosupermanifolds when the distinction between the smooth and the holomor-phic case is immaterial. Moreover if M is a supermanifold, we will denote by O ( M ) the superalgebra O M ( | M | ) of global sections on M .Suppose M is a supermanifold and U is an open subset of | M | . Let J M ( U ) be the ideal of the nilpotent elements of O M ( U ). O M / J M defines asheaf of purely even algebras over | M | locally isomorphic to C ∞ ( R p ) (resp. H ( C p )). Therefore f M := ( | M | , O M / J M ) defines a classical manifold, called7he reduced manifold associated to M . The projection s e s := s + J M ( U ),with s ∈ O M ( U ), is the pullback of the embedding f M → M .In the following we denote by ev x ( s ) := e s ( x ) the evaluation of s at x ∈ U .It is also possible to check that | ϕ | ∗ ( e s ) = ] ϕ ∗ ( s ), so that the morphism | ϕ | is automatically smooth (resp. holomorphic). Moreover since the maximalideal M x in the stalk O M,x is given by the germs of sections whose value at x is zero, we have that the locality condition in the case of supermanifoldmorphisms is automatically satisfied.There are several equivalent ways to assign a morphism between twosupermanifolds. The following result can be found in [19, ch. 4]. Proposition 2.7 (Chart theorem).
Let U and V two smooth or holo-morphic superdomains, i. e. two open subsupermanifolds of K p | q and K m | n respectively. There is a bijective correspondence between (1) superspace morphisms U → V ; (2) the set of pullbacks of a fixed coordinate system on V , i. e. ( m | n ) -uples ( s , . . . , s m , t , . . . , t n ) ∈ O ( U ) m × O ( U ) n such that (cid:0)e s ( x ) , . . . , e s m ( x ) (cid:1) ∈ | V | for each x ∈ | U | . Any supermanifold morphism M → N is then uniquely determined bya collection of local maps, once atlases on M and N have been fixed. Amorphism can hence be given by describing it in local coordinates.In the smooth category a further simplification occurs: we can assign amorphism between supermanifolds by assigning the pullbacks of the globalsections (see [15, § SMan R ( M, N ) ∼ = Hom SAlg R (cid:0) O ( N ) , O ( M ) (cid:1) . (2.1)The essential point here is that, borrowing some terminology from algebraicgeometry, smooth supermanifolds are an “affine” category. By this we meanthat the knowledge of the superalgebra of global sections allows us to fullyreconstruct the supermanifold obtaining its structure sheaf by a localizationprocedure (see, for example, [3] and [6]).To end our very brief summary of supergeometric definitions and resultsin the superspace context, we shall recall the superversion of Hadamard’s8emma, which sheds light on the structure of the stalk of the structure sheafof a supermanifold. Together with the Chart’s theorem, these are key resultsand we shall make frequent use of them in our work. Lemma 2.8 (Hadamard’s lemma).
Suppose M is a (smooth or holomor-phic) supermanifold, x ∈ | M | and { x i , ϑ j } is a system of coordinates in aneighborhood of x . Denote as usual by M x the ideal of the germs of sectionswhose value at x is zero. For each [ s ] ∈ O M,x and k ∈ N there exists apolynomial P in [ x i ] and [ ϑ j ] such that [ s ] − P ∈ M kx .Proof. The holomorphic case is trivial since the stalk at x identifies withconvergent power series. The proof in the smooth case can be found, forexample, in [17, § Due to the presence of nilpotent elements in the structure sheaf of a super-manifold, supergeometry can also be equivalently and very effectively studiedusing the language of functor of points , a very useful tool in algebraic geom-etry applications. We briefly review it; the interested reader can consult [9, § § A and B are two categories, [ A , B ] denotes the category of functors between A and B . Clearly, the morphisms in [ A , B ] are the natural transformations.Moreover we denote by A op the opposite category of A , so that the categoryof contravariant functors between A and B is identified with [ A op , B ]. Formore details we refer to [18]. Definition 2.9.
Given a supermanifold M , we define its functor of points M ( · ) : SMan op −→ Set
9s the functor from the opposite category of supermanifolds to the categoryof sets defined on the objects as M ( S ) := Hom( S, M )and on the morphisms according to M ( ϕ ) : M ( S ) −→ M ( T ) f f ◦ ϕ where ϕ : T → S .The elements in M ( S ) are also called the S -points of M .Given two supermanifolds M and N , Yoneda’s lemma establishes a bijec-tive correspondenceHom SMan ( M, N ) ←→ Hom [ SMan op , Set ] (cid:0) M ( · ) , N ( · ) (cid:1) between the morphisms M → N and the natural transformations M ( · ) → N ( · ) (see [18, ch. 3] or [10, ch. 6]). This allows us to view a morphism ofsupermanifolds as a family of morphisms M ( S ) → N ( S ) depending functo-rially on the supermanifold S . In other words, Yoneda’s lemma provides animmersion Y : SMan −→ [ SMan op , Set ]of
SMan into [
SMan op , Set ] that is full and faithful. There are howeverobjects in [
SMan op , Set ] that do not arise as the functors of points of asupermanifold. We say that a functor
F ∈ [ SMan op , Set ] is representable ifit is isomorphic to the functor of points of a supermanifold.
Observation 2.10.
We first notice that for any supermanifold M the set M ( K | ) = Hom SMan ( K | , M ) ∼ = | M | as sets, since K | is just a point.So the functor of points allows us to recover the set of the points of thetopological space | M | underlying M . The knowledge of the set-theoreticalpoints M ( K | ) however is far from enough to reconstruct the supermanifold M and this is because of two distinct reasons:(1) All the elements of M ( K | ) annihilate the nilpotent part of the sheaf,so they give us no information on the odd part of the structure sheaf.(2) The functor M ( · ) takes values in the category of sets, hence M ( K | )is just a set and does not contain any information on the differentiablestructure of M , even in the classical setting.10he supermanifold M is then recaptured only from the knowledge of its S -points, for all the supermanifolds S .We now want to recall a representability criterion, which allows to singleout, among all the functors from the category of supermanifolds to sets, thosethat are representable, i. e. those that are isomorphic to the functor of pointsof a supermanifold. In order to do this, we need to generalize the notionof open submanifold and of open cover , to fit this more general functorialsetting. Definition 2.11.
Let U and F be two functors SMan op → Set . U is a subfunctor of F if U ( S ) ⊆ F ( S ) for all S ∈ SMan and this inclusion is anatural transformation. We denote it by
U ⊆ F .We say that U is an open subfunctor of F if for all supermanifolds T and all natural transformations α : T ( · ) → F , α − ( U ) = V ( · ), where V isopen in T . If U is also representable we say that U is an open supermanifoldsubfunctor .Let U i be open supermanifold subfunctors of F . We say that { U i } is an open cover of a functor F : SMan op → Set if for all supermanifolds T andall natural transformations α : T ( · ) → F , α − ( U i ) = V i ( · ) and V i cover T .Any functor F : SMan op → Set when restricted to the category of opensubsupermanifolds of a given supermanifold T defines a presheaf over | T | . Definition 2.12.
A functor F : SMan op → Set is said to be a sheaf if it hasthe sheaf property, that is, if { T i } is an open cover of a supermanifold T andwe have a family { α i } , α i ∈ F ( T i ), such that α i | T i ∩ T j = α j | T i ∩ T j , then thereexists a unique α ∈ F ( T ) mapping to each α i (for more details see [25]).In particular, when F is restricted to the open subsupermanifolds of agiven supermanifold T , it is a sheaf over | T | .We are ready to state a representability criterion which gives necessaryand sufficient conditions for a functor from SMan op to Set to be repre-sentable. This is a very formal result and for this reason it holds as it isfor very different categories as smooth and holomorphic supermanifolds andeven for superschemes (for more details on this category see [6]). A completedescription of the classical representability criterion in algebraic geometrycan be found in [8, ch. 1]. For the super setting see [11].11 heorem 2.13 (Representability criterion).
A functor F : SMan op → Set is representable if and only if: (1) F is a sheaf, i. e. it has the sheaf property; (2) F is covered by open supermanifold subfunctors { U i } . A -points In this section we introduce the category
SWA of super Weil algebras. Theseare finite dimensional commutative superalgebras with a nilpotent gradedideal of codimension one. Super Weil algebras are the basic ingredient inthe definition of the Weil–Berezin functor and the Shvarts embedding. Theeasiest examples of super Weil algebras are Grassmann algebras. These arethe only super Weil algebras that can be interpreted as algebras of globalsections of supermanifolds, namely K | q . Given a supermanifold M , we wantto define a functor M ( · ) : SWA → Set assigning to each super Weil algebra A the set of A -points M A . If A is a Grassmann algebra, the A -points of M are identified with the usual K | q -points in the functor of points languagedescribed in the previous section. Unfortunately this functor is not adequateto fully describe the supermanifold M ; as we shall see at the end of thissection, the arriving category needs to have an additional structure in orderfor M ( · ) to contain the same information as M . This is due, as we shall see,to the local nature of M ( · ) . We now define the category of super Weil algebras . The treatment followsclosely that contained in [14, §
35] for the classical case.
Definition 3.1.
We say that A is a (real or complex) super Weil algebra ifit is a commutative unital superalgebra over K and(1) dim A < ∞ ,(2) A = K ⊕ ◦ A ,(3) ◦ A = ◦ A ⊕ ◦ A is a graded nilpotent ideal.12he category of super Weil algebras is denoted by SWA .We also define the height of A as the lowest r such that ◦ A r +1 = 0 and the width of A as the dimension of ◦ A/ ◦ A .Notice that super Weil algebras are local superalgebras, i. e. they containa unique maximal graded ideal. Remark 3.2.
As a direct consequence of the definition, each super Weilalgebra has an associated short exact sequence:0 −→ K j A −→ A = K ⊕ ◦ A pr A −→ A/ ◦ A ∼ = K −→ a ∈ A can be written uniquely as a = e a + ◦ a with e a ∈ K and ◦ a ∈ ◦ A . Example 3.3 (Dual numbers).
The simplest example of super Weil alge-bra in the classical setting is K ( x ) = K [ x ] / h x i the algebra of dual numbers.Here x is an even indeterminate which is nilpotent of degree two. Example 3.4 (Super dual numbers).
The simplest non trivial exampleof super Weil algebra in the super setting is K ( x, ϑ ) = K [ x, ϑ ] / h x , xϑ, ϑ i where x and ϑ are respectively even and odd indeterminates. Example 3.5 (Grassmann algebras).
The polynomial algebra in q oddvariables Λ( ϑ , . . . , ϑ q ) is another example of super Weil algebra. Grassmannalgebras are actually a full subcategory of SWA .Let K [ k | l ] := K [ t , . . . , t k ] ⊗ Λ( ϑ , . . . , ϑ l )denote the superalgebra of polynomials on K in k even and l odd variables. K [ k | l ] is not a super Weil algebra, unless k = 0, however every finite di-mensional graded quotient K [ k | l ] /J , with J graded ideal, is a super Weilalgebra.The next lemma gives a characterization of super Weil algebras, which isgoing to be very important for our treatment.13 emma 3.6. The following are equivalent:1. A is a super Weil algebra;2. A ∼ = K [ k | l ] /I for a suitable graded ideal I ⊇ h t , . . . , t k , ϑ , . . . , ϑ l i k .3. A ∼ = O K p | q , /J for suitable p, q and J graded ideal containing a power ofthe maximal ideal M in the stalk O K p | q , ;Proof. (1) = ⇒ (2). Assume first A is a super Weil algebra, generated by thehomogeneous elements a . . . a n . Then by universality we have a morphism: K [ X . . . X n ] −→ A , X i a i , hence A ∼ = K [ X . . . X n ] /I . Now we show I ⊃ ( X . . . X n ) k . Let n i be such that a n i i = 0. So we have X n i i ∈ I . Let N = max { n i } . We claim that ( X . . . X n ) nN ⊂ I . In fact ( X . . . X n ) nN isgenerated by X j . . . X j n n , with j + · · · + j n = nN . Since j . . . j n are n nonnegative integers whose sum is nN , we have that at least one of them j l isgreater than N , hence X j l l ∈ I , hence X j . . . X j n n ∈ I .(2) = ⇒ (1). If A = K [ X . . . X n ] /I and I ⊃ ( X . . . X n ) k certainly allthe generators X . . . X n are nilpotents, hence A = K ⊕ ( X . . . X n ), where( X . . . X n ) is the maximal ideal generated by the generators (by abuse ofnotation we use the same letter also in the quotient). Clearly this ideal isnot all A , since it consists only of nilpotent elements. Notice that since the X i are nilpotent one can readily check that f X + · · · + f n X n is nilpotent(choosing as exponent nN , N = max { n i } and reasoning as before).(1) = ⇒ (3). By lemma 4.3.2 in [24] pg 140, we have that M in O K p | q , is generated by x . . . x n , where x i are the germs of local coordinates around0. Moreover if f ∈ O K p | q , , for all K , there exists a polinomial P in the x . . . x n such that any f − P ∈ M K by 2.8.If A is a Weil superalgebra and a . . . a n its (nilpotent) homogeneousgenerators, by the previous discussion we have that A = K [ X . . . X n ] /I , I ⊃ ( X . . . X n ) k for a suitable k (specified above). Choose x . . . x n as thecoordinates of K p | q ( n = p + q ). Clearly the polinomial algebra in such coordi-nates K [ x . . . x n ] embeds into O K p | q , (again, with a small abuse of notationwe use x i to denote both the germs and the polynomial coordinates).Let J := < I > be the ideal in O K p | q , generated by the image of I in O K p | q , . Notice that M k ⊂ J , since ( X . . . X n ) k ⊂ I .We claim A ∼ = O K p | q , /J . Certainly we can define a morphism ϕ : K [ X . . . X n ] −→ O K p | q , /J , X i x i . This morphism factors through A =14 [ X . . . X n ] /I since, by the very definition of J , ϕ ( I ) ⊂ J . So we haveobtained a well defined morphism: ψ : A = K [ X . . . X n ] /I −→ O K p | q , /JX i x i We want to show it is surjective and injective. ψ surjective. Let f ∈ O K p | q , /J . By Hadamard’s lemma we have thatthere exists a polinomial P in the x . . . x n such that f − P ∈ M k . Since M k ⊂ J , we have f = P in O K p | q , /J , hence ψ is surjective. ψ injective. Assume ψ ( p ) = 0, that is ψ ( p ) ∈ J . We have that ψ ( p ) = u p + · · · + u n p n , u i ∈ O K p | q , , I = ( p . . . p n ) ⊂ K [ x . . . x n ] ⊂ O K p | q , (again we identify K [ X . . . X n ] with the subring K [ x . . . x n ] of O K p | q , andlook at I as an ideal inside it).Again by Hadamard’s lemma, we have that u i − P i ∈ M K for all K , P i ∈ K [ x . . . x n ], that is u i = P i + m i , with m i ∈ M K . So we have: ψ ( p ) = u p + · · · + u n p n = X i ( P i + m i ) p i = X i P i p i + X i m i p i ∈ I + M K If we set P := P i P i p i ∈ I , we have that ψ ( p ) − P = X i m i p i ∈ M K , ∀ K. Now we want to show that if a polynomial ψ ( p ) − P is in M K , then thepolynomial is also in ( x . . . x n ) K ⊂ K [ x . . . x n ] (let’s not forget that choosingthe correct K we have ( x . . . x n ) K ⊂ I ). But looking at the formula at pg141 in [24] and at the expression of the remainder R K ( x ), we see this is trueat once.(3) = ⇒ (1). Now assume A = O K p | q , /J with J ⊃ M k , and hence offinite codimension. Since A is local we have A = K ⊕ M , where M isgenerated by the nilpotent elements x . . . x n . By then we are done becauseevery element of M is nilpotent, hence we can reason as above.15 .2 A -points We now introduce the notion of A -point of a supermanifold M . Despite theanalogy with the functor of points described previously, the functor associ-ated with the A -points is subtly different. The main difference is that thecollection of the A -points of a supermanifold M , for all super Weil algebras A , will not enable us to recover all the information about the supermanifold M . As we are going to see, in order to transfer all the information from thesupermanifold M to the collection of its A -points it is necessary to endoweach of such sets of an extra structure, that we are going to discuss in sec. 4. Definition 3.7.
Let M be a supermanifold, x ∈ | M | and A a super Weilalgebra. We define the set of A -points near x as M A,x := Hom
SAlg ( O M,x , A )and the set of A -points as M A := G x ∈| M | M A,x .If x A ∈ M A,x , we call e x A := x the base point of x A . Observation 3.8.
Notice that, since O M,x is a local algebra, M K ,x containsonly the evaluation ev x and hence M K is identified with the set of topologicalpoints of M . Moreover, for each A ∈ SWA and x A ∈ M A , we have that x A = ev e x A + L where Im( L ) ⊆ ◦ A .We can consider the functor M ( · ) : SWA −→ Set (3.1)defined on the objects as A M A and on morphisms as ρ ρ , with ρ ∈ Hom
SAlg ( A, B ) and ρ : M A −→ M B x A ρ ◦ x A .16 emark 3.9. Observe that the only local superalgebras which are equal to O ( M ) for some supermanifold M are those of the form Λ K ( ϑ , . . . , ϑ q ) = O ( K | q ). For this reason this functor is quite different from the functor ofpoints borrowed from algebraic geometry and detailed in the previous section. Notation 3.10.
Here we introduce a multiindex notation that we will usein the following. Let { x , . . . , x p , ϑ , . . . , ϑ q } be a system of coordinates. If ν = ( ν , . . . , ν p ) ∈ N p ,we define x ν := x ν x ν · · · x µ p p , ν ! := Q i ν i !, and | ν | := P i ν i . If J = ( j , . . . , j r ) with 1 ≤ j < · · · < j r ≤ q we define ϑ J := ϑ j ϑ j · · · ϑ j r . | J | denotes the cardinality of J .In order to understand the structure of M A we need some preparation.We start with a well known result that holds for smooth supermanifolds andfor Stein superdomains U ⊆ C p | q . Lemma 3.11 (“Super” Milnor’s exercise).
Denote by M either (1) a smooth supermanifold or (2) a Stein superdomain in C p | q .The superalgebra maps O ( M ) → K are exactly the evaluations ev x : s e s ( x ) in the points x ∈ | M | . In other words there is a bijective correspondencebetween Hom
SAlg (cid:0) O ( M ) , K (cid:1) and | M | .Proof. In the smooth case the lemma is a consequence of eq. (2.1), consideringthat O ( R | ) = R and the pullback of a morphism ϕ : R | → M is theevaluation at | ϕ | ( R ).In the holomorphic case it is a again a consequence of the analogousclassical result for Stein manifold (see, for example, [12]). Indeed, if ψ : H ( | M | ) ⊗ Λ q −→ C is an algebra morphism, it uniquely factorizes through the algebra morphism | ψ | : H ( | M | ) −→ C ψ ( f ) = | ψ | ( | f | )Using [13, Proposition 57.1], we are done. Remark 3.12.
Notice that the lemma does not hold for a generic holomor-phic supermanifold.Let ψ ∈ Hom
SAlg (cid:0) O ( M ) , A (cid:1) . Due to the previous lemma, there exists aunique point of | M | , that we denote by e ψ , such that pr A ◦ ψ = ev e ψ , wherepr A is the projection A → K . We thus have a mapHom SAlg (cid:0) O ( M ) , A (cid:1) −→ Hom
SAlg (cid:0) O ( M ) , K (cid:1) ∼ = | M | ψ pr A ◦ ψ = ev e ψ . (3.2)The next proposition establishes the local nature of the functor A M A . Proposition 3.13. (1)
Each element x A of M A is determined by the im-ages of the germs of a system of local coordinates [ x i ] , [ ϑ j ] around e x A .Conversely, given x ∈ | M | , a system of local coordinates { x i } pi =1 , { ϑ j } qj =1 around x and elements { x i } pi =1 , { θ j } qj =1 , x i ∈ A , θ j ∈ A , such that e x i = e x i ( x ) , there exists a unique morphism x A ∈ Hom
SAlg ( O M,x , A ) such that ( x A ([ x i ]) = x i x A ([ ϑ j ]) = θ j . (3.3)(2) Suppose ( U, h ) is a coordinate chart, then there is a bijection U A → | h ( U ) | × ◦ A × ◦ A (3) Suppose U is a coordinate chart in the smooth case, and a Stein coordi-nate chart in the holomorphic case, then there is a bijective correspon-dence U A = G x ∈ U Hom
SAlg ( O M,x , A ) −→ Hom
SAlg (cid:0) O M ( U ) , A (cid:1) .Proof. Let us consider (1). Suppose that x A is given. We want to show thatthe images of the germs of local coordinates x A ([ x i ]), x A ([ ϑ j ]) determine x A completely. This follows noticing that The reader should notice the difference between { x i , ϑ j } and { x i , θ j } . the image of a polynomial section under x A is determined, • there exists k ∈ N such that the kernel of x A contains M kx (see lemma3.6)and using Hadamard’s lemma. We now come to existence. Suppose the eq.(3.3) are given and let [ s ] be a germ at x . We define x A ([ s ]) through a formalTaylor expansion. More precisely let s = X J ⊆{ ,...,q } s J ϑ J be a representative of [ s ] near x , where the s J are smooth (holomorphic)functions in x , . . . , x p . Define x A ( s ) = X ν ∈ N p J ⊆{ ,...,q } ν ! ∂ | ν | s J ∂x ν (cid:12)(cid:12)(cid:12)(cid:12) ( e x ,..., e x p ) ◦ x ν θ J . (3.4)This is the way in which the purely formal expression s ( x A ) = s ( e x + ◦ x , . . . , e x p + ◦ x p , θ , . . . , θ q )is usually understood. Eq. (3.4) has only a finite number of terms due to thenilpotency of the ◦ x i and θ j . It is clear from eq. (3.4) that x A ( s ) does notdepend on the chosen representative. Finally x A so defined is a superalgebramorphism since, for each [ s ] , [ t ] ∈ O M,x , x A ( st ) = X ν ∈ N p K ⊆ J ⊆{ ,...,q } λ ( K, J ) 1 ν ! ∂ | ν | ( s K t J \ K ) ∂x ν ◦ x ν θ J = X ν − µ ∈ N p ,K,J λ ( K, J ) 1 ν ! (cid:18) νµ (cid:19) ∂ | µ | s K ∂x µ ∂ | ν − µ | t J \ K ∂x ν − µ ◦ x ν θ J = X ν,µ,J,K " µ ! ∂ | µ | s K ∂x µ ◦ x µ θ K ν − µ )! ∂ | ν − µ | t J \ K ∂x ν − µ ◦ x ν − µ θ J \ K where (cid:0) νµ (cid:1) = Q i (cid:0) ν i µ i (cid:1) and λ ( K, J ) is defined to be ± θ K θ J \ K = λ ( K, J ) θ J . 192) is just a restatement of (1)Let us now consider (3). Define the map η x : O M ( U ) → O M,x assigningto each section over U the corresponding germ at x ∈ U , and consider G x ∈ U Hom
SAlg ( O M,x , A ) −→ Hom
SAlg (cid:0) O M ( U ) , A (cid:1) x A x A ◦ η e x A .We show that it is invertible. Let ψ ∈ Hom
SAlg (cid:0) O M ( U ) , A (cid:1) . If { x i , ϑ j } is a coordinate system on U , due to (1), the set { ψ ( x i ) , ψ ( ϑ j ) } uniquelydetermines an element in Hom SAlg ( O M, e ψ , A ). It is easy to show that this isthe required inverse.In the smooth category the above setting can be somewhat simplified.This is essentially due to eq. (2.1) and the related discussion. Let us see indetail this point, summarized in prop. 3.15. Lemma 3.14.
Let M be a smooth supermanifold. Let s ∈ O ( M ) and let ψ ∈ Hom
SAlg (cid:0) O ( M ) , A (cid:1) . Assume that s is zero when restricted to a certainneighbourhood of e ψ (see eq. (3.2) ). Then ψ ( s ) = 0 .Proof. Suppose U ∋ e ψ is such that s | U = 0. Let t ∈ O M ( U ) be such thatsupp( t ) ⊂ U and t | V = 1, where the closure of V is contained in U . Then0 = ψ ( st ) = ψ ( s ) ψ ( t ).Hence ψ ( s ) = 0, since ψ ( t ) is invertible being ev e ψ ( t ) = 1. Proposition 3.15. If M is a smooth supermanifold M A ∼ = Hom SAlg R (cid:0) O ( M ) , A (cid:1) in a functorial way. In other words we can equivalently define the functor M ( · ) on the objects as: M ( · ) : SWA → Set A Hom
SAlg R (cid:0) O ( M ) , A (cid:1) . roof. Clearly each x A ∈ M A can be identified with a superalgebra map O ( M ) → A by composing it with the natural map η x : O ( M ) → O M,x . Viceversa let ψ ∈ Hom
SAlg R (cid:0) O ( M ) , A (cid:1) . In the smooth category, given a germ [ s ] ∈O M, e ψ , there exists a global section s ∈ O ( M ) such that η e ψ ( s ) = [ s ]. Sincethe image of s under ψ depends only on the germs of s at e ψ , ψ determinesan element of M A,x . In fact, let s ′ ∈ [ s ] and let U be a neighbourhood of e ψ such that s ′| U = s | U . It is always possible to find a smaller neighbourhood V ⊂ U and u, v , v ∈ O ( M ) such that s = u + v , s ′ = u + v and v i | V = 0.Then, due to the previous lemma, ψ ( s ) = ψ ( u ) = ψ ( s ′ ). The functoriality isclear.The next observation gives us an interesting and very important charac-terization of the A -points of a superdomain. Observation 3.16.
Let (
U, h ) be a chart in a supermanifold M with localcoordinates { x i , ϑ j } . By point (2) of prop. 3.13 we have an injective map U A −→ A p × A q x A (x , . . . , x p , θ , . . . , θ q ) := (cid:0) x A ( x ) , . . . , x A ( ϑ q ) (cid:1) .We can think of it heuristically as the assignment of A -valued coordinates { x i , θ j } on U A . As we are going to see in theorem 4.2 the components ofthe coordinates { x i , θ j } , given by (cid:10) a ∗ k , x i (cid:11) , (cid:10) a ∗ k , θ j (cid:11) with respect to a basis { a k } of A are indeed the coordinates of a smooth or holomorphic manifold.The base point e x A ∈ U has coordinates ( e x , . . . , e x p ). In this language, if ρ : A → B is a super Weil algebra morphism, the corresponding morphism ρ : M A → M B is “locally” given by ρ × · · · × ρ : | U | × ◦ A p × A q −→ | U | × ◦ B p × B q . (3.5)where, with an harmless abuse of notation, we are confusing | h ( U ) | with | U | .This is well defined since ρ does not change the base point.If M = K p | q we can also consider the slightly different identification K p | qA −→ ( A ⊗ K p | q ) x A X i x A ( e ∗ i ) ⊗ e i where { e , . . . , e p + q } denotes a homogeneous basis of K p | q and { e ∗ , . . . , e ∗ p + q } its dual basis. Here a little care is needed as we already remarked at the21eginning of section 2. In the literature the name K p | q is used for two inprinciple different objects: it may indicate the super vector space K p | q = K p ⊕ K q or the superdomain ( K p , O K p ⊗ Λ q ). In the previous equation thefirst K p | q is viewed as a superdomain, while the last as a super vector space.Likewise the { e ∗ i } are interpreted both as vectors and sections of O ( K p | q ).As we shall see in subsection 4.1 the functor A ( A ⊗ K p | q ) recaptures all the information about the superdomain K p | q , so that the two inprinciple different ways of looking at K p | q become then identified. This resultwill hence establish a quite natural way to identify the two objects. Withthis identification, the superdomain morphism ρ : K p | qA → K p | qB correspondsto the super vector space morphism ρ ⊗ : ( A ⊗ K p | q ) −→ ( B ⊗ K p | q ) . A -points In the previous subsection we have seen that, somehow mimicking the func-tor of points approach to supermanifolds, it is possible to associate to eachsupermanifold M a functor SWA −→ Set A M A .Hence we have a functor: B : SMan −→ [ SWA , Set ] .The natural question about such a functor is whether B is a full and faithfulembedding or not. In this subsection, we show that B is not full, in otherwords, there are many more natural transformations between M ( · ) and N ( · ) than those coming from morphisms from M to N . We will show this bygiving a simple example. Then, in prop. 3.20, we will see a characterizationof the natural transformation between two superdomains.Let us start our discussion. We first want to show that the natural trans-formations M ( · ) → N ( · ) arising from supermanifold morphisms M → N ϕ : M → N of superman-ifolds induces a natural transformation between the corresponding functorsof A -points given by ϕ A : M A −→ N A x A x A ◦ ϕ ∗ for all super Weil algebras A . Let M = K p | q and N = K m | n , and denoterespectively by { x i , ϑ j } and { x ′ k , ϑ ′ l } two systems of coordinates over them.With these assumptions, ϕ is determined by the pullbacks of the coordinatesof N , while the A -point ϕ A ( x A ) is determined by(x ′ , . . . , x ′ m , θ ′ , . . . , θ ′ n ) := (cid:0) x A ◦ ϕ ∗ ( x ′ ) , . . . , x A ◦ ϕ ∗ ( ϑ ′ n ) (cid:1) ∈ A m × A n .If (x , . . . , x p , θ , . . . , θ q ) denote the images of the coordinates of M under x A (x = x A ( x ), etc.) and ϕ ∗ ( x ′ k ) = P J s k,J ϑ J ∈ O ( K p | q ) , where the s k,J arefunctions on K p , then we havex ′ k = x A ◦ ϕ ∗ ( x ′ k ) = X ν ∈ N p J ⊆{ ,...,q } ν ! ∂ | ν | s k,J ∂x ν (cid:12)(cid:12)(cid:12)(cid:12) ( e x ,..., e x p ) ◦ x ν θ J (3.6)and similarly for the odd coordinates (see prop. 3.13). Notice that if wepursue the point of view of observation 3.16, i. e. if we consider { x i , θ j } as A -valued coordinates of K p | qA , this equation can be read as a coordinateexpression for ϕ A .Not all the natural transformations M ( · ) → N ( · ) arise in this way. Thishappens also for purely even manifolds, as we see in the next example. Example 3.17.
Let M and N be two smooth manifolds and let ϕ : M → N be a map (smooth or not). The natural transformation α ( · ) : M ( · ) → N ( · ) α A : M A −→ N A x A ev ϕ ( e x A ) is not of the form seen above, even if ϕ is assumed to be smooth, while westill have ϕ = α K .We end this subsection with a technical result, essentially due to A. A.Voronov (see [26]), characterizing all possible natural transformations be-tween the functors of A -points of two superdomains, hence comprehendingalso those not arising from supermanifold morphisms.23 efinition 3.18. Let U be an open subset of K p . We denote by A p | q ( U )the unital commutative superalgebra of formal series with p even and q oddgenerators and coefficients in the algebra F ( U, K ) of arbitrary functions on U , i. e. A p | q ( U ) := F ( U, K )[[ X , . . . , X p , Θ , . . . , Θ q ]].An element F ∈ A p | q ( U ) is of the form F = X ν ∈ N p J ⊆{ ,...,q } f ν,J X ν Θ J where f ν,J ∈ F ( U, K ) and { X i } and { Θ j } are even and odd generators. A p | q ( U ) is a graded algebra: F is even (resp. odd) if | J | is even (resp. odd)for each term of the sum.Let us introduce a partial order between super Weil algebras by sayingthat A ′ (cid:22) A if and only if A ′ is a quotient of A . Lemma 3.19.
The set of super Weil algebras is directed, i. e., if A and A are super Weil algebras, then there exists A such that A i (cid:22) A .Proof. In view of lemma 3.6, choosing carefully k, l ∈ N and J and J idealsof K [ k | l ], we have A i ∼ = K [ k | l ] /J i . If r is the maximum between the heightsof A and A , M r +1 ⊂ J ∩ J . So A ∼ = K [ k | l ] / ( J ∩ J ) is finite dimensionaland then it is a super Weil algebra. Proposition 3.20.
Let U and V be two superdomains in K p | q and K m | n respectively. The set of natural transformations in [ SWA , Set ] between U ( · ) and V ( · ) is in bijective correspondence with the set of elements of the form F = ( F , . . . , F m + n ) ∈ (cid:0) A p | q ( | U | ) (cid:1) m × (cid:0) A p | q ( | U | ) (cid:1) n such that, if F k = P ν,J f kν,J X ν Θ J , (cid:0) f , ∅ ( x ) , . . . , f m , ∅ ( x ) (cid:1) ⊆ | V | ∀ x ∈ | U | . (3.7) Proof.
As above, K p | qA is identified with A p × A q and consequently a map K p | qA → K m | nA consists of a list of m maps A p × A q → A and n maps A p × A q → A . In the same way, U A is identified with | U | × ◦ A p × A q .24et F = ( F , . . . , F m + n ) be as in the hypothesis. A formal series F k determines a map | U | × ◦ A p × A q ⊆ A p × A q → A in a natural way, defining F k (x , . . . , x p , θ , . . . , θ q ) := X ν ∈ N p J ⊆{ ,...,q } f kν,J ( e x , . . . , e x p ) ◦ x ν θ J .The parity of its image is the same as that of F k . Then, in view of therestrictions imposed on the first m F k given by eq. (3.7), F determines amap U A → V A and, varying A ∈ SWA , a natural transformation U ( · ) → V ( · ) ,as it is easily checked.Let us now suppose now that α ( · ) : U ( · ) → V ( · ) is a natural transformation.We will see that it is determined by a unique F in the way just explained.Let A be a super Weil algebra of height r and x A = ( e x + ◦ x , . . . , e x p + ◦ x p , θ , . . . , θ q ) ∈ A p × A q ∼ = K p | qA with e x A ∈ | U | . Let us consider the super Weil algebraˆ A := (cid:0) K [ z , . . . , z p ] ⊗ Λ( ζ , . . . , ζ q ) (cid:1) / M s (3.8)with s > r ( M is as usual the maximal ideal of polynomials without constantterm) and the ˆ A -point y e x A := ( e x + z , . . . , e x + z p , ζ , . . . , ζ q ) ∈ ˆ A p × ˆ A q ∼ = K p | q ˆ A .A homomorphism between two super Weil algebras is clearly fixed by theimages of a set of generators, but this assignment must be compatible withthe relations between the generators. The following assignment is possibledue to the definition of ˆ A . If ρ x A : ˆ A → A denotes the map ( z i ◦ x i ζ j θ j ,then clearly ρ x A ( y e x A ) = x A .Let ( α ˆ A ) k with 1 ≤ k ≤ m + n be a component of α ˆ A , and let( α ˆ A ) k ( y e x A ) = X ν,J a kν,J ( e x A ) z ν ζ J a kν,J ( e x A ) ∈ K and (cid:0) a , ∅ ( e x A ) , . . . , a m , ∅ ( e x A ) (cid:1) ∈ | V | ; the sum is on | J | even(resp. odd), if k ≤ m (resp. k > m ). Due to the functoriality of α ( · ) ( α A ) k ( x A ) = ( α A ) k ◦ ρ x A ( y e x A ) = ρ x A ◦ ( α ˆ A ) k ( y e x A ) = X ν,J a kν,J ( e x A ) ◦ x ν θ J ,so that there exists a non unique F such that F ( x A ) = α A ( x A ). Moreover F ( x A ′ ) = α A ′ ( x A ′ ) for each A ′ (cid:22) A and x A ′ ∈ U A ′ (it is sufficient to use theprojection A → A ′ ).If, by contradiction, F ′ is another list of formal series with this property,there exists a super Weil algebra A ′′ such that F ( x A ′′ ) = F ′ ( x A ′′ ) for some x A ′′ ∈ U A ′′ . Indeed if a component F k differs in f kν,J , it is sufficient to consider A ′′ := K [ p | q ] / M s with s > max (cid:0) | ν | , q (cid:1) . In the previous section we saw that the functor B : SMan −→ [ SWA , Set ]does not define a full and faithful embedding of
SMan in [
SWA , Set ], thecategory of functors from
SWA to Set . Roughly speaking, the root of sucha difficulty can be traced to the fact that the functor B ( M ) : SWA → Set looks only to the local structure of the supermanifold M , hence it loses allthe global information. For the functor of points as we described it in 2.2, weobtain a full and faithfull embedding thanks to the Yoneda’s lemma. If we tryto reproduce its proof in this different setting, we see that the main obstacleis that M can no longer be seen as an M -point of the supermanifold M itself. The following heuristic argument gives a hint on how such a difficultycan be overcome.It is well known (see, for example, [9, § V = V ⊕ V and W = W ⊕ W , there is a bijective correspondencebetween graded linear maps V → W and functorial families of Λ -linear mapsbetween (Λ ⊗ V ) and (Λ ⊗ W ) , for each Grassmann algebra Λ. This resultgoes under the name of even rule principle . Since vector spaces are localmodels for manifolds, the even rule principle seems to suggest that each M A should be endowed with a local structure of A -module. This vague idea is26ade precise with the introduction of the category A Man of A -smoothmanifolds. We then prove that each M A can be endowed in a canonical waywith the structure of A -manifold. This construction allows to specialize thearrival category of the functor of A -points associated to a supermanifold M and to define the Weil–Berezin functor of M as M ( · ) : SWA −→ A Man A M A .In this way we can define a functor S : SMan −→ [[ SWA , A Man ]] M M ( · ) where [[ SWA , A Man ]] is an appropriate subcategory of [
SWA , A Man ] thatwill be specified by definition 4.3. We will call S the Shvarts embedding.As it turns out, this definition of the local functor of points allows torecover the correct natural transformations without any artificial condition.More precisely, lemma 4.8 will show that the natural transformations α ( · ) : M ( · ) → N ( · ) arising from supermanifolds morphisms are exactly those for which α A is A -smooth for each A . S is then a full and faithful embedding. Moreover S preserves the products. In particular this implies that if G is a group objectin SMan , i. e. it is a super Lie group, then S ( G ) is a group object too, i. e.it takes values in the category of the A -smooth Lie groups.Exactly as for the functor of points, the functor S is not an equivalenceof categories, so that the problem of characterizing the functors in the imageof S arises naturally (representability problem). A criterion characterizingthe representable functors is then given in subsection 4.2. A -smooth structure and its consequences Preliminary to everything is the following (rather long) definition of A -man-ifold and of the category A Man . For a more detailed discussion see forexample [22] and references therein.
Definition 4.1.
Fix an even commutative finite dimensional algebra A andlet L be a finite dimensional A -module. Let M be a manifold. An L -chart on M is a pair ( U, h ) where U is open in M and h : U → L is a diffeomorphismonto its image. M is an A -manifold if it admits an L -atlas. By this we27ean a family { ( U i , h i ) } i ∈A where { U i } is an open covering of M and each( U i , h i ) is an L -chart, such that the differentials d ( h i ◦ h − j ) h j ( x ) : T h j ( x ) L ∼ = L −→ L ∼ = T h i ( x ) L are isomorphisms of A -modules for all i , j and x ∈ U i ∩ U j .If M and N are A -manifolds, a morphism ϕ : M → N is a smooth mapwhose differential is A -linear at each point. We also say that such morphismis A -smooth . We denote by A Man the category of A -manifolds.We define also the category A Man in the following way. The objects of A Man are manifolds over generic finite dimensional commutative algebras.The morphisms in the category are defined as follows. Denote by A and B two commutative finite dimensional algebras, and let ρ : A → B be analgebra morphism. Suppose M and N are A and B manifolds respectively,we say that a morphism ϕ : M → N is ρ -smooth if ϕ is smooth and( dϕ ) x ( av ) = ρ ( a )( dϕ ) x ( v )for each x ∈ M , v ∈ T x ( M ), and a ∈ A .Notice that A -linearity always implies K -linearity, in particular, in thecomplex case, A -manifolds are holomorphic.The above definition is motivated by the following theorems. In order toease the exposition we first give the statements of the results and we postponetheir proofs to the last part of this subsection. Theorem 4.2.
Let M be a smooth (resp. holomorphic) supermanifold, andlet A be a real (resp. complex) super Weil algebra. (1) M A can be endowed with a unique A -manifold structure such that, foreach open subsupermanifold U of M and s ∈ O M ( U ) the map definedby ˆ s : U A −→ Ax A x A ( s ) is A -smooth. (2) If ϕ : M → N is a supermanifold morphism, then ϕ A : M A −→ N A x A x A ◦ ϕ ∗ is an A -smooth morphism. If B is another super Weil algebra and ρ : A → B is an algebra mor-phism, then ρ : M A −→ M B x A ρ ◦ x A is a ρ | A -smooth map. The above theorem says that supermanifolds morphisms give rise to mor-phisms in the A Man category. From this point of view the next definitionis quite natural.
Definition 4.3.
We call [[
SWA , A Man ]] the subcategory of [
SWA , A Man ]whose objects are the same and whose morphisms α ( · ) are the natural trans-formations F → G , with F , G : SWA → A Man , such that α A : F ( A ) −→ G ( A )is A -smooth for each A ∈ SWA .Theorem 4.2 allows us to give more structure to the arrival category of thefunctor of A -points. More precisely we have the following definition, whichis the central definition in our treatment of the local functor of points. Definition 4.4.
Let M be a supermanifold. We define the Weil–Berezinfunctor of M as M ( · ) : SWA −→ A Man A M A . (4.1)Moreover we define the Shvarts embedding S : SMan −→ [[ SWA , A Man ]] M M ( · ) .We can now state one of the main results in this paper; it tells that theWeil-Berezin functor M ( · ) recaptures all the information contained in thesupermanifold M . Theorem 4.5. S is a full and faithful embedding, i. e. if M and N are twosupermanifolds, and M ( · ) and N ( · ) their Weil–Berezin functors, then Hom
SMan ( M, N ) ∼ = Hom [[ SWA , A Man ]] ( M ( · ) , N ( · ) ) . orollary 4.6. Two supermanifolds are isomorphic if and only if their Weil-Berezin functors are isomorphic.
Observation 4.7.
If we considered the bigger category [
SWA , A Man ] in-stead of [[
SWA , A Man ]], the above theorem is no longer true. In example3.17 we examined a natural transformation between functors from
SWA to Set , which did not come from a supermanifolds morphism. If, in the sameexample, ϕ is chosen to be smooth, we obtain a morphism in [ SWA , A Man ]that is not in [[
SWA , A Man ]]. Indeed, it is not difficult to check that if π A : A → A is given by a e a , then α A (in the example) is π A -linear.We now examine the proofs of theorems 4.2 and 4.5. First we need toprove theorem 4.5 in the case of two superdomains U and V in K p | q and K m | n respectively (lemma 4.8). As usual, if A is a super Weil algebra, U A and V A are identified with | U | × ◦ A p × A q and | V | × ◦ A m × A n (see observation 3.16).Then they have a natural structure of open subsets of A -modules. The nextlemma is due to A. A. Voronov in [26] and it is the local version of theorem4.5. Lemma 4.8.
A natural transformation α ( · ) : U ( · ) → V ( · ) comes from a su-permanifold morphism U → V if and only if α A : U A → V A is A -smooth foreach A .Proof. Due to prop. 3.20 we know that α ( · ) is determined by m even and n odd formal series of the form F k = P ν,J f kν,J X ν Θ J with f kν,J arbitraryfunctions in p variables satisfying eq. (3.7). Moreover as we have seen in thediscussion before example 3.17 a supermanifold morphism ϕ : U → V givesrise to a natural transformation ϕ A : U A → V A whose components are of theform of eq. (3.6).Let us suppose that α A is A -smooth. This clearly happens if and only ifall its components are A -smooth and the smoothness request for all A forcesall coefficients f kν,J to be smooth.Let ( α A ) k be the k -th component of α A and let i ∈ { , . . . , p } . We wantto study ω : A −→ A j x i ( α A ) k (x , . . . , x i , . . . , x p , θ , . . . , θ q ),supposing the other coordinates fixed ( j = 0 if 1 ≤ k ≤ p or j = 1 if p < k ≤ p + q ). Since ◦ x i ∈ A commutes with all elements of A , ω (x i ) = X t ≥ a t ( e x i ) ◦ x ti (4.2)30ith a t ( e x i ) := X ν,Jν i = t f kν,J ( e x , . . . , e x i , . . . , e x p ) ◦ x ( ν − tδ i ) θ J (4.3)( tδ i is the element of N p with t at the i -th component and 0 elsewhere).If y = e y + ◦ y ∈ A and ω is A -smooth ω (x i + y) − ω (x i ) = dω x i (y) + o (y) = ( e y + ◦ y) dω x i (1 A ) + o (y) (4.4)(where 1 A is the unit of A ). On the other hand, from eq. (4.2) and defining a ′ t ( e x i ) := X ν,Jν i = t ∂ i f kν,J ( e x , . . . , e x i , . . . , e x p ) ◦ x ( ν − tδ i ) θ J (4.5)( ∂ i denotes the partial derivative respect to the i -th variable), we have ω (x i + y) − ω (x i ) = X t ≥ a t ( e x i + e y)( ◦ x i + ◦ y) t − X t ≥ a t ( e x i ) ◦ x ti = X t ≥ (cid:0) a ′ t ( e x i ) e y ◦ x ti + a t ( e x i ) t ◦ x t − i ◦ y + o (y) (cid:1) = e y X t ≥ a ′ t ( e x i ) ◦ x ti + ◦ y X t ≥ ( t + 1) a t +1 ( e x i ) ◦ x ti + o (y). (4.6)Thus, comparing eq. (4.4) and (4.6), we get that the identity( e y + ◦ y) dω x i (1 A ) = e y X t ≥ a ′ t ( e x i ) ◦ x ti + ◦ y X t ≥ ( t + 1) a t +1 ( e x i ) ◦ x ti must hold and, consequently, also the following relations must be satisfied: X t ≥ a ′ t ( e x i ) ◦ x ti = X t ≥ ( t + 1) a t +1 ( e x i ) ◦ x ti and then, from eq. (4.3) and (4.5), X ν,J ∂ i f kν,J ( e x , . . . , e x p ) ◦ x ν θ J = X ν,J ( ν i + 1) f kν + δ i ,J ( e x , . . . , e x p ) ◦ x ν θ J .31et us fix ν ∈ N p and J ⊆ { , . . . , q } . If A = K [ p | q ] / M s with s > max( | ν | + 1 , q ) ( M is as usual the maximal ideal of polynomials without con-stant term), we note that necessarily, due to the arbitrariness of (x , . . . , θ q ), ∂ i f kν,J = ( ν i + 1) f kν + δ i ,J and, by recursion, ( α A ) k is of the form of (3.6) with s k,J = f k ,J .Conversely, let ( α A ) k be of the form of eq. (3.6). It is A -smooth if andonly if it is A -smooth in each variable. It is A -smooth in the even variablesfor what has been said above and in the odd variables since it is polynomialin them.In particular the above discussion shows also that any superdiffeomor-phism U → U gives rise, for each A , to an A -smooth diffeomorphism U A → U A and then each U A admits a canonical structure of A -manifold.We now use the results obtained for superdomains in order to prove the-orems 4.2 and 4.5 in the general supermanifold case. We need to recall thefollowing elementary result from ordinary differential geometry. Lemma 4.9.
Let X be a set. Suppose a countable covering { U i } and acollection of injective maps h i : U i → K n are given, satisfying the followingconditions: (1) for each i and j , h i ( U i ∩ U j ) is open in K n ; (2) h j ◦ h − i : h i ( U i ∩ U j ) → h j ( U i ∩ U j ) is a diffeomorphism; (3) for each x and y in X , x = y , there exists V x ⊆ U i and V y ⊆ U j suchthat x ∈ V x , y ∈ V y , V x ∩ V y = ∅ , h i ( V x ) and h j ( V y ) both open.Then X admits a unique smooth manifold structure such that { ( U i , h i ) } defines an atlas over it. We leave the proof of this lemma to the reader and we return to the proofsof theorems 4.2 and 4.5.
Proof of theorem 4.2.
Let { ( U i , h i ) } be an atlas over M and p | q the dimen-sion of M . Each chart ( U i , h i ) of such an atlas induces a chart (cid:0) ( U i ) A , ( h i ) A (cid:1) ,( U i ) A = F x ∈ U i M A,x , over M A given by( h i ) A : ( U i ) A −→ U p | qA x A x A ◦ h ∗ i .32ith U p | q open subset of K p | q .The coordinate changes are easily checked to be given, with some abuseof notation, by ( h i ◦ h − j ) A , which are A -smooth due to lemma 4.8. Theuniqueness of the A -manifold structure is clear. This proves the first point.The other two points concern only the local behavior of the considered mapsand are clear in view of lemma 4.8 and eq. (3.5). Proof of theorem 4.5.
Lemma 4.8 accounts for the case in which M and N are superdomains. For the general case, let us suppose we have α ∈ Hom [[ SWA , A Man ]] (cid:0) M ( · ) , N ( · ) (cid:1) .Fixing a suitable atlas of both supermanifolds, we obtain, in view of lemma4.8, a family of local morphisms. Such a family will give a morphism M → N if and only if they do not depend on the choice of the coordinates. Let ussuppose that U and V are charts on M and N respectively, U ∼ = U p | q ⊆ K p | q , V ∼ = V m | n ⊆ K m | n , such that α K ( | U | ) ⊆ | V | , and h i : U −→ U p | q k i : V −→ V m | n i = 1 , U and V respectively. The naturaltransformations( ˆ ϕ i ) ( · ) := ( k i ) ( · ) ◦ (cid:0) α ( · ) (cid:1) | U ( · ) ◦ (cid:0) h − i (cid:1) ( · ) : U p | q ( · ) −→ V m | n ( · ) give rise to two morphisms ˆ ϕ i : U p | q → V m | n . If ϕ i := k − i ◦ ˆ ϕ i ◦ h i : U −→ V ,we have ϕ = ϕ since ( ϕ i ) ( · ) = (cid:0) α ( · ) (cid:1) | U ( · ) and two morphisms that give riseto the same natural transformation on a superdomain are clearly equal.We end this subsection with the next proposition stating that the Shvartsembedding preserves products and, in consequence, group objects. Proposition 4.10.
For all supermanifolds M and N , S ( M × N ) ∼ = S ( M ) × S ( N ) .Moreover S ( K | ) is a terminal object in the category [[ SWA , A Man ]] . roof. The fact that ( M × N ) A ∼ = M A × N A for all A can be checked easily.Indeed, let z A ∈ ( M × N ) A with e z A = ( x, y ), we have that O x and O y naturallyinject in O e z A . Hence z A defines, by restriction, two A -points x A ∈ M A and y A ∈ N A . Using prop. 3.13 and rectangular coordinates over M × N it iseasy to check that such a correspondence is injective, and is also a naturaltransformation. Conversely, if x A ∈ M A,x and y A ∈ N A,y , they define a map z A : O x ⊗ O y → A through z A ( s ⊗ s ) = x A ( s ) · y A ( s ). Using again prop.3.13, it is not difficult to check that this requirement uniquely determines anelement in ( M × N ) A, ( x,y ) and that this correspondence defines an inverse forthe morphism ( M × N ) ( · ) → M ( · ) × N ( · ) defined above.Along the same lines it can be proved that, a similar condition for themorphisms holds. Finally S ( K | ) is a terminal object, since K | A = K forall A .It is easy to check that the stated result is equivalent to the fact that S preserves finite products for arbitrary many objects.We now consider a super Lie group G , i. e. a group object in the categoryof (smooth or holomorphic) supermanifolds. First we recall briefly the notionof group object. Definition 4.11. A group object in some category with finite products andterminal object T , is an object G with three arrows µ G : G × G −→ G i G : G −→ G e G : T −→ G satisfying the usual commutative diagrams for multiplication, inverse andunit respectively. Observation 4.12.
In a locally small category C we have that, equiva-lently, a group is an object G whose functor of points G ( · ) takes value inthe category of groups Grp , i. e. G is a group object if there exists a functor C op → Grp that, composed with the forgetful functor
Grp → Set , equals G ( · ). Corollary 4.13. If G is a super Lie group, S ( G ) with the arrows S ( µ G ) , S ( i G ) and S ( e G ) is a group object in [[ SWA , A Man ]] . This means thatthe Weil–Berezin functor of G takes values in the category of A -smooth Liegroups.Proof. This is an immediate consequence of prop. 4.10.34or more information on group objects and product preserving functors,see [25].
Next definition is the natural generalization of the classical one to the Weil–Berezin functor setting.
Definition 4.14.
We say that a functor F : SWA −→ A Man is representable if there exists a supermanifold M F such that F ∼ = ( M F ) ( · ) in [[ SWA , A Man ]].Notice that we are abusing the category terminology, that considers afunctor F to be representable if and only if F is isomorphic to the Homfunctor.Due to theorem 4.5, if a functor F is representable, then the supermani-fold M F is unique up to isomorphism. Next example shows that there existsnon representable functors. Example 4.15.
Consider the constant functor
SWA → A Man defined as A K on the objects ( K ∼ = A/ ◦ A is an A -module) and ρ K on themorphisms. This functor is not representable in the sense explained above.In this subsection we look for conditions ensuring the representability fora functor F : SWA → A Man .Since F ( K ) is a manifold, we can consider an open set U ⊆ F ( K ). If A isa super Weil algebra and pr A := F (pr A ), where pr A is the projection A → K ,pr − A ( U ) is an open A -submanifold of F ( A ). Moreover, if ρ : A → B is asuperalgebra map, since pr B ◦ ρ = pr A , ρ := F ( ρ ) can be restricted to ρ pr − A ( U ) : pr − A ( U ) −→ pr − B ( U ).We can hence define the functor F U : SWA −→ A Man A pr − A ( U ) ρ ρ pr − A ( U ) .35 roposition 4.16 (Representability). A functor F : SWA −→ A Man is representable if and only if there exists an open cover { U i } of F ( K ) suchthat F U i ∼ = ( ⌢ V i ) ( · ) with ⌢ V i superdomains in a fixed K p | q .Proof. The necessity is clear due to the very definition of supermanifold.Let us prove sufficiency. We have to build a supermanifold structure onthe topological space |F ( K ) | . Let us denote by ( h i ) ( · ) : F U i → ( ⌢ V i ) ( · ) thenatural isomorphisms in the hypothesis. On each U i , we can put a super-manifold structure ⌢ U i , defining the sheaf O ⌢ U i := [( h − i ) K ] ∗ O ⌢ V i . Let k i be theisomorphism ⌢ U i → ⌢ V i and ( k i ) ( · ) the corresponding natural transformation.If U i,j := U i ∩ U j , consider the natural transformation ( h i,j ) ( · ) defined by thecomposition( k − i ) ( · ) ◦ ( h i ) ( · ) ◦ ( h − j ) ( · ) ◦ ( k j ) ( · ) : ( U i,j , O ⌢ U j | U i,j ) ( · ) −→ ( U i,j , O ⌢ U i | U i,j ) ( · ) where in order to avoid heavy notations we didn’t explicitly indicate the ap-propriate restrictions. Each ( h i,j ) ( · ) is a natural isomorphism in [[ SWA , A Man ]]and, due to lemma 4.8, it gives rise to a supermanifold isomorphism h i,j : ( U i,j , O ⌢ U j | U i,j ) −→ ( U i,j , O ⌢ U i | U i,j ).The h i,j satisfy the cocycle conditions h i,i = and h i,j ◦ h j,k = h i,k (restrictedto U i ∩ U j ∩ U k ). This follows from the analogous conditions satisfied by( h i,j ) A for each A ∈ SWA . The supermanifolds ⌢ U i can hence be glued (formore information about the construction of a supermanifold by gluing seefor example [9, ch. 2] or [24, § M F the manifold thusobtained. Moreover it is clear that F is represented by the supermanifold M F . Indeed, one can check that the various ( h i ) ( · ) glue together and give anatural isomorphism h ( · ) : F → ( M F ) ( · ) . Remark 4.17.
The supermanifold M F admits a more synthetic characteri-zation. In fact it is easily seen that | M F | := |F ( K ) | and O M F ( U ) := Hom [[ SWA , A Man ]] (cid:0) F U , K | · ) (cid:1) .36 .3 The functors of Λ -points In this subsection we want to give a brief exposition of the original approach ofA. S. Shvarts and A. A. Voronov (see [23, 26]). In their work they consideredonly Grassmann algebras instead of all super Weil algebras. There are someadvantages in doing so: Grassmann algebras are many fewer, moreover, aswe noticed in remark 3.9, they are the sheaf of the super domains K | q andso the restriction to Grassmann algebras of the local functors of points canbe considered as a true restriction of the functor of points. Finally the use ofGrassmann algebras is also used by A. S. Shvarts to formalize the languagecommonly used in physics.On the other hand the use of super Weil algebras has the advantage thatwe can perform differential calculus on the Weil–Berezin functor as we shallsee in section 5. Indeed prop. 5.3 is valid only for the Weil–Berezin functorapproach, since not every point supported distribution can be obtained usingonly Grassmann algebras. Also theorem 5.5 and its consequences are validonly in this approach, since purely even Weil algebras are considered.If M is a supermanifold and Λ denotes the category of Grassmann alge-bras, we can consider the two functors Λ −→ Set Λ M Λ Λ −→ A Man Λ M Λ in place of those introduced by eq. (3.1) and eq. (4.1) respectively. As inthe case of A -points, with a slight abuse of notation we denote by M Λ theΛ-points for each of the two different functors. What we have seen previouslystill remains valid in this setting, provided we substitute systematically SWA with Λ ; in particular theorems 4.2 and 4.5 still hold true. They are based onprop. 3.20 and lemma 4.8 that we state here in their original formulation asit is contained in [26]. Proposition 4.18.
The set of natural transformations between Λ K p | q Λ and Λ K m | n Λ is in bijective correspondence with (cid:0) A p | q ( K p ) (cid:1) m × (cid:0) A p | q ( K p ) (cid:1) n .A natural transformation comes from a supermanifold morphism K p | q → K m | n if and only if it is Λ -smooth for each Grassmann algebra Λ . roof. The proof is the same as in prop. 3.20 and lemma 4.8. The onlydifference is in the first proof. Indeed the algebra (3.8) is not a Grassmannalgebra. So, if A = Λ n = Λ( ε , . . . , ε n ), we have to considerˆ A := Λ p ( n − q = Λ( η i,a , ξ i,a , ζ j )(1 ≤ i ≤ p , 1 ≤ j ≤ q , 1 ≤ a ≤ n − n -point can be written as x Λ n = u + X aa ε b k i,a,b ζ j κ j to each component. The nilpotent part of each even component of y e x Λ n can beviewed as a formal scalar product between ( η i, , . . . , η i,n − ) and ( ξ i, , . . . , ξ i,n − ).This is stable under formal rotations, where η i,a and ε b have to be thought ascoordinates in an n − η i,a and ξ i,a can occur in the image onlyas a polynomial in P a η i,a ξ i,a . In other words the image of y e x Λ n (and then of x Λ n ) is polynomial in the nilpotent part of the coordinates. In this section we want to examine Batchelor’s approach to supergeometry(see [2], Sec. 1) and relate it with what was done in the previous sections.In such an approach, a supermanifold is modelled on what is called a( r, s )-dimensional super Euclidean space E r,s , which is given a non Hausdorfftopology. Despite the apparently very different setting, the choice of an38ppropriate set of morphisms, makes the category of such objects equivalentto the category of supermanifolds as we have defined in Sec. 2.1. We shall alsosee the interpretation of Batchelor’s definition in term of our local functorsof points.Throughout this section, by a supermanifold we mean a differentiablesupermanifold , since Batchelor’s approach cannot be extended (as it is) tothe holomorphic category. Definition 4.19.
Let r , s < L be integers. We define ( r, s ) -dimensionalsuper Euclidean space E r,s to be: E r,s = Λ R ( ϑ . . . ϑ L ) r ⊕ Λ R ( ϑ . . . ϑ L ) s where Λ R ( ϑ . . . ϑ L ) as usual denotes the real exterior algebra in the variables ϑ . . . ϑ L , while Λ R ( ϑ . . . ϑ L ) r means we are taking the direct product of r copies of the even part of Λ R ( ϑ . . . ϑ L ) and similarly for Λ R ( ϑ . . . ϑ L ) s .We endow E r,s with the following topology: U ⊂ E r,s is open if and onlyif U = ε − ( V ), for V open in R r where ε : E r,s −→ R r ( u . . . u r , v . . . v s ) ( u , . . . u r )where u i is the component of Z -degree 0 of the element u i ∈ Λ R ( ϑ . . . ϑ L ) .It is clear that the topology so defined is non Hausdorff. In fact we havethat, as vector spaces, E r,s ∼ = R N , for N > r , hence if we define the topologyvia the projection ε : E r,s −→ R r , we are unable to separate points whichlie in the same fiber above the same point in R r . In other words, two points( u . . . u r , v . . . v s ), ( u ′ . . . u ′ r , v ′ . . . v ′ s ) ∈ E r,s with ( u . . . u r ) = ( u ′ . . . u r ′ )have the same neighbourhoods.Let us clarify in the next example the relation of E r,s with the Λ-pointsand more in general with the T -points of a supermanifold. Example 4.20.
Let us consider the superspace R r | s as in example 2.3 andthe T -points of R r | s for T = R | L . Notice that topologically | T | = R , that is, | T | is a point. By the Chart theorem 2.7 we have that R r | s ( T ) is in bijectivecorrespondence with the ( r | s )-uples of elements in O ( T ) r × O ( T ) s . In other39ords: R r | s ( R | L ) := Hom( R | L , R r | s ) = Hom( O ( R r | s ) , O ( R | L )) == { ϕ : C ∞ ( R r ) ⊗ Λ R ( η . . . η s ) −→ Λ R ( ϑ . . . ϑ L ) } = { ( u . . . u r , ν . . . ν s ) ∈ Λ R ( ϑ . . . ϑ L ) r × Λ R ( ϑ . . . ϑ L ) s } ∼ = E r,s . Hence we have that set-theoretically E r,s are the R | L -points of R r | s .Consequently in the language of Λ-points of the previous sections, E r,s are the Λ R ( ϑ . . . ϑ L )-points of the supermanifold R r | s .We shall now give the definition of supermanifolds according to Batchelor. Definition 4.21.
Let | M | be a topological space. We define a super-Euclideanchart on | M | as a pair ( U, ϕ ) where U ⊂ | M | and ϕ : U −→ E r,s is an home-omorphism onto its image. We say that { ( U α , ϕ α ) } is a smooth superatlason | M | if the U α form an open cover of | M | and ϕ α · ϕ − β : ϕ β ( U α ∩ U β ) −→ ϕ α ( U α ∩ U β )is a superdiffeomorphism .A B -supermanifold is a topological space together with a maximal smoothsuperatlas.In this definition we have not specified what a superdiffeomorphism is.Certainly it is a diffeomorphism in the ordinary sense (we are in R N forsome N ), but it is necessary to require further conditions in order to obtainthe correct set of arrows, that will give the equivalence of categories be-tween supermanifolds and B -supermanifolds. Given the scope of the presentpaper, we are unable to provide a characterization of the morphisms of B -supermanifolds and so to properly define the category of B -supermanifolds.Furthermore it must be noticed the role of L in the construction. For L ′ < L we obtain a subcategory of B -supermanifolds, which embeds into the cate-gory constructed above. Consequently in order to take into account all thepossible values of L it is necessary to consider a direct limit. We shall notpursue this point furtherly in our note.Despite our necessarily short treatment, the next observation will give anoverview of how this construction stands with respect to the others we haveexamined so far. 40 bservation 4.22. Let M = ( | M | , O M ) be a supermanifold defined as asuperspace together with a local model. The functor of points of M is bydefinition a functor: M ( · ) : SMan op −→ Set , M ( S ) := Hom( S, M ) . One should notice that the arriving category is
Set , which has a very simplestructure. We know that M ( · ) characterizes M , in the sense that super-manifolds and their functors of points correspond bijectively to each other(Yoneda’s Lemma). The functor of Λ-points on the other hand, weakensconsiderably the category we start from, in the sense that we are consid-ering only supermanifolds with underlying topological space made by justone point: namely those supermanifolds whose superalgebra of global sec-tions is a Grassmann algebra. If we want such a functor to characterize thesupermanifold, we are forced to increase the hypothesis on the arrival cate-gory, namely we need to ask that the set M Λ is in A Man , the category of A -manifolds, and consequently the morphisms appear to be more compli-cated as we have seen in the previous sections. In Batchelor’s approach thisstrategy reaches its limit: we are taking into exam just one Λ-point for aspecial Grassmann algebra, which has room enough to accomodate the odddimensions: this the meaning of the condition on the integer L to be greaterthan the odd dimension s . Consequently in order to achieve the equivalenceof categories, in other words, in order for this special Λ-point to completelycharacterize the supermanifold, it is necessary to add extra hypotheses on thearrival category, and this is the meaning of the complicated non Hausdorfflocal model and of the difficulty in describing the correct set of arrows, whichwe have not detailed. The Weil-Berezin functor of points represents, in ouropinion, a good compromise between the functor of points M ( · ), the functorof Λ-points and Batchelor’s approach. The morphisms appear in a naturaland reasonably simple form, while we still have not to deal with the categoryof SMan .In the next section we want to show why we believe the Weil-Berezinfunctor of points shows a definite advantage, with respect to the other equiv-alent descriptions we have examined so far, when dealing with differentialgeometry issues. 41
Applications to differential calculus
In this section we discuss some aspects of super differential calculus on super-manifolds using the language of the Weil–Berezin functor. In particular weestablish a relation between the A -points of a supermanifold M and the finitesupport distributions over it, which play a crucial role in Kostant’s seminalapproach to supergeometry.We also prove the super version of the Weil transitivity theorem, whichis a key tool for the study of the infinitesimal aspects of supermanifolds, andwe apply it in order to define the “tangent functor” of A M A . A -points In this subsection we want to introduce and discuss Kostant’s approach (see[15]) using the Weil–Berezin functor formalism.Let ( | M | , O M ) a supermanifold of dimension p | q and x ∈ | M | . As in [15, § x . Definition 5.1. If O ′ M,x is the algebraic dual of the stalk at x , the distribu-tions with support at x of order k are defined as: O k ∗ M,x := (cid:8) v ∈ O ′ M,x (cid:12)(cid:12) v (cid:0) M k +1 x (cid:1) = 0 (cid:9) where M x is, as usual, the maximal ideal of the germs of sections that arezero if evaluated at x . Clearly O k ∗ M,x ⊆ O k +1 ∗ M,x . The distributions with supportat x are given by the union O ∗ M,x := ∞ [ k =0 O k ∗ M,x . Observation 5.2.
The distributions O k ∗ M,x form a super vector space: aneven distribution is 0 on an odd germ and vice versa. If x , . . . , x p , ϑ , . . . , ϑ q are coordinates in a neighbourhood of x , a distribution of order k is of theform v = X ν ∈ N p J ⊆{ ,...,q }| ν | + | J |≤ k a ν,J ev x ∂ | ν | ∂x ν ∂ | J | ∂ϑ J a ν,J ∈ K . This is immediate since we have the following isomorphisms: O k ∗ M,x ∼ = (cid:0) O | M | ,x ⊗ Λ( ϑ , . . . , ϑ q ) (cid:1) ∗ ∼ = O ∗| M | ,x ⊗ Λ( ϑ , . . . , ϑ q ) ∗ and O ∗| M | ,x = P a ν,J ev x ∂ | ν | ∂x ν because of the classical theory. Proposition 5.3.
Let A be a super Weil algebra and A ∗ its dual. Let x A : O M,x −→ A be an A -point near x ∈ | M | . If ω ∈ A ∗ , then ω ◦ x A ∈ O ∗ M,x .Moreover each element of O k ∗ M,x can be obtained in this way with A = O M,x / M k +1 x ∼ = K [ p | q ] / M k +10 (see lemma 3.6).Proof. If A has height k , since x A ( M x ) ⊆ ◦ A , ω ◦ x A ∈ O k ∗ M,x . If vice versa v ∈ O k ∗ M,x , it factorizes through O M,x pr −→ O M,x / M k +1 x ω −→ K for a suitable ω .In the next observation we relate the finite support distributions, togetherwith their interpretation via the Weil–Berezin functor, with the tangent su-perspace. Observation 5.4.
Let us first recall that the tangent superspace to a su-permanifold M at a point x is the super vector space consisting of all thederivations of the stalk at x : T x ( M ) := { v : O M,x −→ K | v is a derivation } .As in the classical setting we can recover the tangent space by using the al-gebra of super dual numbers . Let us consider A = K ( e, ε ) = K [ e, ε ] / h e , eε, ε i be the super Weil algebra of super dual numbers (see example 3.4). If x A ∈ M A,x and s, t ∈ O
M,x , we have x A ( st ) = ev x ( st ) + x e ( st ) e + x ε ( st ) ε x e , x ε : O M,x → K . On the other hand x A ( st ) = x A ( s ) x A ( t )= ev x ( s )ev x ( t ) + (cid:0) x e ( s )ev x ( t ) + ev x ( s ) x e ( t ) (cid:1) e + (cid:0) x ε ( s )ev x ( t ) + ev x ( s ) x ε ( s ) (cid:1) ε .Then x e (resp. x ε ) is a derivation of the stalk that is zero on odd (resp. even)elements and so x e ∈ T x ( M ) (resp. x ε ∈ T x ( M ) ). The map T ( M ) := G x ∈| M | T x ( M ) −→ M K ( e,ε ) v + v ev x + v e + v ε (with v i ∈ T x ( M ) i ) is an isomorphism of vector bundles over f M ∼ = M K ,where f M is the classical manifold associated with M , as in subsection 2.1(see also [14, ch. 8] for an exhaustive exposition in the classical case). Thereader should not confuse T ( M ), which is the classical bundle obtained bythe union of all the tangent superspaces at the different points of | M | , with T M which is the super vector bundle of all the derivations of O M . We now want to give a brief account on how we can perform differentialcalculus using the language of A -points. The essential ingredient is the superversion of the transitivity theorem that we discuss below.In the following, when classical smooth (resp. holomorphic) manifolds areconsidered, O denotes the corresponding sheaf of smooth (resp. holomorphic)functions. Theorem 5.5 (Weyl transitivity theorem).
Let M be a smooth (resp.holomorphic) supermanifold, A a super Weil algebra and B a purely evenWeil algebra, both real (resp. complex). Then ( M A ) B ∼ = M A ⊗ B as ( A ⊗ B ) -manifolds. roof. Let O M A and O AM A be the sheaves of smooth (resp. holomorphic) mapsfrom the classical manifold M A to K and A respectively. Clearly O AM A ∼ = A ⊗O M A through the map f P i a i ⊗ (cid:10) a ∗ i , f (cid:11) , where { a i } is a homogeneousbasis of A .If x A ∈ M A , let τ x A : O M, e x A −→ O AM A ,x A ∼ = A ⊗ O M A ,x A [ s ] e x A [ˆ s ] x A where, if s ∈ O M ( U ) and e x A ∈ U ,ˆ s : y A y A ( s )for all y A ∈ M A such that e y A ∈ U (it is not difficult to show that this mapdescends to a map between stalks).Recalling that ( M A ) B := G x A ∈ M A Hom
SAlg ( O M A ,x A , B ) M A ⊗ B := G x ∈| M | Hom
SAlg ( O M,x , A ⊗ B ),we can define a map ξ : ( M A ) B −→ M A ⊗ B X ξ ( X )setting ξ ( X ) : [ s ] ee X ( A ⊗ X ) τ e X ([ s ] ee X ).This definition is well-posed since ξ ( X ) is a superalgebra map, as one caneasily check.Fix now a chart ( U, h ), h : U → K p | q , in M and denote by ( U A , h A ), (cid:0) ( U A ) B , ( h A ) B (cid:1) and ( U A ⊗ B , h A ⊗ B ) the corresponding charts lifted to M A ,( M A ) B and M A ⊗ B respectively. If { e , . . . , e p + q } is a homogeneous basisof K p | q , we have (here, according to observation 3.16, we tacitly use theidentification K p | qA ∼ = ( A ⊗ K p | q ) )( h A ) B : ( U A ) B −→ ( A ⊗ B ⊗ K p | q ) X X i,j a i ⊗ X (cid:0) h ∗ A ( a ∗ i ⊗ e ∗ j ) (cid:1) ⊗ e j h A ⊗ B : U A ⊗ B −→ ( A ⊗ B ⊗ K p | q ) Y X k Y (cid:0) h ∗ ( e ∗ k ) (cid:1) ⊗ e k .Then, since ξ ( X ) (cid:0) h ∗ ( e ∗ k ) (cid:1) = ( ⊗ X ) (cid:0) \ h ∗ ( e ∗ k ) (cid:1) = ( ⊗ X ) (cid:0) P i a i ⊗ h ∗ A ( a ∗ i ⊗ e ∗ k ) (cid:1) ,we have h A ⊗ B ◦ ξ ◦ ( h A ) − B = ( h A ) B (( U A ) B ) .This entails in particular that ξ is a local ( A ⊗ B )-diffeomorphism. Thefact that it is a global diffeomorphism follows noticing that it is fibered overthe identity, being ξ ( X ) ∈ M A ⊗ B , ee X .From now on we assume all supermanifolds to be smooth.We want to briefly explain some applications of the Weil transitivity the-orem to the smooth category. Let M be a smooth supermanifold and let A be a real super Weil algebra. As we have seen in the previous section, M A has a natural structure of classical smooth manifold and, due to prop. 3.15,we can identify M A with the space of superalgebra maps O ( M ) → A . Definition 5.6. If x A ∈ M A , we define the space of x A -linear derivations of M ( x A -derivations for short) as the A -module Der x A (cid:0) O ( M ) , A (cid:1) := n X ∈ Hom (cid:0) O ( M ) , A (cid:1) (cid:12)(cid:12)(cid:12) ∀ s, t ∈ O ( M ) ,X ( st ) = X ( s ) x A ( t ) + ( − p ( X ) p ( s ) x A ( s ) X ( t ) o . Proposition 5.7.
The tangent superspace at x A in M A canonically identifieswith Der x A (cid:0) O ( M ) , A (cid:1) . We recall that if V = V ⊕ V and W = W ⊕ W are super vector spaces thenHom( V, W ) denotes the super vector space of all linear morphisms between V and W with the gradation Hom( V, W ) := Hom( V , W ) ⊕ Hom( V , W ), Hom( V, W ) :=Hom( V , W ) ⊕ Hom( V , W ). roof. If R ( e ) is the algebra of dual number (see example 3.3), ( M A ) R ( e ) isisomorphic, as a vector bundle, to the tangent bundle T ( M A ), as we haveseen in observation 5.4. Due to theorem 5.5, we thus have an isomorphism ξ : T ( M A ) ∼ = ( M A ) R ( e ) −→ M A ⊗ R ( e ) .On the other hand, it is easy to see that x A ⊗ R ( e ) ∈ M A ⊗ R ( e ) can be writtenas x A ⊗ R ( e ) = x A ⊗ v x A ⊗ e , where x A ∈ M A and v x A : O ( M ) → A is aparity preserving map satisfying the following rule for all s, t ∈ O ( M ): v x A ( st ) = v x A ( s ) x A ( t ) + x A ( s ) v x A ( t ).Then each tangent vector on M A at x A canonically identifies a even x A -deriva-tion and, vice versa, each such derivation canonically identifies a tangentvector at x A .We conclude studying more closely the structure of Der x A (cid:0) O ( M ) , A (cid:1) . Thefollowing proposition describes it explicitly.Let K be a right A -module and let L be a left B -module for some algebras A and B . Suppose moreover that an algebra morphism ρ : B → A is given.One defines the ρ -tensor product K ⊗ ρ L as the quotient of the vector space K ⊗ L with respect to the equivalence relation k ⊗ b · l ∼ k · ρ ( b ) ⊗ l for all k ∈ K , l ∈ L and b ∈ B .Moreover, if M is a supermanifold, we denote by T M the super tangentbundle of M , i. e. the sheaf defined by T M := Der( O M ). Proposition 5.8.
Let M be a smooth supermanifold and let x ∈ | M | . De-note T M,x the germs of vector fields at x . One has the identification of left A -modules Der x A (cid:0) O ( M ) , A (cid:1) ∼ = A ⊗ T e x A ( M ) ∼ = A ⊗ x A T M, e x A . This result is clearly local so that it is enough to prove it in the case M isa superdomain. Next lemma do this for the first identification. The seconddescends from eq. (5.1), since T M, e x A = O M, e x A ⊗ T e x A ( M ).47 emma 5.9. Let U be a superdomain in R p | q with coordinate system { x i , ϑ j } , A a super Weil algebra and x A ∈ U A . To any list of elements f = ( f , . . . , f p , F , . . . , F q ) f i , F j ∈ A there corresponds a x A -derivation X f : O ( U ) −→ A given by X f ( s ) = X i f i x A (cid:18) ∂s∂x i (cid:19) + X j F j x A (cid:18) ∂s∂ϑ j (cid:19) . (5.1) X f is even (resp. odd) if and only if the f i are even (resp. odd) and the F j areodd (resp. even). Moreover any x A -derivation is of this form for a uniquelydetermined f .Proof. That X f is a x A -derivation is clear. That the family f is uniquelydetermined is also immediate from the fact that they are the value of X f onthe coordinate functions.Let now X be a generic x A -derivation. Define f i = X ( x i ), F j = X ( ϑ j ),and X f = f i x A ◦ ∂∂x i + F j x A ◦ ∂∂ϑ j .Let D = X − X f . Clearly D ( x i ) = D ( ϑ j ) = 0. We now show that thisimplies D = 0. Let s ∈ O ( U ). Due to lemma 2.8, for each x ∈ U and foreach integer k ∈ N there exists a polynomial P in the coordinates such that[ s ] x − [ P ] x ∈ M k +1 x . Due to Leibniz rule D ( s − P ) ∈ ◦ A k and, since clearly D ( P ) = 0, D ( s ) is in ◦ A k for arbitrary k . So we are done.The previous result gives the following corollary. Corollary 5.10.
We have the identification T x A M A ∼ = (cid:0) A ⊗ T e x A ( M ) (cid:1) ∼ = (cid:0) A ⊗ x A T M, e x A (cid:1) . eferences [1] L. Balduzzi, C. Carmeli, R. Fioresi. The local functors of points of su-permanifolds . Exp. Math. Vol. 28(3):201–217, 2010.[2] Marjorie Batchelor. Two approaches to supermanifolds.
Trans. Amer.Math. Soc. , 258(1):257–270, 1980.[3] C. Bartocci, U. Bruzzo, and D. Hern´andez Ruip´erez.
The geometry ofsupermanifolds , volume 71 of
Mathematics and its Applications . KluwerAcademic Publishers Group, Dordrecht, 1991.[4] F. A. Berezin.
Introduction to superanalysis , volume 9 of
MathematicalPhysics and Applied Mathematics . D. Reidel Publishing Co., Dordrecht,1987. Edited and with a foreword by A. A. Kirillov, With an appendixby V. I. Ogievetsky, Translated from the Russian by J. Niederle and R.Koteck´y, Translation edited by Dimitri Le˘ıtes.[5] F. A. Berezin and D. A. Le˘ıtes. Supermanifolds.
Dokl. Akad. NaukSSSR , 224(3):505–508, 1975.[6] C. Carmeli, L. Caston and R. Fioresi
Mathematical Foundation of Su-persymmetry , with an appendix with I. Dimitrov, EMS Ser. Lect. Math.,European Math. Soc., Zurich 2011.[7] B. DeWitt.
Supermanifolds . Cambridge Monographs on MathematicalPhysics. Cambridge University Press, Cambridge, 1984.[8] M. Demazure and P. Gabriel.
Groupes alg´ebriques. Tome I: G´eom´etriealg´ebrique, g´en´eralit´es, groupes commutatifs . Masson & Cie, ´Editeur,Paris, 1970. Avec un appendice ıt Corps de classes local par MichielHazewinkel.[9] P. Deligne and J. W. Morgan. Notes on supersymmetry (followingJoseph Bernstein). In
Quantum fields and strings: a course for math-ematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997) , pages 41–97. Amer.Math. Soc., Providence, RI, 1999.[10] D. Eisenbud and J. Harris.
The geometry of schemes , volume 197 of
Graduate Texts in Mathematics . Springer-Verlag, New York, 2000.4911] R. Fioresi, M. A. Lled´o, and V. S. Varadarajan. The Minkowski andconformal superspaces.
J. Math. Phys. , 48(11):113505, 27, 2007.[12] L.H¨ormander.
An introduction to complex analysis in several variables.Third edition . North-Holland Mathematical Library, 7. North-HollandPublishing Co., Amsterdam, 1990[13] L.Kaup, B.Kaup.
Holomorphic functions of several variables. An intro-duction to the fundamental theory. . de Gruyter Studies in Mathematics,3. Walter de Gruyter Co., Berlin, 1983.[14] I. Kol´aˇr, P. W. Michor, and J. Slov´ak.
Natural operations in differentialgeometry . Springer-Verlag, Berlin, 1993.[15] B. Kostant. Graded manifolds, graded Lie theory, and prequantization.In
Differential geometrical methods in mathematical physics (Proc. Sym-pos., Univ. Bonn, Bonn, 1975) , pages 177–306. Lecture Notes in Math.,Vol. 570. Springer, Berlin, 1977.[16] J.-L. Koszul. Differential forms and near points on graded manifolds.In
Symplectic geometry (Toulouse, 1981) , volume 80 of
Res. Notes inMath. , pages 55–65. Pitman, Boston, MA, 1983.[17] D. A. Le˘ıtes. Introduction to the theory of supermanifolds.
UspekhiMat. Nauk , 35(1(211)):3–57, 255, 1980.[18] S. MacLane.
Categories for the working mathematician . Springer-Verlag,New York, 1971. Graduate Texts in Mathematics, Vol. 5.[19] Yu. I. Manin.
Gauge field theory and complex geometry , volume 289 of
Grundlehren der Mathematischen Wissenschaften [Fundamental Prin-ciples of Mathematical Sciences] . Springer-Verlag, Berlin, 1988. Trans-lated from the Russian by N. Koblitz and J. R. King.[20] A. Rogers. A global theory of supermanifolds.
J. Math. Phys. ,21(6):1352–1365, 1980.[21] A. Rogers.
Supermanifolds . World Scientific Publishing Co. Pte. Ltd.,Hackensack, NJ, 2007. Theory and applications.5022] V. V. Shurygin. The structure of smooth mappings over Weil algebrasand the category of manifolds over algebras.
Lobachevskii J. Math. ,5:29–55 (electronic), 1999.[23] A. S. Shvarts. On the definition of superspace.
Teoret. Mat. Fiz. ,60(1):37–42, 1984.[24] V. S. Varadarajan.
Supersymmetry for mathematicians: an introduc-tion , volume 11 of
Courant Lecture Notes in Mathematics . New YorkUniversity Courant Institute of Mathematical Sciences, New York, 2004.[25] A. Vistoli. Notes on Grothendieck topologies, fibered categories anddescent theory. Preprint, arXiv:math/0412512 , 2007.[26] A. Voronov. Maps of supermanifolds.
Teoret. Mat. Fiz. , 60(1):43–48,1984.[27] A. Weil. Th´eorie des points proches sur les vari´et´es diff´erentiables. In