The concept of super-scheme is the generalization of that of scheme from commutative algebra to supercommutative superalgebra: where an ordinary scheme is a space locally modeled on the formal dual of a commutative ring (i.e. an affine scheme), so a superscheme is a space locally modeled on the formal dual of a supercommutative algebra (i.e. on affine superschemes, def. below).

Affine superschemes

The following is a detailed and conceptual introduction of the concept of affine super-schemes. It is taken from geometry of physics – superalgebra.

The key idea of supercommutative superalgebra is that it is nothing but plain commutative algebra but “internalized” not in ordinary vector spaces, but in super vector spaces. This is made precise by def. and ef. below.

The key idea then of supergeometry is to define super-spaces to be spaces whose algebras of functions are supercommutative superalgebras. This is not the case for any “ordinary” space such as a topological space or a smooth manifold. But these spaces may be characterized dually via their algebras of functions, and hence it makes sense to generalize the latter.

For smooth manifolds the duality statement is the following:


(embedding of smooth manifolds into formal duals of R-algebras)

The functor

C ():SmoothMfdAlg op C^\infty(-) \;\colon\; SmoothMfd \longrightarrow Alg_{\mathbb{R}}^{op}

which sends a smooth manifold (finite dimensional, paracompact, second countable) to (the formal dual of) its \mathbb{R}-algebra of smooth functions is a full and faithful functor.

In other words, for two smooth manifolds X,YX,Y there is a natural bijection between the smooth functions XYX \to Y and the \mathbb{R}-algebra homomorphisms C (X)C (Y)C^\infty(X)\leftarrow C^\infty(Y).

A proof is for instance in (Kolar-Slovak-Michor 93, lemma 35.8, corollaries 35.9, 35.10).

This says that we may identify smooth manifolds as the “formal duals” of certain associative algebras, namely those in the image of the above full embedding. Accordingly then, any larger class of associative algebras than this may be thought of as the class of formal duals to a generalized kind of manifold, defined thereby. Given any associative algebra AA, then we may think of it as representing a space Spec(A)Spec(A) which is such that it has AA as its algebra of functions.

This duality between certain spaces and their algebras of functions is profound. In physics it has always been used implicitly, in fact it was so ingrained into theoretical physics that it took much effort to abstract away from coordinate functions to discover global Riemannian geometry in the guise of“general relativity”. As mathematics, an early prominent duality theorem is Gelfand duality (between topological spaces and C*-algebras) which served as motivation for the very definition of algebraic geometry, where affine schemes are nothing but the formal duals of commutative rings/commutative algebras. Passing to non-commutative algebras here yields non-commutative geometry, and so forth. In great generality this duality between spaces and their function algebras appears as “Isbell duality” between presheaves and copresheaves.

In supergeometry we are concerned with spaces that are formally dual to associative algebras which are “very mildly” non-commutative, namely supercommutative superalgebras. These are in fact commutative algebras when viewed internal to super vector spaces (def. below). The corresponding formal dual spaces are, depending on some technical details, super schemes or supermanifolds. In the physics literature, such spaces are usually just called superspaces.

We now make this precise.


Given a monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def ), then a monoid internal to (𝒞,,1)(\mathcal{C}, \otimes, 1) is

  1. an object A𝒞A \in \mathcal{C};

  2. a morphism e:1Ae \;\colon\; 1 \longrightarrow A (called the unit)

  3. a morphism μ:AAA\mu \;\colon\; A \otimes A \longrightarrow A (called the product);

such that

  1. (associativity) the following diagram commutes

    (AA)A a A,A,A A(AA) Aμ AA μA μ AA μ A, \array{ (A\otimes A) \otimes A &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}& A \otimes (A \otimes A) &\overset{A \otimes \mu}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{\mu \otimes A}}\downarrow && && \downarrow^{\mathrlap{\mu}} \\ A \otimes A &\longrightarrow& &\overset{\mu}{\longrightarrow}& A } \,,

    where aa is the associator isomorphism of 𝒞\mathcal{C};

  2. (unitality) the following diagram commutes:

    1A eid AA ide A1 μ r A, \array{ 1 \otimes A &\overset{e \otimes id}{\longrightarrow}& A \otimes A &\overset{id \otimes e}{\longleftarrow}& A \otimes 1 \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\mu}} & & \swarrow_{\mathrlap{r}} \\ && A } \,,

    where \ell and rr are the left and right unitor isomorphisms of 𝒞\mathcal{C}.

Moreover, if (𝒞,,1)(\mathcal{C}, \otimes , 1) has the structure of a symmetric monoidal category (def. ) (𝒞,,1,τ)(\mathcal{C}, \otimes, 1, \tau) with symmetric braiding τ\tau, then a monoid (A,μ,e)(A,\mu, e) as above is called a commutative monoid in (𝒞,,1,B)(\mathcal{C}, \otimes, 1, B) if in addition

  • (commutativity) the following diagram commutes

    AA τ A,A AA μ μ A. \array{ A \otimes A && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} && A \otimes A \\ & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}} \\ && A } \,.

A homomorphism of monoids (A 1,μ 1,e 1)(A 2,μ 2,f 2)(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2) is a morphism

f:A 1A 2 f \;\colon\; A_1 \longrightarrow A_2

in 𝒞\mathcal{C}, such that the following two diagrams commute

A 1A 1 ff A 2A 2 μ 1 μ 2 A 1 f A 2 \array{ A_1 \otimes A_1 &\overset{f \otimes f}{\longrightarrow}& A_2 \otimes A_2 \\ {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}} \\ A_1 &\underset{f}{\longrightarrow}& A_2 }


1 𝒸 e 1 A 1 e 2 f A 2. \array{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,.

Write Mon(𝒞,,1)Mon(\mathcal{C}, \otimes,1) for the category of monoids in 𝒞\mathcal{C}, and CMon(𝒞,,1)CMon(\mathcal{C}, \otimes, 1) for its subcategory of commutative monoids.


A monoid object according to def. in the monoidal category of vector spaces from example is equivalently an ordinary associative algebra over the given ground field. Similarly a commutative monoid in VectVect is an ordinary commutative algebra. Moreover, in both cases the homomorphisms of monoids agree with usual algebra homomorphisms. Hence there are equivalences of categories.

Mon(Vect k)Alg k Mon(Vect_k) \simeq Alg_k
CMon(Vect k)CAlg k. CMon(Vect_k) \simeq CAlg_k \,.

For GG a group, then a GG-graded associative algebra is a monoid object according to def. in the monoidal category of GG-graded vector spaces from example .

Alg k GMon(Vect k G). Alg_k^G \simeq Mon(Vect_k^G) \,.

This means that a GG-graded algebra is

  1. a GG-graded vector space A=gGA gA = \underset{g\in G}{\oplus} A_g

  2. an associative algebra structure on the underlying vector space AA

such that for two elements of homogeneous degree, i.e. a 1A g 1Aa_1 \in A_{g_1} \hookrightarrow A and a 2A g 2Aa_2 \in A_{g_2} \hookrightarrow A then their product is in degre g 1g 2g_1 g_2

a g 1a g 2A g 1g 2A. a_{g_1} a_{g_2} \in A_{g_1 g_2} \hookrightarrow A \,.

Example motivates the following definition:


A supercommutative superalgebra is a commutative monoid (def. ) in the symmetric monoidal category of super vector spaces (def. ). We write sCAlg ksCAlg_k for the category of supercommutative superalgebras with the induced homomorphisms between them:

sCAlg kCMon(sVect k). sCAlg_k \;\coloneqq\; CMon(sVect_k) \,.

Unwinding what this means, then a supercommutative superalgebra AA is

  1. a /2\mathbb{Z}/2-graded associative algebra according to example ;

  2. such that for any two elements a,ba, b of homogeneous degree, their product satisfies

    ab=(1) deg(a)deg(b)ba. a b \; = \; (-1)^{deg(a) deg(b)}\, b a \,.

In view of def. we might define a not-necessarily supercommutative superalgebra to be a monoid (not necessarily commutative) in sVect, and write

sAlg kMon(sVect). sAlg_k \coloneqq Mon(sVect) \,.

However, since the definition of not-necessarily commutative monoids (def. ) does not invoke the braiding of the ambient tensor category, and since super vector spaces differ from /2\mathbb{Z}/2-graded vector spaces only via their braiding (example ), this yields equivalently just the /2\mathbb{Z}/2-graded algebras froom example :

sAlg kAlg k /2. sAlg_k \simeq Alg_k^{\mathbb{Z}/2} \,.

Hence the heart of superalgebra is super-commutativity.


The supercommutative superalgebra which is freely generated over kk from nn generators {θ i} i=1 n\{\theta_i\}_{i = 1}^n is the quotient of the tensor algebra T nT^\bullet \mathbb{R}^n, with the generators θ i\theta_i in odd degree, by the ideal generated by the relations

θ iθ j=θ jθ i \theta_i \theta_j = - \theta_j \theta_i

for all i,j{1,,n}i,j \in \{1, \cdots, n\}.

This is also called a Grassmann algebra, in honor of (Grassmann 1844), who introduced and studied the super-sign rule in def. a century ahead of his time.

We also denote this algebra by

( n)sCAlg . \wedge^\bullet_{\mathbb{R}}(\mathbb{R}^n) \;\in\; sCAlg_{\mathbb{R}} \,.

Given a homotopy commutative ring spectrum EE (i.e., via the Brown representability theorem, a multiplicative generalized cohomology theory), then its stable homotopy groups π (E)\pi_\bullet(E) inherit the structure of a super-commutative ring.

See at Introduction to Stable homotopy theory in the section 1-2 Homotopy commutative ring spectra this proposition.

The following is an elementary but fundamental fact about the relation between commutative algbra and supercommutative superalgebra. It is implicit in much of the literature, but maybe the only place where it has been made explicit before is (Carchedi-Roytenberg 12, example 3.18).


There is a full subcategory inclusion

CAlg k sCAlg k = = CMon(Vect k) CMon(sVect k) \array{ CAlg_k &\hookrightarrow& sCAlg_k \\ = && = \\ CMon(Vect_k) &\hookrightarrow& CMon(sVect_k) }

of commutative algebras (example ) into supercommutative superalgebras (def. ) induced via prop. from the full inclusion

i:VectksVect k i \;\colon\; Vectk \hookrightarrow sVect_k

of vector spaces (def. ) into super vector spaces (def. ), which is a braided monoidal functor by prop. . Hence this regards a commutative algebra as a superalgebra concentrated in even degree.

This inclusion functor has both a left adjoint functor and a right adjoint functor , (an adjoint triple exibiting a reflective subcategory and coreflective subcategory inclusion, an “adjoint cylinder”):

CAlg k() even()/() oddsCAlg k. CAlg_k \underoverset {\underset{(-)_{even}}{\longleftarrow}} {\overset{(-)/(-)_{odd}}{\longleftarrow}} {\hookrightarrow} sCAlg_k \,.


  1. the right adjoint () even(-)_{even} sends a supercommutative superalgebra to its even part AA evenA \mapsto A_{even};

  2. the left adjoint ()/() even(-)/(-)_{even} sends a supercommutative superalgebra to the quotient by the ideal which is generated by its odd part AA/(A odd)A \mapsto A/(A_{odd}) (hence it sets all elements to zero which may be written as a product such that at least one factor is odd-graded).


The full inclusion ii is evident. To see the adjunctions observe their characteristic natural bijections between hom-sets: If A ordinaryA_{ordinary} is an ordinary commutative algebra regarded as a superalgeba i(A ordinary)i(A_{ordinary}) concentrated in even degree, and if BB is any superalgebra,

  1. then every super-algebra homomorphism of the form A ordinaryBA_{ordinary} \to B must factor through B evenB_{even}, simply because super-algebra homomorpism by definition respect the /2\mathbb{Z}/2-grading. This gives a natual bijection

    Hom sCAlg k(i(A ordinary),B)Hom CAlg k(A ordinary,B even), Hom_{sCAlg_k}(i(A_{ordinary}), B) \simeq Hom_{CAlg_k}(A_{ordinary,B_{even}}) \,,
  2. every super-algebra homomorphism of the form Bi(A ordinary)B \to i(A_{ordinary}) must send every odd element of BB to 0, again because homomorphism have to respect the /2\mathbb{Z}/2-grading, and since homomorphism of course also preserve products, this means that the entire ideal generated by B oddB_{odd} must be sent to zero, hence the homomorphism must facto through the projection BB/B oddB \to B/B_{odd}, which gives a natural bijection

    Hom sCalg k(B,i(A ordinary))Hom Alg k(B/B odd,A ordinary). Hom_{sCalg_k}(B, i(A_{ordinary})) \simeq Hom_{Alg_k}(B/B_{odd}, A_{ordinary}) \,.

It is useful to make explicit the following formally dual perspective on supercommutative superalgebras:


For 𝒞\mathcal{C} a symmetric monoidal category, then we write

Aff(𝒞)CMon(𝒞) op Aff(\mathcal{C}) \coloneqq CMon(\mathcal{C})^{op}

for the opposite category of the category of commutative monoids in 𝒞\mathcal{C}, according to def. .

For RCMon(𝒞)R \in CMon(\mathcal{C}) we write

Spec(A)Aff(𝒞) Spec(A) \in Aff(\mathcal{C})

for the same object, regarded in the opposite category. We also call this the affine scheme of AA. Conversely, for XAff(𝒞)X \in Aff(\mathcal{C}), we write

𝒪(X)CMon(𝒞) \mathcal{O}(X) \in CMon(\mathcal{C})

for the same object, regarded in the category of commutative monoids. We also call this the algebra of functions on XX.


For the special case that mathalC=\mathal{C} = sVect (def. ) in def. , then we say that the objects in

Aff(sVect k)=scAlg k op=CMon(sVect k) op Aff(sVect_k) = scAlg_k^{op} = CMon(sVect_k)^{op}

are affine super schemes over kk.


For ACAlg A \in CAlg_{\mathbb{R}} an ordinary commutative algebra over \mathbb{R}, then of course this becomes a supercommutative superalgebra by regarding it as being concentrated in even degrees. Accordingly, via def. , ordinary affine schemes fully embed into affine super schemes (def. )

Aff(Vect k)Aff(sVect k). Aff(Vect_k) \hookrightarrow Aff(sVect_k) \,.

In particular for p\mathbb{R}^p an ordinary Cartesian space, this becomes an affine superscheme in even degree, under the above embedding. As such, it is usually written

p|0Aff(sVect k). \mathbb{R}^{p \vert 0} \in Aff(sVect_k) \,.

The formal dual space, according to def. (example ) to a Grassmann algebra ( q)\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q) (example ) is to be thought of as a space which is “so tiny” that the coefficients of the Taylor expansion of any real-valued function on it become “so very small” as to be actually equal to zero, at least after the qq-th power.

For instance for q=2q = 2 then a general element of ( q)\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q) is of the form

f=a 0+a 1θ 1+a 2θ 2+a 12θ 1θ 2 ( q). f = a_0 + a_1 \theta_1 + a_2 \theta_2 + a_{12} \theta_1 \theta_2 \;\;\;\in \wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q) \,.

for a 1,a 2,a 12a_1,a_2, a_{12} \in \mathbb{R}, to be compared with the Taylor expansion of a smooth function g: 2g \colon \mathbb{R}^2 \to \mathbb{R}, which is of the form

g(x 1,x 2)=g(0)+gx 1(0)x 1+gx 2(0)x 2+ 2gx 1x 2(0)x 1x 2+. g(x_1, x_2) = g(0) + \frac{\partial g}{\partial x_1}(0)\, x_1 + \frac{\partial g}{\partial x_2}(0)\, x_2 + \frac{\partial^2 g}{\partial x_1 \partial x_2}(0) \, x_1 x_2 + \cdots \,.

Therefore the formal dual space to a Grassmann algebra behaves like an infinitesimal neighbourhood of a point. Hence these are also called superpoints and one writes

0|qSpec( ( q)). \mathbb{R}^{0\vert q} \coloneqq Spec(\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q)) \,.

Combining example with example , and using prop. , we obtain the affine super schemes

p|q p|0× 0|qSpec(C ( p) q). \mathbb{R}^{p \vert q} \coloneqq \mathbb{R}^{p\vert 0} \times \mathbb{R}^{0\vert q} \simeq Spec\left( \underbrace{C^\infty(\mathbb{R}^p)} \otimes_{\mathbb{R}} \wedge^\bullet_{\mathbb{R}} \mathbb{R}^q \right) \,.

These may be called the super Cartesian spaces. The play the same role in the theory of supermanifolds as the ordinary Cartesian spaces do for smooth manifolds. See at geometry of physics – supergeometry for more on this.


Given a supercommutative superalgebra AA (def. ), its parity involution is the algebra automorphism

par:AA par \;\colon\; A \overset{\simeq}{\longrightarrow} A

which on homogeneously graded elements aa of degree deg(a){even,odd}=/2deg(a) \in \{even,odd\} = \mathbb{Z}/2\mathbb{Z} is multiplication by the degree

a(1) deg(a)a. a \mapsto (-1)^{deg(a)}a \,.

(e.g. arXiv:1303.1916, 7.5)

Dually, via def. , this means that every affine super scheme has a canonical involution.

Here are more general and more abstract examples of commutative monoids, which will be useful to make explicit:


Given a monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), then the tensor unit 11 is a monoid in 𝒞\mathcal{C} (def. ) with product given by either the left or right unitor

1=r 1:111. \ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1 \,.

By lemma , these two morphisms coincide and define an associative product with unit the identity id:11id \colon 1 \to 1.

If (𝒞,,1)(\mathcal{C}, \otimes , 1) is a symmetric monoidal category (def. ), then this monoid is a commutative monoid.


Given a symmetric monoidal category (𝒞,,1)(\mathcal{C}, \otimes, 1) (def. ), and given two commutative monoids (E i,μ i,e i)(E_i, \mu_i, e_i), i{1,2}i \in \{1,2\} (def. ), then the tensor product E 1E 2E_1 \otimes E_2 becomes itself a commutative monoid with unit morphism

e:111e 1e 2E 1E 2 e \;\colon\; 1 \overset{\simeq}{\longrightarrow} 1 \otimes 1 \overset{e_1 \otimes e_2}{\longrightarrow} E_1 \otimes E_2

(where the first isomorphism is, 1 1=r 1 1\ell_1^{-1} = r_1^{-1} (lemma )) and with product morphism given by

E 1E 2E 1E 2idτ E 2,E 1idE 1E 1E 2E 2μ 1μ 2E 1E 2 E_1 \otimes E_2 \otimes E_1 \otimes E_2 \overset{id \otimes \tau_{E_2, E_1} \otimes id}{\longrightarrow} E_1 \otimes E_1 \otimes E_2 \otimes E_2 \overset{\mu_1 \otimes \mu_2}{\longrightarrow} E_1 \otimes E_2

(where we are notationally suppressing the associators and where τ\tau denotes the braiding of 𝒞\mathcal{C}).

That this definition indeed satisfies associativity and commutativity follows from the corresponding properties of (E i,μ i,e i)(E_i,\mu_i, e_i), and from the hexagon identities for the braiding (def. ) and from symmetry of the braiding.

Similarly one checks that for E 1=E 2=EE_1 = E_2 = E then the unit maps

EE1ideEE E \simeq E \otimes 1 \overset{id \otimes e}{\longrightarrow} E \otimes E
E1Ee1EE E \simeq 1 \otimes E \overset{e \otimes 1}{\longrightarrow} E \otimes E

and the product map

μ:EEE \mu \;\colon\; E \otimes E \longrightarrow E

and the braiding

τ E,E:EEEE \tau_{E,E} \;\colon\; E \otimes E \longrightarrow E \otimes E

are monoid homomorphisms, with EEE \otimes E equipped with the above monoid structure.

Monoids are preserved by lax monoidal functors:


Let (𝒞, 𝒞,1 𝒞)(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}}) and (𝒟, 𝒟,1 𝒟)(\mathcal{D}, \otimes_{\mathcal{D}},1_{\mathcal{D}}) be two monoidal categories (def. ) and let F:𝒞𝒟F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} be a lax monoidal functor (def. ) between them.

Then for (A,μ A,e A)(A,\mu_A,e_A) a monoid in 𝒞\mathcal{C} (def. ), its image F(A)𝒟F(A) \in \mathcal{D} becomes a monoid (F(A),μ F(A),e F(A))(F(A), \mu_{F(A)}, e_{F(A)}) by setting

μ F(A):F(A) 𝒞F(A)F(A 𝒞A)F(μ A)F(A) \mu_{F(A)} \;\colon\; F(A) \otimes_{\mathcal{C}} F(A) \overset{}{\longrightarrow} F(A \otimes_{\mathcal{C}} A) \overset{F(\mu_A)}{\longrightarrow} F(A)

(where the first morphism is the structure morphism of FF) and setting

e F(A):1 𝒟F(1 𝒞)F(e A)F(A) e_{F(A)} \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}}) \overset{F(e_A)}{\longrightarrow} F(A)

(where again the first morphism is the corresponding structure morphism of FF).

This construction extends to a functor

Mon(F):Mon(𝒞, 𝒞,1 𝒞)Mon(𝒟, 𝒟,1 𝒟) Mon(F) \;\colon\; Mon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}}) \longrightarrow Mon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}})

from the category of monoids of 𝒞\mathcal{C} (def. ) to that of 𝒟\mathcal{D}.

Moreover, if 𝒞\mathcal{C} and 𝒟\mathcal{D} are symmetric monoidal categories (def. ) and FF is a braided monoidal functor (def. ) and AA is a commutative monoid (def. ) then so is F(A)F(A), and this construction extends to a functor

CMon(F):CMon(𝒞, 𝒞,1 𝒞)CMon(𝒟, 𝒟,1 𝒟). CMon(F) \;\colon\; CMon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}}) \longrightarrow CMon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}}) \,.

This follows immediately from combining the associativity and unitality (and symmetry) constraints of FF with those of AA.


General accounts include

  • Carmeli-Caston-Fioresi: Mathematical foundations of supersymmetry, EMS Series of Lectures in Mathematics, EMS 2011

  • Mikhail KapranovEric Vasserot, Supersymmetry and the formal loop space, Advances in Math. 227 (2011), 1078-1128 (arXiv:1005.4466)

Discussion of crystalline cohomology of super-schemes:

  • Martin Luu, Chrystalline cohomology of superschemes (pdf)

Last revised on March 3, 2017 at 03:37:20. See the history of this page for a list of all contributions to it.