superalgebra and (synthetic ) supergeometry
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).
The following is a detailed and conceptual introduction of the concept of affine super-schemes. This 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
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 $X,Y$ there is a natural bijection between the smooth functions $X \to Y$ and the $\mathbb{R}$-algebra homomorphisms $C^\infty(X)\leftarrow C^\infty(Y)$.
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 $A$, then we may think of it as representing a space $Spec(A)$ which is such that it has $A$ 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 $(\mathcal{C}, \otimes, 1)$ (def ), then a monoid internal to $(\mathcal{C}, \otimes, 1)$ is
an object $A \in \mathcal{C}$;
a morphism $e \;\colon\; 1 \longrightarrow A$ (called the unit)
a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the product);
such that
(associativity) the following diagram commutes
where $a$ is the associator isomorphism of $\mathcal{C}$;
(unitality) the following diagram commutes:
where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$.
Moreover, if $(\mathcal{C}, \otimes , 1)$ has the structure of a symmetric monoidal category (def. ) $(\mathcal{C}, \otimes, 1, \tau)$ with symmetric braiding $\tau$, then a monoid $(A,\mu, e)$ as above is called a commutative monoid in $(\mathcal{C}, \otimes, 1, B)$ if in addition
(commutativity) the following diagram commutes
A homomorphism of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism
in $\mathcal{C}$, such that the following two diagrams commute
and
Write $Mon(\mathcal{C}, \otimes,1)$ for the category of monoids in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ 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 $Vect$ is an ordinary commutative algebra. Moreover, in both cases the homomorphisms of monoids agree with usual algebra homomorphisms. Hence there are equivalences of categories.
For $G$ a group, then a $G$-graded associative algebra is a monoid object according to def. in the monoidal category of $G$-graded vector spaces from example .
This means that a $G$-graded algebra is
a $G$-graded vector space $A = \underset{g\in G}{\oplus} A_g$
an associative algebra structure on the underlying vector space $A$
such that for two elements of homogeneous degree, i.e. $a_1 \in A_{g_1} \hookrightarrow A$ and $a_2 \in A_{g_2} \hookrightarrow A$ then their product is in degre $g_1 g_2$
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 $sCAlg_k$ for the category of supercommutative superalgebras with the induced homomorphisms between them:
Unwinding what this means, then a supercommutative superalgebra $A$ is
In view of def. we might define a not-necessarily supercommutative superalgebra to be a monoid (not necessarily commutative) in sVect, and write
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 $\mathbb{Z}/2$-graded vector spaces only via their braiding (example ), this yields equivalently just the $\mathbb{Z}/2$-graded algebras froom example :
Hence the heart of superalgebra is super-commutativity.
The supercommutative superalgebra which is freely generated over $k$ from $n$ generators $\{\theta_i\}_{i = 1}^n$ is the quotient of the tensor algebra $T^\bullet \mathbb{R}^n$, with the generators $\theta_i$ in odd degree, by the ideal generated by the relations
for all $i,j \in \{1, \cdots, n\}$.
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
Given a homotopy commutative ring spectrum $E$ (i.e., via the Brown representability theorem, a multiplicative generalized cohomology theory), then its stable homotopy groups $\pi_\bullet(E)$ 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
of commutative algebras (example ) into supercommutative superalgebras (def. ) induced via prop. from the full inclusion
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”):
Here
the right adjoint $(-)_{even}$ sends a supercommutative superalgebra to its even part $A \mapsto A_{even}$;
the left adjoint $(-)/(-)_{even}$ sends a supercommutative superalgebra to the quotient by the ideal which is generated by its odd part $A \mapsto A/(A_{odd})$ (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 $i$ is evident. To see the adjunctions observe their characteristic natural bijections between hom-sets: If $A_{ordinary}$ is an ordinary commutative algebra regarded as a superalgeba $i(A_{ordinary})$ concentrated in even degree, and if $B$ is any superalgebra,
then every super-algebra homomorphism of the form $A_{ordinary} \to B$ must factor through $B_{even}$, simply because super-algebra homomorpism by definition respect the $\mathbb{Z}/2$-grading. This gives a natual bijection
every super-algebra homomorphism of the form $B \to i(A_{ordinary})$ must send every odd element of $B$ to 0, again because homomorphism have to respect the $\mathbb{Z}/2$-grading, and since homomorphism of course also preserve products, this means that the entire ideal generated by $B_{odd}$ must be sent to zero, hence the homomorphism must facto through the projection $B \to B/B_{odd}$, which gives a natural bijection
It is useful to make explicit the following formally dual perspective on supercommutative superalgebras:
For $\mathcal{C}$ a symmetric monoidal category, then we write
for the opposite category of the category of commutative monoids in $\mathcal{C}$, according to def. .
For $R \in CMon(\mathcal{C})$ we write
for the same object, regarded in the opposite category. We also call this the affine scheme of $A$. Conversely, for $X \in Aff(\mathcal{C})$, we write
for the same object, regarded in the category of commutative monoids. We also call this the algebra of functions on $X$.
For the special case that $\mathal{C} =$ sVect (def. ) in def. , then we say that the objects in
are affine super schemes over $k$.
For $A \in CAlg_{\mathbb{R}}$ 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. )
In particular for $\mathbb{R}^p$ an ordinary Cartesian space, this becomes an affine superscheme in even degree, under the above embedding. As such, it is usually written
The formal dual space, according to def. (example ) to a Grassmann algebra $\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q)$ (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 $q$-th power.
For instance for $q = 2$ then a general element of $\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q)$ is of the form
for $a_1,a_2, a_{12} \in \mathbb{R}$, to be compared with the Taylor expansion of a smooth function $g \colon \mathbb{R}^2 \to \mathbb{R}$, which is of the form
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
Combining example with example , and using prop. , we obtain the affine super schemes
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 $A$ (def. ), its parity involution is the algebra automorphism
which on homogeneously graded elements $a$ of degree $deg(a) \in \{even,odd\} = \mathbb{Z}/2\mathbb{Z}$ is multiplication by the degree
(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 $(\mathcal{C}, \otimes, 1)$ (def. ), then the tensor unit $1$ is a monoid in $\mathcal{C}$ (def. ) with product given by either the left or right unitor
By lemma , these two morphisms coincide and define an associative product with unit the identity $id \colon 1 \to 1$.
If $(\mathcal{C}, \otimes , 1)$ is a symmetric monoidal category (def. ), then this monoid is a commutative monoid.
Given a symmetric monoidal category $(\mathcal{C}, \otimes, 1)$ (def. ), and given two commutative monoids $(E_i, \mu_i, e_i)$, $i \in \{1,2\}$ (def. ), then the tensor product $E_1 \otimes E_2$ becomes itself a commutative monoid with unit morphism
(where the first isomorphism is, $\ell_1^{-1} = r_1^{-1}$ (lemma )) and with product morphism given by
(where we are notationally suppressing the associators and where $\tau$ denotes the braiding of $\mathcal{C}$).
That this definition indeed satisfies associativity and commutativity follows from the corresponding properties of $(E_i,\mu_i, e_i)$, and from the hexagon identities for the braiding (def. ) and from symmetry of the braiding.
Similarly one checks that for $E_1 = E_2 = E$ then the unit maps
and the product map
and the braiding
are monoid homomorphisms, with $E \otimes E$ equipped with the above monoid structure.
Monoids are preserved by lax monoidal functors:
Let $(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ and $(\mathcal{D}, \otimes_{\mathcal{D}},1_{\mathcal{D}})$ be two monoidal categories (def. ) and let $F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}$ be a lax monoidal functor (def. ) between them.
Then for $(A,\mu_A,e_A)$ a monoid in $\mathcal{C}$ (def. ), its image $F(A) \in \mathcal{D}$ becomes a monoid $(F(A), \mu_{F(A)}, e_{F(A)})$ by setting
(where the first morphism is the structure morphism of $F$) and setting
(where again the first morphism is the corresponding structure morphism of $F$).
This construction extends to a functor
from the category of monoids of $\mathcal{C}$ (def. ) to that of $\mathcal{D}$.
Moreover, if $\mathcal{C}$ and $\mathcal{D}$ are symmetric monoidal categories (def. ) and $F$ is a braided monoidal functor (def. ) and $A$ is a commutative monoid (def. ) then so is $F(A)$, and this construction extends to a functor
This follows immediately from combining the associativity and unitality (and symmetry) constraints of $F$ with those of $A$.
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)
Ugo Bruzzo, Daniel Hernandez Ruiperez, Alexander Polishchuk, Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes (arXiv:2008.00700)
Discussion of crystalline cohomology of super-schemes:
Last revised on February 26, 2021 at 10:54:20. See the history of this page for a list of all contributions to it.