Appendix: Even rules

P. Deligne

Let A be a commutative super ring and M, N be A-modules. A morphism f: M → N is a parity respecting map f such that f(am) = af(m).

The “even rules” point of view is that

(a) one should think to M in terms of the functor which to any extension of scalars A → A′ attach the even part (M′)^{+} of M′ := A′ ⊗_{A} M;

(b) it amounts to the same to give a morphism of modules f: M → N or to give for each commutative super A-algebra A′ a (A′)^{+}-linear morphism F [A′] from (M′)^{+} to (N′)^{+}, functorial in A′.

More precisely, the map f ↦ F:

F [A′] = f′|M′^{+} ,

with f′ deduced from f by extension of scalars, is bijective.

Proof. To a functorial F [A′]: (M′)^{+} → (N′)^{+} one attaches a morphism f: M → N as follows. On M^{+}, f := F [A]. Take now A′ = A[θ] with θ odd. An element n of N′^{+} can be written uniquely in the form n_{0} + θn_{1} with n_{0} ∈ N^{+} and n_{1} ∈ N^{−}. Define φ_{0} and φ_{1} on M^{−} by

F [A′](θm) = φ_{0}(m) + θ_{1}(m),

and define f on M^{−} to be φ_{1}.

Apply functoriality to A′ → A: θ ↦ 0. The corresponding map from (M′)^{+} to M^{+} maps θm (m ∈ M^{−}) to zero, and it follows that φ_{0} = 0: for A′ = A[θ], we have

F (A′)(m_{0} + θm_{1}) = f(m_{0}) + θf(m_{1})

for m_{0} in M^{+} and m_{1} in M^{−}.

We now check that f is A-linear. The A^{+}-linearity is clear. If a is inA^{−} and m in M^{+},

θf(am) = F (A′)(θam) = θaF (A′)(m) = θaf(m) .

For m in M^{−}, applying functoriality to A′ → A: θ → a, we get f(am) = af(m) as well. If F is obtained from a morphism f, one has for A′ = A[θ]

f′(m_{0} + θm_{1}) = f(m_{0}) + θf(m_{1})

and it follows that this construction gives back f.

Conversely, the f deduced from F gives back F by extensions of scalars: for a′ ∈ A′ and m ∈ M even,

F (a′m) = a′F (m) = a′f(m)

by A → A′ functoriality. For a′ and m odd, functoriality for A[θ] → A′: θ ↦ a′ gives

F (a′m) = a′f(m)

as well.

A similar, more complicated argument, gives the

Theorem. Let M_{i} be a finite family of A-modules. The data of a morphism

f: ⊗M_{i} → N

amounts to the data of a A′^{+}-multilinear map, functorial in A′, of the (M′_{i})^{+} to (N′)^{+}.

The proof is left to the reader.

Corollary. Let B be an A-module. To give on B a structure of A algebra amounts to give on (B′)^{+} a structure of (A′)^{+} algebra, functorially in A′. The algebra B will be associative (resp. commutative, resp. Lie, …) if and only if the algebras (B′)^{+} are.

Note that the (B′)^{+} are purely even objects: at the cost of having to think functorially, which one wants to do anyway, this removes signs from the definitions.

Example. Let X be a supermanifold and L be an (even) function on X. The locus where L is stationary is the subspace S of X (a sub-supermanifold in good cases) defined by the vanishing of the section dL of the cotangent bundle of X.

When thinking functorially, one thinks to X as determined by the functor which to any supermanifold B attaches the set B-points of X, where a B-point is a morphism x: B → X. Intuitively: a family of points of X parametrized by B. In this language, describing S means describing which B-points x: B → X map to S, i.e. factors through S ↪ X. Such B-points will be called stationary.

By definition, a B-point x is stationary if for any even or odd section s on B of x^{∗}T_{X} (the “fiber of T_{X} at x”), ⟨dL, s⟩ = 0 on B. If s is odd, let B′ = B × ℝ^{0,1}: we add an odd parameter θ. Then, θs is even and ⟨dL, s⟩ = 0 on B if and only if ⟨dL, θs⟩ = 0 on B′. We see that a B-point x is stationary if and only if after any change of basis B′ → B, the B′-point B′ → B → X, still noted x, is such that for any even section s of x^{∗}T_{M}, ⟨dL, s⟩ = 0 on B′.

This can be rephrased as follows: after any change of basis B′ → B, one wants that for any family of points x(t) (−ε < t < ε): B′×] − ε, ε[→ X, one has

\frac{d}{dt} L(x) = 0 at t = 0

or, more cumbersomely written: the pull back of L to B′×] − ε, ε[ is such that \frac{∂L}{∂t} = 0 at t = 0.

This formulation is convenient when X is “infinite dimensional”, and when what makes good sense is not the space X and the function L, but the notion of map x: B → X and of the pull back by x of L.

Minority report: Odd rules.

The notion of super Lie algebra can be expressed nicely as follows: Data: after any extension of scalars A → A′, a quadratic map x^{2} from L^{−} to L^{+}, functorial in A′.

Applying the theorem to the associated bilinear form, from (ΠL)^{+} ⊗ (ΠL)^{+} to L^{+}, we see that x ↦ x^{2} defines a bracket [x, y]: L ⊗ L → L such that for x and y in L^{−}, [x, y] = (x + y)^{2} − x^{2} − y^{2}. In particular, for x in L^{−}, [x, x] = 2x^{2}.

Axiom: after any extension of scalars A → A′, for any x in L^{−}, one has

[x, x^{2}] = 0 .

Provided 6 is invertible in A, this definition is equivalent to the one in term of the Jacobi identity (which is a polarized version of the axiom).