En el sector agropecuario

LetG= _n≥0G_n be any graded algebra. A linear mapδ: G →G is called aderivation when it satisfies the product rule

δ(xy) =δ(x)y+xδ(y),

for all x and y in G. A derivation δ is said to lower degree by 1 when δ(G_n)⊆G_n−1, that is, when δ maps every elementx that is homogeneous of

degreen to an elementδ(x) that is homogeneous ofdegreen−1.

We have been studying the algebra offorms Sym( ˆA∗), which is a graded algebra. For any anchor p over A, let δ_p: Sym( ˆA∗) →Sym( ˆA∗) be the map that contracts on p, so that δ_p(f) := f p. For any fixed anchor p over A, the map δ_p is a derivation that lowers degree by 1.

It turns out that every derivation δ: Sym( ˆA∗) → Sym( ˆA∗) that lowers degree by 1 is ofthe formδ =δ_p, for some anchorpoverA. Here is why. Since δlowers degree by 1, δ must map 1-forms to real numbers; soδ restricts to a linear functional on coanchors. But every such linear functional corresponds to pairing with some anchor. So there exists some anchorpwithδ(h) =h, p, for all coanchors h. Rephrasing this, we have δ(f) =f p=δ_p(f) for every 1-form f on A. We also have δ(1) = 1 p = δ_p(1) = 0, since the only way that δ can lower the degree ofthe 0-form 1 is by taking it to 0, the unique (−1)-form. (See also Exercise 7.10-1.) But every form f on A can be written as a linear combination ofproducts ofzero or more coanchors. The derivations δ and δ_p agree on the empty product 1, they agree on all coanchors, they are both linear, and they both satisfy the product rule; so we can conclude that δ(f) =δ_p(f) for all formsf.

Exercise 7.10-1 Letδ:G→Gbe any derivation ofan algebraG. Without any assumption about what the derivation δ does to degrees, show that δ(1) = 0. (Hint: Substitute x:=y:= 1 in the product rule.)

Exercise 7.10-2 Define a map δ: Sym( ˆA∗)→Sym( ˆA∗) by setting δ(f) := nf, for every n-form f. Show that δ is a derivation ofthe algebra offorms that leaves degree unchanged.

Answer: For ann-formf and anm-formg, we haveδ(f g) = (n+m)f g = nf g+mf g =δ(f)g+f δ(g).

So every derivation ofthe algebra offorms that lowers degree by 1 simply contracts on some anchor. Those derivations have the additional pleasant property that they all commute with each other. For any anchors p and q and any form f, we have (f p) q = (f q) p = f (pq); so we have δ_p◦δ_q =δ_q◦δ_p. Indeed, the differential operatorsD_pandD_qactually commute with each other more generally; we have D_p(D_q(f)) = D_q(D_p(f)), not just

7.10. DERIVATIONS AND DIFFERENTIAL OPERATORS 99 for the functions f in Poly( ˆA,R), but at least for all real-valued functions f: ˆA→R that are twice continuously differentiable.

Since the derivationsδ_p, for anchorspin ˆA, commute with each other, we can use them to build up a commutative algebra: the algebra ofall differential operators that can be expressed as polynomials in the derivations (δ_p)_p∈Aˆ.

Note that two different polynomials in the variables (δ_p)_p∈Aˆ may denote the

same operator. For example, if E := (Q+R +S)/3 is the centroid ofa reference triangle QRS in A, then 3δ_E = δ3E = δQ +δR +δS. Sound

familiar? Indeed, this algebra of differential operators is simply the algebra ofsites in disguise. Any site s on A gives us such a differential operator by contraction, by the rule f →f s.

Thus, ifwe already understand the algebra offorms, one way to con- struct the algebra ofsites is as a certain algebra ofdifferential operators on forms. For example, suppose that Ais an affine plane. Working in Cartesian coordinates, we could define an n-site over A to be a polynomial that is ho- mogeneous ofdegree n, not in the three anchors (C, ϕ, ψ), but in the three derivations (∂/∂w, ∂/∂u, ∂/∂v). We would then pair an n-site s with an n-formf by applying, to the formf, the differential operator thats denotes. I’ve never seen anyone do so, but it would make equal sense to treat sites as basic and to define forms as certain differential operators on sites, replacing the three coanchors (w, u, v) by the derivations (∂/∂C, ∂/∂ϕ, ∂/∂ψ).

People who define sites to be differential operators on forms get the right answers, but they obscure the fundamental symmetry between forms and sites. Suppose that we have somehow defined the algebra offorms Sym( ˆA∗). Whatever technique we used to algebrize the linear space ˆA∗ ofcoanchors would surely work, equally well, to algebrize the space ˆA ofanchors, thus producing the algebra ofsites Sym( ˆA). It seems more natural to produce forms and sites via the same technology, rather than to exploit differential operators to define one ofthem in terms ofthe other.

It also seems strange, when talking about differential operators, to restrict ourselves to operators that are polynomials in the three derivations ∂/∂w, ∂/∂u, and ∂/∂v. Typically, when defining differential operators, we also allow multiplying byw,u, orv; for example,u(∂/∂u) is a common differential operator that preserves degree. Ofcourse, differential operators ofthis more general type typically don’t commute; for example, the operator u(∂/∂u) first partials with respect to uand then multiplies by u, not the reverse.

Exercise 7.10-3 People who define sites to be differential operators on forms are naturally led to one of the two possible pairings. Which one is it, the summed pairing or the averaged pairing?

Answer: The summed pairing. For example, they compute the real num- ber wn, Cn = wn,(∂/∂w)n by applying the operator (∂/∂w)n to the n-formwn_{, getting (∂/∂w)}n_(wn_{) =n! , rather than 1.}

100 CHAPTER 7. THE PAIRED-ALGEBRAS FRAMEWORK

Math remark: The derivations that we have defined, maps from an algebra to itself, are a special case. There is a more general notion of a derivation as a linear map δ: G → H that satisfies the product rule, where G is a commutative algebra and H is a G-module. Generalizing the notion ofa derivation in this way lets us construct, for any commutative algebra, a universal derivation ofthat algebra. For a concrete example, suppose that G = Sym( ˆA∗) = R[w, u, v] is a polynomial algebra in three variables. The universal derivation ofG is the map d: G→H defined by

d(f) := ∂f ∂wdw+ ∂f ∂udu+ ∂f ∂vdv,

where H is the free G-module with basis (dw, du, dv). Any derivation of G can then be achieved by substituting appropriate values for the three symbols dw, du, and dv. For example, ifwe substitute scalars p_w,p_u, and p_v fordw, du, and dv, we get a derivation δ: G → G that lowers degree by 1; in fact, we get the derivationδ_p associated with the anchor p=p_wC+p_uϕ+p_vψ = p_w(∂/∂w) +p_u(∂/∂u) +p_v(∂/∂v). For another example, ifwe substitute w, u, and v for dw, du, and dv, we get w(∂/∂w) + u(∂/∂u) +v(∂/∂v), the degree-preserving derivation ofExercise 7.10-2.