4.9 Linearization revisited
4.9.5 So what is an anchor, really?
With this universal mapping condition in mind, we can now give the truest and deepest answer to the question, “What is an anchor?” Answer: An anchor overAis an element ofsome concrete linearization ofA, but with the understanding that, iftwo different linearizations ofA ever get involved in the same argument, we are required to use the unique isomorphism between them to identify each element ofone with the corresponding element ofthe other. That is, we agree not to distinguish between different linearizations. So all ofour earlier answers were correct simultaneously. An anchor over A is a linear combination of(C, ϕ1, . . . , ϕd). It’s also a linear combination
of(R0, . . . , Rd). It’s also a huge equivalence class oflinear combinations of
points of A. And it’s a linear functional on coanchors, and it’s a vector field ofa certain type, and so on. Speaking loosely, an anchor overAis an element of“the” linearization ˆA of A.
Keep in mind that these same issues are going to arise again in defining sites. A site over A is, speaking loosely, an element of“the” algebrization Sym( ˆA) of the linear space ˆA ofanchors. Given any linear spaceX, there is a universal mapping condition that determines when a commutative algebra is an algebrization of X. Since it follows from this condition that any two algebrizations ofX are isomorphic in a unique way, we typically pretend that the algebrization Sym(X) is uniquely determined.
In fact, the same issues arose already in defining forms, although we didn’t comment about them at the time. The algebra offorms is, we claim,
48 CHAPTER 4. THE HOMOGENIZED FRAMEWORK Sym( ˆA∗) Sym0( ˆA∗) A Sym2( ˆA∗) Sym3( ˆA∗) C, ϕ, ψ ˆ A Sym0( ˆA∗)∗ Sym2( ˆA∗)∗ Sym3( ˆA∗)∗ 1 w, u, v Sym1( ˆA∗) = ˆA∗ w2, wu, wv, u2, uv, v2 w3, w2u, w2v, wu2, wuv, wv2, u3, u2v, uv2, v3
Figure 4.3: The homogenized framework with abstract labels
the algebrization Sym( ˆA∗) of the linear space ˆA∗ ofcoanchors. That claim should be plausible, because a form is, roughly speaking, a polynomial whose variables are coanchors. We introduced the algebra offorms in Section 4.7 as the algebra Poly( ˆA,R) ofall polynomial, real-valued functions on ˆA. But that is simply one concrete construction ofthe abstract algebra Sym( ˆA∗). Indeed, for any linear space X, it turns out that we can exploit duality to construct the algebrization Sym(X) concretely as Poly(X∗,R). So one concrete model for the algebra of forms Sym( ˆA∗) is the algebra offunctions Poly( ˆA∗∗,R) = Poly( ˆA,R).
Figure 4.3 shows the homogenized framework again, just as in Figure 4.2, except that the spaces are now labeled abstractly. For example, the space of n-forms on A, which used to be labeled Polyn( ˆA,R), is now labeled Symn( ˆA∗). (Many authors write Symn(X) f or the nth graded slice ofthe
algebra Sym(X), with the n as a superscript. We make the n a subscript just for consistency with the notations Polyn( ˆA,R) and Rn[w, u, v].)
Chapter 5
The Separate-Algebras
Framework
The homogenized framework has brought us closer to symmetry, in the sense that, in Figure 4.3, neither the primal spaces (Symn( ˆA∗))n≥0 nor the dual
spaces (Symn( ˆA∗)∗)n≥0 are nested. But we still haven’t achieved symmetry.
The primal spaces fit together to make up the algebra offorms, while each dual space stands alone. Our eventual goal is thepaired-algebras framework, in which the dual spaces fit together, in similar way, to make up the algebra ofsites. But it is going to take us two steps to get there.
In the first ofthose two steps, we achieve symmetry in a brute-force way by treating the linear space ˆA ofanchors exactly as the homogenized framework treats the space ˆA∗ ofcoanchors. The result is the separate- algebras framework, shown in Figure 5.1. This framework has the serious drawback that there are four linear spaces associated with each degreen, the space ofn-forms Symn( ˆA∗) and its dual Symn( ˆA∗)∗ being joined by the space of n-sites Symn( ˆA) and its dual Symn( ˆA)∗.
In the second step, we shall choose a sequence ofpairing maps, thenth of which pairs the space ofn-forms with the space ofn-sites, thereby allowing us to use each ofthose spaces to represent the dual ofthe other. This yields the paired-algebras framework, with just two spaces on the nth level once again, rather than four. The reason that we delay taking this second step until Chapter 7 is that it entails a contentious choice about an annoying factor of n! . There are two sequences ofpairing maps, in which thenth maps differ by a f actor of n! . Consider ann-form and ann-site, both ofwhich happen to be perfect powers — say then-formhn and then-site pn, where his a coanchor
and p is an anchor. With one pairing, we have hn, pn = h, pn; with the other, we have hn, pn = n!h, pn. Sad to say, adopting either convention
leaves us with annoying factors in many of our formulas, as we discuss in Appendix B. For now, let’s get as far as we can using the separate-algebras framework, before tackling the annoying n! .
50 CHAPTER 5. THE SEPARATE-ALGEBRAS FRAMEWORK Sym( ˆA) Sym( ˆA∗)
Sites
Forms
Sym0( ˆA) Sym0( ˆA∗) A Sym2( ˆA) Sym2( ˆA∗) Sym3( ˆA) Sym3( ˆA∗) Sym0( ˆA)∗ Sym2( ˆA)∗ Sym3( ˆA)∗ Sym0( ˆA∗)∗ Sym2( ˆA∗)∗ Sym3( ˆA∗)∗ 1 C, ϕ, ψ ˆ A C2, Cϕ, Cψ, ϕ2, ϕψ, ψ2 C3, C2ϕ, C2ψ, Cϕ2, Cϕψ, Cψ2, ϕ3, ϕ2ψ, ϕψ2, ψ3 1 w, u, v ˆ A∗ w2, wu, wv, u2, uv, v2 w3, w2u, w2v, wu2, wuv, wv2, u3, u2v, uv2, v35.1. THE ALGEBRA OF SITES 51
5.1The algebra of sites
The triangle on the right in Figure 5.1 is the algebra offorms Sym( ˆA∗) = Poly( ˆA,R), just as in the homogenized framework. In a completely symmet- ric way, the triangle on the left is the algebra Sym( ˆA) = Poly( ˆA∗,R), which we christen the algebra of sites.
This may be as good a time as any to discuss why I chose the word “site”. I wanted a noun that would accept a numeric prefix, so that I could talk about n-sites as being dual to n-forms; that strongly suggested a one- syllable noun. I also wanted a noun that means something like “point”. The nouns “place” and “site” met those criteria. Unfortunately, both of those words have preexisting meanings in algebraic geometry. A place on a curve is an equivalence class ofirreducible parameterizations — roughly speaking, a point on a branch ofthe curve [1, 48]. From the Encyclopedic Dictionary ofMathematics [35], I learned that a site is a category in which each object comes equipped with a covering family of morphisms that fit together to form a Grothendieck topology. I hope that Grothendieck topologies are high- powered enough that no confusion will arise between that meaning of “site” and sites as the duals offorms.
The equality Sym( ˆA) = Poly( ˆA∗,R) suggests that a site is a real-valued, polynomial function on coanchors; and indeed, that is one of various equiva- lent ways to define a site. But viewing sites from that perspective is not the best way to get to know them. Keep in mind that, roughly speaking, sites are polynomials whose variables are anchors, just as forms are polynomials whose variables are coanchors. Let’s refer to such polynomials as anchor polynomials and coanchor polynomials.
For definiteness, let’s assume once again that A is an affine plane, of dimension d = 2. What is a site over A, more precisely? For that matter, what is a form on A? Both questions have an abstract answer, given on the first line ofTable 5.1, and a variety ofconcrete answers, three ofwhich are given on the following lines. Table 5.2 shows the four different ways in which we shall denote the linear space ofall n-sites over A and the linear space of alln-forms onA, corresponding to the four lines in Table 5.1. So each of the bottom three lines names a concrete construction for the linear space that the top line names abstractly.
5.1.1
Imposing a universal mapping condition
The linearization ˆAofan affine spaceAis a linear space that satisfies a certain universal mapping condition. In a similar way, the algebrization Sym(X) of a linear space X is a commutative algebra that satisfies a universal mapping condition. We shall discuss that condition and related issues in Chapter 9. Until then, just keep in mind that there is an abstract characterization of
52 CHAPTER 5. THE SEPARATE-ALGEBRAS FRAMEWORK
site form
an element ofthe algebrization Sym( ˆA) ofthe linear space ˆA
an element ofthe algebrization Sym( ˆA∗) of the linear space ˆA∗ a polynomial in the anchor vari-
ables (C, ϕ, ψ)
a polynomial in the coanchor vari- ables (w, u, v)
an equivalence class ofanchor poly- nomials whose variables are arbi- trary anchors
an equivalence class ofcoanchor polynomials whose variables are ar- bitrary coanchors
a real-valued function on coanchors that can be defined by some anchor polynomial — in fact, by an equiv- alence class ofanchor polynomials
a real-valued function on anchors that can be defined by some coanchor polynomial — in fact, by an equiva- lence class ofcoanchor polynomials
Table 5.1: What are sites and forms?
the algebra ofsites that determines it up to a unique isomorphism. So which concrete construction we adopt for that algebra doesn’t matter.