Graded commutative algebras are an important object of study in algebraic geometry (lying more on the algebraic side but admitting a geometric interpretation). We’ll offer two perspectives here on the role that they play, first a more conventional perspective (labeled here “the lowbrow story”), and then a more category-theoretic perspective (“the highbrow story”).
A “projective variety” over a field can be described by giving a “homogeneous system of equations”; that is, a list of variables and a list of homogeneous polynomial equations in those variables. A point in the projective space belongs to just in case its “homogeneous coordinates” give a solution of the system. (The homogeneous coordinates of form a vector in the vector space which is specified only up to multiplication by a scalar, but the homogeneity of the equations ensures that the property of being a solution of the system doesn’t depend on the scalar.)
More abstractly, can be described by giving a “graded ideal” in a polynomial algebra (namely the ideal generated by in the polynomial algebra generated by ), or still more abstractly by giving a graded commutative -algebra (namely the algebra obtained by modding out the polynomial algebra by the graded ideal).
(To make it easier to introduce certain motivating ideas from analysis and geometry, assume for the moment that is the field of complex numbers and that the projective variety is “non-singular” so that it can be thought of as a complex-analytic manifold.)
What is the conceptual significance for the projective variety of the elements in grade of this graded commutative algebra? They’re not (global) holomorphic functions on ; unlike affine varieties, (irreducible) projective varieties lack non-constant holomorphic functions. Rather, they’re holomorphic sections of a line bundle over , namely of the th tensor power of the dual of the line bundle which tautologically carries by virtue of being embedded in a projective space.
In this way, the holomorphic sections of line bundles over a non-singular complex projective variety step forward to take over the task which the largely non-existent holomorphic functions on the variety can’t perform: to describe the variety by algebraic means.
This hints at what’s really going on: the category of (holomorphic) line bundles over the projective variety knows everything about and thus serves as an effective stand-in for . in the next section we abandon the lowbrow approach and start over with this viewpoint, to found algebraic geometry on the principle that a variety is to be understood by understanding its category of line bundles (or of objects analogous to line bundles).
An object in a symmetric monoidal category is called a line object if it has an inverse object with respect to tensor product, and if the canonical “switching” morphism is the identity morphism. A section of a line object is a morphism from the unit object. We will be mainly but not exclusively interested in the case where is enriched over the category of vector spaces.
In a symmetric monoidal category of vector bundles, the line objects are the line bundles.
Another name for the study of categories of line objects is “dimensional analysis”. In dimensional analysis, a physical theory is described by specifying an abelian group of “dimensions” (these are the line objects) together with a commutative algebra of “quantities” (these are the sections of the line objects) which is graded by the dimension group. We’ll call a physical theory described in this way a “dimensional algebra”, but the fundamental fact about a dimensional algebra is that it’s equivalent to a dimensional category, which is a symmetric monoidal category where all objects are line objects.
Let be the abelian group freely generated by the dimensions “mass” and “velocity”. Let be the -graded commutative algebra generated by the six quantities “mass of particle #1”, “mass of particle #2”, “initial velocity of particle #1”, “initial velocity of particle #2”, “final velocity of particle #1”, and “final velocity of particle #2”, subject to the two relations “conservation of momentum” and “conservation of energy”.
(The names given to the dimensions, quantities, and relations here are informally meant to suggest both a physical situation and the precise dimensional algebra used to describe it. The physical context suggests taking the base field of the dimensional algebra to be the field of real numbers, but there are advantages and no disadvantages to leaving the field unspecified for now; even requiring it to be a field may be over-definite.)
For example, there are seven linearly independent quantities that live in the dimension “momentum” (defined as mass times velocity) in this dimensional algebra: multiplying the mass of either particle by the initial or final velocity of either particle gives eight quantities, but conservation of momentum cuts it down to seven. Thus in the corresponding dimensional category, the hom-space from the “dimensionless” dimension to the “momentum” dimension is a 7-dimensional vector space.
As a rough first approximation, we’ll construe algebraic geometry as the study of dimensional algebras, but from the perspective that it’s better to think of them as dimensional categories. The benefit to interpreting them as categories in this way is that it’s often useful to think of categories equipped with a certain kind of extra structure as “theories” of a sort, and to think of functors that preserve the extra structure as “models of the domain theory in the environment provided by the co-domain”. (This philosophy will be explained in more detail in the next chapter, on “doctrines”.) Thus we can now interpret a dimensional algebra as a “theory”, and consider “models” of this theory in our favorite “environment”, and consider isomorphism classes of such models as “points in a moduli space” (“moduli space” being a name with a certain pedigree for a set of isomorphism classes of objects of some kind).
It’s in this way that a dimensional algebra gets a “geometric interpretation” as a kind of “space”, but moreover a moduli space. Thus in this approach to algebraic geometry, a preoccupation with “moduli spaces” is not grafted on as an afterthought, but rather built in from the start, in that every “algebraic variety” is inherently a “moduli space” in a natural way (except that a more systematic treatment as in the next chapter (on “doctrines”) leads to replacing the idea of “moduli space” by the concept of “classifying topos”.)
Besides reading this paper, you can also watch Jim’s lectures on this subject:
There are also some related materials, some of which will eventually become part of this paper:
There is also some old stuff: