What is an exponential object?

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One interpretation for a power of natural numbers, such as 10^6, is that it is counting a certain collection of functions: in this case, the number of functions from a six-element set to a ten-element set. (One way to think about it is that 10^6 is the number of six-digit combinations from 000000 to 999999, and each combination corresponds to a function from \{1,\ldots,6\} to \{0,\dots,9\}: the first digit is the image of 1, the second digit is the image of 2, etc.)

In general, the number m^n is the number of functions from an n-element set to a m-element set. This is somewhat analogous to standard interpretations of other arithmetical operations:

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What is exponentiation?

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Last time, we explored how to extend multiplication from an operation on positive integers to an operation on arbitrary real numbers. The three tools we had at our disposal were:

  1. Find a nice interpretation for the operation, and see how broadly it applies.
    That’s how we moved from multiplying positive integers to multiplying any real number by a positive integer.
  2. Write down the laws the operation satisfies, and try to use them to deduce extra cases where the operation must give a certain value.
    For example, the law of associativity let us deduce the result of multiplying by a rational number, and the law of distributivity gave us the result of multiplying by zero or a negative number.
  3. Use continuity, monotonicity, or other “nice” properties of functions to decide what the operation should give for even more values.
    Knowing how to multiply by rational numbers doesn’t tell you how to multiply by irrational ones, but there’s at most one way to do so that makes the multiplication operation continuous.

Today we’ll explore these ideas for exponentiation instead of multiplication, and the story has a twist that finally explains why (a^2)^{1/2} doesn’t always equal a.

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What is a free module?

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Among modules over a commutative ring R, the module R^n has a nice feature: homomorphisms from R^n to another module M are entirely determined by where they send the n elements (1,0,\dots,0), (0,1,0,\dots, 0), up to (0,\dots,0,1). Moreover, any choice of where we want those basis elements to be sent in M extends uniquely to a homomorphism R^n\to M. Thus homomorphisms R^n\to M correspond bijectively to n-tuples of elements of M. This is an example of a universal property: R^n is the universal module equipped with an n-tuple of elements, also called the free module on n elements.

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What is a universal property?

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The following three constructions have something in common:

  • Kernels: If f:G\to H and g:H\to K are two group homomorphisms, then the composite g\circ f is the trivial homomorphism G\to K if and only if the image of f is contained in the kernel of g.
  • Polynomial rings: If A is any \mathbb{R}-algebra, then an \mathbb{R}-algebra homomorphism \mathbb{R}[x]\to A is entirely determined by where it sends x.
  • Topological products: The product topology on a product of topological spaces \prod_{n=1}^\infty X_n is a little counterintuitive—at first glance, the box topology seems more natural—but the product topology has the property that a function (f_1,f_2,\dots):A\to\prod_{n=1}^\infty X_n is continuous if and only if each component map f_n:A\to X_n is continuous.

Do you already see the similarity? Let me rephrase them to make it more obvious:

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Interlude: principal ideal rings

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Recently I mentioned that Noetherian rings can be characterized either by the fact that their finitely generated modules don’t have non-finitely generated submodules, or by the a priori weaker condition that their ideals are all finitely generated. In other words, if you want a counterexample to the statement that a submodule of a finitely generated module M is again finitely generated, it’s sufficient to consider the case M = R.

I also claimed that the function \mu_R sending an R-module to its minimum number of generators is not generally increasing: for R-modules N\subseteq M we don’t necessarily have \mu_R(N)\leq \mu_R(M), even when R is Noetherian. For example, over R = \mathbb{C}[x,y], the ideal I = (x,y)\subseteq R has \mu_R(I)=2, but \mu_R(R) = 1.

These two facts led me to ask the following question:

If we want a commutative ring R to have the property that \mu_R is increasing, is it again sufficient to look for counterexamples among the ideals of R? In other words, are the rings whose ideals can all be generated by single elements (the principal ideal rings) exactly the rings with this property?

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What is an exact sequence?

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I’ve heard it said that the concept of modules was invented to put ideals of rings on the same footing as quotient rings. Both the inclusion of an ideal I into a ring R and the quotient map R \to R/I are homomorphisms of R-modules. In fact, these two homomorphisms have a special relationship: the elements of R that are sent to zero in R/I are exactly those in the image of I\to R.

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