Slick ways to make annoying verifications

Question

There are many times in mathematics that one needs to make verifications that are annoying and distract from the main point of the argument. Often, there are lemmas that can make this much easier, at least in many important cases.

For instance, in topology, it can be quite annoying to verify directly from the definition that a particular quotient space is what you think it is, and not something else with the same underlying set. (In fact, I suspect that many topologists habitually skip this verification.) However, the following lemma can in many cases make this verification much simpler:

Lemma: If $X$ is compact, $Y$ is Hausdorff, and $f \colon X \to Y$ is surjective, then $f$ is a quotient map.

This lemma can be made more powerful using the fact that it suffices to show a map is locally a quotient map.

Another such difficulty is to verify that a category is abelian; if you go directly from the definition, there is an annoyingly long list of things to verify. However, unless I am mistaken, once you have an abelian category $\mathcal{A}$, there are a number of other categories that are guaranteed to give other abelian categories. These (I think) include the category of functors into $\mathcal{A}$ from a fixed other category, the category of sheaves in $\mathcal{A}$ on a (topological space? other category?) (assuming $\mathcal{A}$ is nice enough for this to make sense), and any full subcategory of $\mathcal{A}$ that is closed under 0, $\oplus$, kernels, and cokernels. Using these in combination, together with the fact that $R$-mod is an abelian category for every ring $R$, I believe one can get to every abelian category used in Hartshorne. (Note: I am not too confident in this example, so if someone wanted to elaborate this in an answer, it would be appreciated.)

EDIT: As is pointed out in the comments below, the category of $\mathcal{O}_X$-modules is not of this form. (I came up with this example while writing the question, and did not think it through too carefully.) Thus, I would doubly appreciate a good answer specifically addressing, "How do you show a category is abelian?"

Question: What are some more of these useful lemmas / collections of lemmas, and how are they used?

Answer 1

In functional analysis, the closed graph theorem is a good example of this. If you have two Banach spaces $X,Y$ and you define some linear operator $A : X \to Y$ on all of $X$, you often need to verify that it is continuous. Naively you would say, "let $x_n$ be a sequence in $X$ converging to some $x$; I have to show (1) that $A x_n$ converges and (2) its limit is $Ax$." But the closed graph theorem asserts that any everywhere defined operator between Banach spaces with a closed graph is in fact continuous. An operator has a closed graph iff for any sequence $x_n$ in $X$ such that $x_n$ and $A x_n$ converge, one has $\lim A x_n = A \lim x_n$. So in the argument above, you can skip the verification of (1) and treat it instead as an assumption.

There is also the closely related open mapping theorem, which says that a continuous linear bijection $B : X \to Y$ of Banach spaces is automatically a homeomorphism; i.e. you get to skip the verification that $B^{-1}$ is continuous. It's reminiscent of the theorem that any continuous bijection $f : K_1 \to K_2$ of compact Hausdorff spaces is automatically a homeomorphism.

Answer 2

Proofs exploiting universality often provide nice examples of slick ways to avoid annoying special cases. For example, the matrix identities below have trivial algebraic proofs by proceeding "generically", i.e. let the matrix entries $a_{ij}, b_{ij}$ be indeterminates and perform the proof over the polynomial ring $\mathbb Z[a_{ij}, b_{ij}]$.

$\rm\quad\; det(I-AB) = det(I-BA)\;\:$ by taking $\;\rm det\;$ of $\;\;\rm (I-AB)\;A = A\;(I-BA)\;$ then canceling $\;\rm det \:A$

$\rm\quad\quad det(adj \:A) = (det \:A)^{n-1}\quad$ by taking $\;\rm det\;$ of $\;\rm\quad A\;(adj\: A) = (det\: A) \;I\quad\;\;$ then canceling $\;\rm det \:A$

Contrast these absolutely trivial algebraic proofs with the more complex and more frequently presented topological proofs via density arguments. See my post [1] and its comments for some further discussion.

Answer 3

The probabilistic method is a good source of slick proofs of things that are either very hard to prove or even not known to be provable in any other way. For example, suppose you are asked to prove that it is possible to find infinitely many points in general position on the unit sphere in R^d. If you take any d-1 points, then the probability that a random point lies in the subspace they generate is zero. So if you randomly pick an infinite sequence of points, then the probability that it fails to be an example is zero. Actually thinking up an example and proving that it worked would be pretty tedious.

Answer 4

• The Yoneda Lemma (used everywhere!)
• In algebraic geometry, the vanishing locus of a map of vector bundles on a scheme $X$ is a closed subscheme of $X$.

I'll give an example where we use both. Say we want to show that there is such a thing as a scheme of "flags" on a vector space $V$, and that it is a closed subscheme of projective space. Such a flag scheme $\text{Fl}_{i_1, ..., i_k}$ represents the functor sending a scheme $S$ to the set of $M_1\hookrightarrow ...\hookrightarrow M_k\hookrightarrow V\otimes \mathcal{O}_S$ where $M_j$ is a vector bundle on $S$ of rank $i_j$ and where all the maps are injective bundle maps. We could do this by working with coordinates, but that would be a huge pain, so let's use our lemmas.

It suffices to do this for $k=2$, as larger flag schemes can be written as fiber products of $\text{Fl}_{i_1, i_2}$. We embed $\text{Fl}_{i_1, i_2}$ as a closed subscheme of the scheme $T:=\text{Gr}(i_1, V)\times \text{Gr}(i_2, V)$. We know that this is a closed subscheme of $\mathbb{P}(\Lambda^{i_1}(V)\otimes \Lambda^{i_2}(V))$ so this suffices.

But indeed, letting $M_j$ be the canonical rank $i_j$ bundle over $\text{Gr}(i_j, V)$ (induced by the identity map through the universal property of the Grassmannian and Yoneda), we have that the flag bundle $\text{Fl}_{i_1, i_2}$ is the vanishing locus of the map $p_1^*M_1\to \text{coker}(p_2^*M_2\to V\otimes \mathcal{O}_T)$. It's easy to check that this represents the desired functor.

Answer 5

One can often work out the exponents in some identity or inequality by using dimensional analysis or by plugging in key examples, and this is often faster than deriving the identity or inequality painstakingly by hand. I discuss this in more detail at

http://terrytao.wordpress.com/2008/12/27/tricks-wiki-use-basic-examples-to-calibrate-exponents/

Answer 6

To show that an object S has property P, first show that the collection of all objects satisfying P is closed under a bunch of operations, prove that certain very simple objects have property P, and show that S can be "decomposed" or "filtered" or somehow unscrewed into these simple objects using the operations mentioned above.

This is a sort of induction, and it is used all the time to turn annoying verifications into verifying that something is true for like... a point. Maybe a shorthand for this "slick method" would be "think like Grothendieck."

For lots of examples of this see any proof in Higher Topos Theory or Higher Algebra by Lurie.

Answer 7

Riffing off of Nate's answer, another result in functional analysis (often proven in the same breath as the others) is the Uniform Boundedness Principle. This is exceedingly useful in getting uniform estimates from pointwise ones... and it's pretty magical. It says that if we have a collection $\mathcal{F}$ of continuous linear operators from a Banach space to a normed vector space, then these are uniformly bounded (i.e. $\sup_{T \in \mathcal{F}} \Vert T \Vert < \infty$) if they are pointwise bounded (i.e., for every $x$ we have $\sup_{T \in\mathcal{F}} \Vert T(x)\Vert < \infty$).

Answer 8

My example is perhaps not exactly what you had in mind, but I hope that will make it more interesting, thus my reason for posting it. I thought of mentioning the (very elementary) but delightfully clever method of proving polynomials equal by comparing them evaluated at finitely many points (which is applied in, for example in A Proof in the Spirit of Zeilberger of an Amazing Identity of Ramanujan) but instead I thought I would point out the notion of reflective proof from dependent type theory:

To write a formal (computer checkable proof) you must justify every single step of reasoning all the way down to the axioms of the foundation. Of course that would make proving simple things like $((ab)cd)e = a(bc)(de)$ a terrible chore requiring repeated applications of the associativity (infact proving something as simple as $1 + 100 = 101$ could require $100$ applications of the definition of plus!). The idea of reflection is to reduce trivial reasoning steps to computation, This is described in Section 4 of Henk Barendregt's Proofs of correctness in Mathematics and Industry, but it was also applied heavily in a recent formal proof of the four color theorem.

Answer 9

Sometimes in elementary analysis there are things that are a pain to check, but one can at least minimize the pain. For example, if you want to prove that for every $\delta > 0$ the sequence $(1+\delta)^n$ is unbounded, you can use the lemma that it is at least $1+n\delta$, and more generally if you want to prove that it grows faster than any power you can use the lemma that it is at least $\binom n k\delta^k$, both of which follow from the binomial expansion.

Another one I like from elementary analysis. Suppose you want to prove rigorously that $\cos(x)$ is always at least $1-x^2/2$. You can do it as follows. First, $\cos(x)$ is always at most 1. It follows by integrating that $\sin(x)$ is at most $x$ when $x$ is positive. And then by integrating again we get the result for positive $x$, and evenness does the rest. (The integral shows that $-\cos(x)+\cos(0)$ is always at most $x^2/2$, and rearranging proves the inequality.) Iterating this argument gives you the inequality you expect wherever you truncate the Taylor expansion.

Answer 10

Before algebraic geometry was developed sufficiently and the Riemann-Roch was proved with its present power, the Lefschetz principle was used to dispose of many statements in algebraic geometry.

For instance: Define an elliptic curve over a field as a curve in the Weierstrass form with nonzero determinant. Upon this define the addition and inverse laws using the chord-and-tangent process, obtaining algebraic expressions. To show that the elliptic curve is a group, you have to show the addition is associative. One way is a very tedious verification of the identities.

Another way is to use elliptic functions to prove the identity in the complex case. Since the algebraic group law holds true over the complex numbers, it is satisfied by an infinite number of algebraically independent solutions, and therefore the group law must be true in universality, over any field whatsoever. Of course this needs to be made precise with Lefschetz principle.

But later algebraic geometry developed and it was possible to prove statements without relying on the Lefschetz principle. For instance, the group law on elliptic curve is always a consequence of the Riemann-Roch.

Answer 11

Elementary but still useful is the regular value theorem or the submersion theorem:

Let $f \colon M \to N$ be smooth and $n \in N$. If $T(f)_m \colon T_m M \to T_{f(m)} N $ is onto for all $m \in f^{-1}(n)$ then $f^{-1}(n) \subset M$ is a submanifold of dimension $\text{dim}(N) - \text{dim}(M)$.

Answer 12

A really elementary one :

A linear map between two vector spaces of the same finite dimension is an isomorphism if and only if its kernel is zero.

As an application, I like the proof of the existence of Lagrange interpolation polynomials.

Answer 13

My example comes from algebraic geometry, specifically in the theory of $\lambda$-rings, which are used in K-theory and representation theory (and possibly other places of which I am unaware). If you look in Lecture Notes in Mathematics 308 $\lambda$-rings and the Reprsentation Theory of the Symmetric Group by Donald Knutson, on pp 27, there is a theorem which is called--quite appropriately--the "Verification Principle", and it is used to do exactly the kind of thing you're asking about. The statement is:

If $\mu$ is a $\lambda$-ring operation, then $\mu$ is uniquely a polynomial in the $\lambda$-operations and for any particular polynomial $f(\lambda^1,\lambda^2,\ldots , \lambda^n,\ldots)$ it is sufficient to check that $\mu = f$, operating on a sum $\xi_1+\ldots + \xi_r$ of elements of degree 1, for all $r>0$. If you read the first part of the book (i.e. up to page 27 where you read this) you learn that this is a very handy thing indeed, especially given the generating function polynomials that show up in the study of these objects and how much easier it is to check for things which are polynomials in the $\lambda$-operations (as they are called). It is this that so nicely ties this kind of ring in with the representation theory of the symmetric group, in particular the elementary symmetric functions, as Knutson mentions on p. 10 of the same publication.

Another useful technique is one that comes up in algebra, when you check something for general elements by checking on a basis. Eg. testing something for all the elements of your tensor or exterior algebra by testing it on the "pure" or "completely decomposable" tensors which form a generating set.

Answer 14

A surjective map between two modules (over a commutative ring) of the same rank is a bijection.

Answer 15

A group homomorphism from a topologically finitely generated profinite group to any profinite group is continuous. (Nikolov-Segal, arxiv:math/0604399v1)

Answer 16

How about this (presumably) common practice:

Suppose $f:M\to \mathbb{R}$ a smooth function from some manifold $M$ (that maybe has extra structure $S$) and you are trying to prove some property $P$ (about $M$, $S$ or $f$). Suppose also that your life would be much easier if you could assume that $f$ was Morse.

You are often saved from much tedium by (hopefully) finding that your result is ``generic". That is:

1) There is a family of smooth functions $f_\epsilon:M\to \mathbb{R}$ with $f_0=f$ and for $\epsilon>0$ $f_\epsilon$ is Morse.

2) You can then easily prove some properties $P_\epsilon$ (modulo adjusting your structure to $S_\epsilon$) about ($M$, $S_\epsilon$ or $f_\epsilon$) that have the additional property of implying $P$ "by letting $\epsilon\to 0$".

Of course there are always those times (and I speak from personal experience) where things are too ``special" for such an approach to work. Then you have to get your hands dirty.

Answer 17

To show that $E\to M$ is a $G$-bundle (say in the fin-dim manifold category), one needs to check local triviality using some charts on $M$ and $E$ and that transitions between them are controlled by the group $G$. However, if you have a principal $G$-bundle on your hands, you can easily use the associated bundle construction to build the total space and the projection maps that satisfy the bundle property given only how $G$ acts on an abstract typical fiber.