# What is the relationship amongst all the different kinds of spectra?

The word "spectrum" gets tossed around a lot in mathematics, and there seem to be a number of different concepts to which it applies. There is of course a physical connotation to the word which is commonly associated with scattering processes, rainbows, etc. : http://en.wikipedia.org/wiki/Spectrum

However, in mathematics the term seems to be quite loaded. To name a just a few concepts which could be called spectral, here is a big list:

1. Eigenvalues/eigenvectors for normal matrices

2. Jordan normal form for matrices

3. Spectrum of a bounded linear operator

4. Spectrum of a C* algebra

5. Spectrum of a commutative ring / Scheme

6. Characters of an abelian group

7. Irreducible unitary representations of a group

8. Spectrum of a graph

9. Spectrum of a Riemannian manifold

10. Spectral sequences from cohomology theory

Of course some of these concepts are more general than others. For example, normal matrix < bounded linear operator < c* algebra < commutative ring. However, it doesn't seem like any one of these definitions is sufficiently encompassing to give the whole story. As a counter example, the irreducible representations of a non-commutative group are an instance of the Jordan normal form (for finite groups anyway), but are not really captured by the corresponding notion of the spectra of a commutative ring. Similarly, the spectrum of a graph and a Riemannian manifold don't seem to have much to do with the spectrum of schemes, but yet they are related to the spectra of linear operators. And then there are spectral sequences which are just a bit weird...

I won't profess to completely understand the general idea here, but there do seem to be some patterns. A common theme seems to be decomposability', for example when finding the eigenvalues/vectors of a matrix one attempts to split apart the domain of the matrix into independent components. This is similar to splitting a space into points; and suggests that there is perhaps a correspondence between eigenvectors/eigenvalues and prime ideals/local rings. The non-commutative picture is of course more complicated, but perhaps the concept might be equally described in terms of irreducible modules and some type of generalized localization at an irreducible module (which is a hazy concept I must admit). In a perfect world, it would be nice if this same concept could even extend to things like the Gauss map from differential geometry, or the Legendre transform from statistical mechanics/convex optimization (of course that might not be feasible).

Also, I marked the question community wiki in case anyone else has some other good examples of spectral' concepts in mathematics.

• These questions cover some of the same ground: mathoverflow.net/questions/24090/… mathoverflow.net/questions/17357/… – user5117 Jul 16 '11 at 20:59
• "Decomposition into primary constituents, out of which the original can be reconstructed" – Yemon Choi Jul 16 '11 at 22:07
• Mikola -- all examples that you mention but the last one are related and none of them has anything to do with spectral sequences, as far as I can tell. Finally, neither spectra of rings nor spectral sequences have anything to do with spectra in topology that give rise to generalized cohomology theories. – algori Jul 16 '11 at 22:33
• What is the question, exactly? The one in the title? It would take a few books to answer... Maybe the question could be made more focused in order to turn it into something more answerable? – Mariano Suárez-Álvarez Jul 17 '11 at 0:42
• Mikola, why do you think that they should be building towards the same thing, rather than inspired by similar instincts/urges? – Yemon Choi Jul 17 '11 at 17:42

As far as I can tell, every example except the last is representation theory, understood in a suitably general sense. For example studying the eigenvalues of an operator is the same as studying the corresponding representation of $\mathbb{C}[X]$ or some suitable enlargement of it (e.g. the $C^{\ast}$-algebra generated by $X$), so 1, 2, 3, 8, and 9 are morally special cases of 4. 4 and 5 are morally special cases of the representation theory of various types of rings and algebras, while 6 and 7 also fall under this category thanks to the existence of various types of group algebras.

(I can't tell if you know this or not, but the spectrum of a commutative ring is also just representation theory. After all, every prime ideal $P$ of a ring $R$ gives rise to a homomorphism $R \to R/P \to \text{Frac}(R/P)$ which may be understood as a one-dimensional representation of $R$.)

As far as I know, spectral sequences and spectra in homotopy theory are unrelated to spectra in the above sense.

• The spectrum of a unital commutative $C^*$ algebra $A$ is defined such that $Hom_{top}(Spec(A), \mathbb{C}) = A$, the same holds for the spectra in homotopy theory with different hom. So I always understand this as a question about representable functors. Does this make sense? – Marc Palm Jul 17 '11 at 0:56
• I understand the first part of what you are saying, except the part about spectrums of commutative rings. It is obvious that localization at a prime ideal gives you an R-module, but why should I call that a representation? I know how to define a representation of a group, monoid or a category, but it's not clear to me how to extend that to a ring. One possibility might be to equate representations with modules and irr. reps with spectra, but this causes some problems for schemes since there are in general more irreducible modules in a ring than simply the prime ideals. – Mikola Jul 17 '11 at 4:01
• @Mikola: a representation of a ring is a module. As I said, one should understand the term "representation" in a suitably general sense. Of course for different types of objects, different types of representations are most appropriate to consider. – Qiaochu Yuan Jul 17 '11 at 5:00
• @Mikola: if you know how to define a representation of a monoid, then you already know how to define a representation of a ring, since a ring is nothing more than a monoid object in the category of abelian groups and all of the definitions transfer immediately. Anyway, maybe you are asking a more precise question than I thought you were - I am just talking about a basic intuition but maybe you literally want a definition that correctly specializes to all of these cases? – Qiaochu Yuan Jul 17 '11 at 5:04
• @Qiaochu Yuan: Sure, if there is a general definition I would love to hear it! I'm just not completely convinced that the definition of "set of all irreducible modules" = spectrum for rings correctly recovers the notion of a scheme in all cases (and in fact it seems like a scheme/Spec of a ring is a more restrictive/orthogonal concept that only appears to be well defined for commutative rings). Of course it is also possible that I am just very dense and there is something obvious that I am missing here... – Mikola Jul 17 '11 at 5:42

I'll extend my comment to an answer. Let $L$ be a complete lattice. Then a prime element of $L$ is an element p such that $a\wedge b\leq p$ implies $a\leq p$ or $b\leq p$. These elements are in bijection with maps from $L$ to the 2-element lattice preserving all sups and finite infs. For example, the prime elements of the lattice of ideals in a commutative ring are the prime ideals. If $A$ is a separable C*-algebra, then the prime elements of the lattice of closed 2-sided ideals are the primitive ideals (kernels of irreducible representations). If $A$ is commutative, these are the maximal ideals.

The prime elements of a lattice $L$ form a space $spec(L)$ called the spectrum of $L$. The topology has as open sets the sets D(a) with $a\in L$ where $D(a)$ consists of all prime elements $p$ with $a\nleq p$. For example, if $L$ is the lattice of ideals, this is the Zariski spectrum. If $A$ is a separable C*-algebra, then the spectrum of the closed 2-sided ideal lattice is the primitive ideal spectrum. So many of your examples are spectra of lattices.

Recall a space $X$ is sober if each irreducible closed subset has a unique generic point. Sober spaces are precisely the spectra of complete lattices. The proof $spec(L)$ is sober amounts to showing that the irreducible closed subsets are the complements of the sets D(p) with p prime and p is generic. Conversely, if X is sober, take the lattice of closed subspaces ordered by reverse inclusion. The prime elements are the irreducible closed subsets, which can be identified with points of X by taking generic points.