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## Some examples of graded algebras

Often in mathematics we work in an algebra with the property that the “degree” of an element has a multiplicative property. For example, in a polynomial ring in $n$ variables we can define the degree of a monomial to be the vector of its degrees with respect to each variable, and the product of monomials corresponds to the sum of vectors. More typically we can define the degree of a monomial to be its total degree (the sum of the components of the above vector); this degree is also multiplicative.

Algebras with this additional property are called graded algebras, and they show up surprisingly often in mathematics. As Alexandre Borovik notes, when schoolchildren work with units such as “apples” and “people” they are really working in a $\mathbb{Z}^n$-graded algebra, and one could argue that the study of homogeneous elements (that is, elements of the same degree) in $\mathbb{Z}^n$-graded algebras is the entire content of dimensional analysis.

At this point, I should give some definitions.

To define a graded algebra, we want to generalize the definition of a monomial. To say that a polynomial in one variable can be uniquely written as a sum of monomials is equivalent to giving a direct sum decomposition

$\mathbb{C}[x] \simeq \mathbb{C} \oplus \mathbb{C} x \oplus \mathbb{C} x^2 \oplus ...$

where $\mathbb{C} x^k$ denotes the monomials of the form $a_k x^k$. Since the product of a monomial of degree $n$ with a monomial of degree $m$ is a monomial of degree $n+m$, and $n$ ranges over the non-negative integers, we call this a $\mathbb{Z}_{\ge 0}$-graded algebra.

In general, given a semigroup $G$, a $G$-graded algebra is an algebra $A$ with a direct sum decomposition

$\displaystyle A = \bigoplus_{g \in G} A_g$

with the property that the multiplication sends the product of an element of $A_g$ and an element of $A_h$ to an element of $A_{gh}$. The elements of the factors $A_g$ are called the homogeneous elements, and the value of $g$ is called the degree. If you’re unfamiliar with direct sums, just remember that it means that any element of $A$ can be written uniquely as a sum of homogeneous elements. Because polynomial rings are the prototypical example, the case $G = \mathbb{Z}_{\ge 0}$ is referred to as “graded.”

Note that a “polynomial” (a sum of homogeneous elements of different degree) doesn’t necessarily have a well-defined degree; we aren’t requiring that $G$ have an ordering.

Graded algebras seem to appear whenever symmetry or homogeneity are important, although I don’t have much experience with their more sophisticated uses. Below are a few examples.

Fourier transforms

Every function $f : \mathbb{C} \to \mathbb{C}$ can be uniquely written as the sum of an even function and an odd function. Generically, this takes the form

$\displaystyle f(x) = \frac{f(x) + f(-x)}{2} + \frac{f(x) - f(-x)}{2}$.

This gives the set of functions $\mathbb{C} \to \mathbb{C}$ the structure of a $\mathbb{Z}/2\mathbb{Z}$-graded algebra; the direct sum decomposition is into the even and odd functions.

More generally, let $\omega$ be a primitive $n^{th}$ root of unity and say that a function $f : \mathbb{C} \to \mathbb{C}$ has weight $k$ if $f (\omega x) = \omega^k f(x)$. This gives the set of functions $\mathbb{C} \to \mathbb{C}$ the structure of a $\mathbb{Z}/n\mathbb{Z}$-graded algebra; the direct sum decomposition is into the functions of weight $k, k = 0, 1, ... n-1$. You might know this as the discrete Fourier transform or as the decomposition of a representation of $\mathbb{Z}/n\mathbb{Z}$ into its irreducible one-dimensional representations. For example, for $n = 3$ the decomposition into functions of weight $0, 1, 2$ takes the form

$\displaystyle f(x) = \frac{f(x) + f(\omega x) + f(\omega^2 x)}{3} \\ + \frac{f(x) + \omega^2 f(\omega x) + \omega f(\omega^2 x)}{3} \\ + \frac{f(x) + \omega f(\omega x) + \omega^2 f(\omega^2 x)}{3}$.

Even more generally, let $G$ be a locally compact abelian group and let $\hat{G}$ denote its Pontryagin dual, i.e. the continuous homomorphisms $G \to \mathbb{C}$. Let $X$ be a space on which $G$ acts continuously, and given a character $\chi : G \to \mathbb{C}$, say that a function $f : X \to \mathbb{C}$ has weight $\chi$ if $f(gx) = \chi(g) f(x)$ for every $x \in X$. Subject to technical assumptions, this gives the space of functions $X \to \mathbb{C}$ the structure of a $\hat{G}$-graded algebra; the direct sum decomposition is into the functions of weight $\chi, \chi \in \hat{G}$. (If $\hat{G}$ isn’t discrete; the direct sum is replaced by an integral.) For more details, see Terence Tao’s notes on the Fourier transform. With $G = \mathbb{T}$ (the circle group), $X = S^1$ (the circle), and $\hat{G} = \mathbb{Z}$ we recover the usual gradation on the space of Fourier series of functions on the circle (equivalently, the space of Fourier series of periodic functions on the real line).

Commutative algebra

A polynomial ring $F[x_1, ... x_n]$ is a graded algebra under total degree. This allows us to focus our attention on homogeneous polynomials, since those are the important ones in algebraic geometry. Given a graded algebra $A = \bigoplus_{n \ge 0} A_n$, define $H(A, n) = \dim A_n$ and define the Hilbert series

$\displaystyle H_A(t) = \sum_{n \ge 0} H(A, n) t^n$.

One can verify that when $A = F[x_1, ... x_n]$ the Hilbert series is $\frac{1}{(1 - t)^n}$, and this should be familiar if you did the exercise about symmetric functions awhile back. Hilbert series behave well under the obvious operations: they are additive under direct sum and multiplicative under tensor product, provided the degree of a tensor product is defined appropriately. One can think of this as a “linearization” of the properties of combinatorial generating functions under disjoint union and Cartesian product. It is therefore reasonable to expect that the Hilbert series of a graded algebra encodes information about its structure.

Hilbert series can be used to study algebraic varieties, as follows: given a projective variety $V$ defined over $\mathbb{C}^n$, the ring of polynomial functions $V \to \mathbb{C}$ is a quotient of $\mathbb{C}[x_1, ... x_n]$ by the ideal $I(V)$ of functions in $x_1, ... x_n$ vanishing on $V$, hence inherits a gradation. The Hilbert series of a variety $V$ can be used to define its dimension, as follows.

Theorem: There exists a polynomial (the Hilbert polynomial) $P(V, n)$ such that $H(V, n) = P(V, n)$ for all sufficiently large $n$. The degree of this polynomial is the dimension of $V$.

Intuitively, the degree of the Hilbert polynomial measures the number of “degrees of freedom” that polynomial functions on $V$ have. For $V = \mathbb{C}^d$ the ring of functions is $\mathbb{C}[x_1, ... x_d]$ and the Hilbert polynomial is $\displaystyle {n+d-1 \choose d}$. For $V \subset \mathbb{C}^4$ the Segre variety $xy - zw = 0$, the ring of functions is $\mathbb{C}[x, y, z, w]/(xy - zw)$. Its Hilbert series begins

$\displaystyle 1 + 4t + 9t^2 + ...$

and we can compute its Hilbert polynomial as follows: after replacing the factor $xy$ by $zw$ in every monomial, the space of monomials of degree $n$ consists of

• Monomials in $z, w$; there are $n+1$ of these.
• Monomials in $y, z, w$ with a factor of $y$; there are ${n+2 \choose 2} - (n+1)$ of these.
• Monomials in $x, z, w$ with a factor of $x$; there are ${n+2 \choose 2} - (n+1)$ of these.

This gives the Hilbert polynomial $2 {n+2 \choose 2} - (n+1) = (n+1)^2$, hence $V$ has dimension $2$; in fact, it’s a doubly ruled surface.

I haven’t checked, but I believe this generalizes: the Segre embedding might correspond to the Hadamard product of Hilbert series in general.

Supersymmetry

Now we enter the realm of things I don’t understand. A $\mathbb{Z}/2\mathbb{Z}$-graded algebra is called a superalgebra. Superalgebras have an even part and an odd part, as we have seen. A good example of a superalgebra is the ring of invariants of the alternating group $A_n$ acting on $\mathbb{C}[x_1, ... x_n]$ by permutation of the variables. The even part consists of the polynomials invariant under $S_n$, the symmetric polynomials, and the odd part consists of the polynomials that gain the sign of a permutation under permutation, the alternating polynomials.

Supersymmetry is an idea from physics relating bosons to fermions. According to Masoud Khalkhali, if $H$ is the Hilbert space of states of a single boson, then the Hilbert space of states of $n$ bosons is the symmetric tensor power $S^n H$. If $H$ is instead the Hilbert space of states of a single fermion, then the Hilbert space of states of $n$ fermions is the exterior power $\wedge^n H$; this is the Pauli exclusion principle.

The exterior algebra of a $d$-dimensional vector space $V$ has Hilbert series $(1 + t)^d$, since by Pauli exclusion a monomial of degree $k$ corresponds to a subset of size $k$ of $d$ basis vectors. In other words, fermion = subset. The symmetric algebra of $V$ can be identified with the space of polynomials in $d$ variables, so as we saw before it has Hilbert series $\frac{1}{(1 - t)^d}$. In other words, boson = multiset. The “supersymmetry” relating bosons and fermions is hinted at by the following:

$\displaystyle [x^k] (1 + t)^d = {d \choose k}$
$\displaystyle [x^k] \frac{1}{(1 - t)^d} = (-1)^k {-d \choose k}$.

The next GILA post will attempt to discuss these issues from a combinatorial perspective.

### 15 Responses

1. Thanks for a very helpful post. Is it not more typical that the binary operation of the semigroup G (in the formal definition) be denoted additively i.e. the multiplication sends the product of an element of A_g and an element of A_h to an element of A_{g+h} (as opposed to A_{gh})?

• Maybe?

2. […] is preserved under addition and multiplication, and since symmetric polynomials inherit the grading from , the symmetric polynomials form a graded -algebra. General results in algebra then guarantee […]

3. Since you mentioned the discrete Fourier transform, I thought I’d mention this post by Tim Gowers, together with Terry Tao’s first comment. It has a nice perspective on solvability by radicals.

4. The (perhaps inferior) way I understood it was that the Hilbert polynomial of a finitely generated graded module $M = \bigoplus M_n$ over $k[x_1, \dots, x_n]$ doesn’t have to agree with the appropriate dimensions even when $n=0$, because there might be some “garbage” in the first few dimensions. But one proves the Hilbert polynomial theorem for sheaves on schemes, one does noetherian induction on the support, and when the support is $\emptyset$, the dimensions are always trivially zero.

Incidentally I know a few people who are working indirectly with Professor Etingof on topics related to Hilbert series.

5. No, don’t apologize; as I said, I don’t really know anything about the sophisticated uses of Hilbert series, so I really appreciate all the comments from people who actually know what they’re talking about đź™‚ I was introduced to Hilbert series as part of a research project nominally under Pavel Etingof which I later dropped because I didn’t understand what was going on! I just like the construction because it appeals to my sensibilities regarding “weighted counting.”

• Given your interests in combinatorics, you may appreciate this paper: Hilbert polynomials in combinatorics by Francesco Brenti.

Hilbert’s approach to Hilbert series was through the Hilbert syzygy theorem: given a graded module M over a polynomial ring $K[x_1, \dots, x_n]$ (K is a field), write down a graded free resolution for it, and use additivity to write the Hilbert polynomial of M as an alternating sum of Hilbert series for graded shifts of free modules. Problem: can we find a finite free resolution? The answer is yes, and we only need at most n+1 free modules in the resolution.

Some places to look if you like this: Eisenbud’s Commutative Algebra and Eisenbud’s The Geometry of Syzygies.

• I wrote a reply a while back but it never showed up I guess. Anyway, if you like combinatorics and thinking of Hilbert functions as “weighted counting”, you might like this paper:

F. Brenti, Hilbert functions in combinatorics
http://www.mat.uniroma2.it/~brenti/16.dvi

The paper addresses the question: what kinds of polynomial in combinatorics (like chromatic polynomials) are Hilbert functions for graded algebras?

• My apologies; I didn’t realize this until today, but both your comments were caught in the spam filter. It seems as if Akismet doesn’t like gmail accounts.

6. You probably already know this, but it’s worth mentioning. When I first learned about Hilbert polynomials, I was always very confused about the fact that it agrees with the Hilbert polynomial only for sufficiently large n. My first exposure to generating functions was the Ehrhart series of an integral polytope. Since the number of lattice points in dilates of an integral polytope is exactly a polynomial (Ehrhart polynomial), I thought it was weird that for graded algebras, there is some error, especially because the Ehrhart polynomial is a special case of a Hilbert polynomial.

What was going on? Well, the Hilbert polynomial is NOT counting the dimensions of the graded pieces of your algebra. The real statement goes as follows. Pick a projective variety X over a field K embedded in projective space, and let O(1) denote the hyperplane section, and pick a coherent sheaf F on X. We can define a function $\chi_F(d) = \sum_{i \ge 0} (-1)^i \dim_K \mathrm{H}^i(X; F(d))$ where H denotes sheaf cohomology, and $F(d) = F \otimes O(1)^{\otimes d}$ for all integers d. The cohomology groups are finite-dimensional since X is projective and F is coherent, and the sum is always finite by Grothendieck vanishing. Then this is a polynomial function on the nose. If I’m not mistaken, we could have picked O(1) to be any ample line bundle, and then $\chi_F(d)$ would be a quasi-polynomial.

How does it relate to the original story? A finitely generated graded K-algebra A which is generated in degree 1 over $A_0 = K$ can be written as a quotient of a polynomial ring, which gives us an embedded projective variety X and hyperplane section O(1). Then $\mathrm{H}^0(X; O(d)) = A_d$ for nonnegative d. By Serre’s theorem, the higher cohomology of O(d) vanishes for sufficiently large d, and hence we get back the theorem you mentioned. And hence the obstruction to the Hilbert polynomial matching the Hilbert function comes from some cohomology. I was very happy when I learned this fact, because it cleared up the mystery of what the phrase “sufficiently large d” was really saying!

Going back to polytopes, the varieties we get are always toric, and ample line bundles on toric varieties never have higher cohomology, so that’s why there was no qualifier “sufficiently large”. What’s a bit strange is that the graded ring associated to a polytope need not be generated in 1, but everything works out anyway.

If I just told you a bunch of stuff you already knew, then I apologize!

• On second reading, what I wrote is not 100% accurate. We need to assume that the ring we started with is an integrally closed domain to get the conclusion ${\rm H}^0({\rm Proj } A, \mathcal{O}(d)) = A_d$. For example, from Hartshorne Ex. II.5.14, the ring $\bigoplus_{n \ge 0} {\rm H}^0({\rm Proj } A, \mathcal{O}(n))$ is the integral closure of Proj A if A is a domain (and f.g. in degree 1) and if the stalks of Proj A are integrally closed.

But dropping the hypothesis of integrally closed domain, from Hartshorne Ex. II.5.9(b), we do know that $A_d = {\rm H}^0({\rm Proj } A, \mathcal{O}(d))$ for $d \gg 0$. But I haven’t digested what the obstruction in this case is.

So we kind of have an explanation, but it’s still a little mysterious… bummer!

• Here’s the real story. We can use sheaf cohomology in the case of integrally closed domains, but in general, the actual obstruction is coming from local cohomology.

Let A be a standard graded polynomial ring in n variables and let $\mathfrak{m} = A_+$ be the irrelevant ideal. Then for any f.g. A-module M, we have that

$H_M(d) - P_M(d) = \sum_{i \ge 0} (-1)^i \dim {\rm H}^i_{\mathfrak{m}}(M)_d$

where $H_M$ is the Hilbert function of M and $P_M$ is the Hilbert polynomial of M, and ${\rm H}^*_{\mathfrak{m}}$ is local cohomology with support in $\mathfrak{m}$ (which inherits a grading from M). Now for our application, we take M to be A/I for some homogeneous ideal I.

There are analogues of Grothendieck/Serre vanishing for local cohomology, so everything is fine.

(This is Corollary A1.15 in Eisenbud’s _The Geometry of Syzygies_)

• On second reading, what I wrote is not 100% accurate. We need to assume that the ring we started with is an integrally closed domain to get the conclusion {\rm H}^0({\rm Proj } A, \mathcal{O}(d)) = A_d. For example, from Hartshorne Ex. II.5.14, the ring \bigoplus_{n \ge 0} {\rm H}^0({\rm Proj } A, \mathcal{O}(n)) is the integral closure of Proj A if A is a domain (and f.g. in degree 1) and if the stalks of Proj A are integrally closed.

But dropping the hypothesis of integrally closed domain, from Hartshorne Ex. II.5.9(b), we do know that A_d = {\rm H}^0({\rm Proj } A, \mathcal{O}(d)) for $d \gg 0$. But I havenâ€™t digested what the obstruction in this case is.

So we kind of have an explanation, but itâ€™s still a little mysteriousâ€¦ bummer!

7. Thanks for the comments, Theo. I don’t have a reference available for this stuff, so I’m just working off a few things I’ve read online and a discussion I had with Todd Trimble.

8. Nice post!

(1) There is a little more to the word “superalgebra” than “$Z/2$ graded algebra”. The best definition is to entirely describe the symmetric monoidal category of supervector spaces, and then internalize the word “algebra” to that category. That category is as follows: As a monoidal category, it is the category of $Z/2$-graded vector spaces (with, of course, grading-preserving maps). But the canonical “flip” isomorphism $V\otimes W \to W\otimes V$ acts by $-1$ on the odd \otimes odd part (and as $+1$ on the other homogeneous components).

Why does this matter? You don’t need the flip map to define an algebra, so your definition is not wrong. But you do need the flip map to define a commutative algebra. A “commutative superalgebra” is one where if $a$ and $b$ are homogeneous elements, then $ab = \pm ba$, where the sign is $+$ unless both $a$ and $b$ are odd, in which case the sign is even. A “commutative $Z/2$-graded algebra” is where $ab = ba$. A related use of the flip map: it’s required to define a “Lie algebra”, and so the words “super Lie algebra” and “$Z/2$-graded Lie algebra” are different.

(2) The Hilbert Series you described is closely related to the following. Let $g$ be a semisimple Lie algebra over $C$. Let $h \leq g$ be a Cartan subalgebra. Let $V$ be a $g$-module. Then $h$ acts diagonalizably on $V$ — i.e. $V$ decomposes as an $h$-module into a direct sum $V = \bigoplus V_\lambda$ where the $h$-action on $V_\lambda$ is given by some linear functional $\lambda: h \to C$. The $\lambda$ for a particular module are called the “weights” of the module. If $V$ is finite-dimensional, then the weights of $V$ necessarily lie in a lattice in $h^* = hom(g,C)$, called the “weight lattice” $P$.

So, this means that the category of finite-dimensional $g$-modules is $P$-graded, and the category of all modules if $h^*$-graded. To each module, I want to write down a generating function like your Hilbert Series: if $\lamdba \in h^*$, I want the coefficient of $x^\lambda$ to be $\dim V_\lambda$. But this is bad unless I require that my modules be “locally finite”. Then, though, my category is not closed under tensor products. So let’s pick a direction in $h^*$ (or rather, first prove that $P$ lies in a real form of $h^*$, restrict the weights to that real form, and pick a direction, i.e. a vector in $h$, in it, and best to make the choice oblique to every vector in $P$). Then we can require that the modules have only finitely many weights in the negative (say) direction.

Great. Let $h^*_+$ be the semigroup of positive elements of $h^*$, and $P_+ = h^*_+ \cap P$; then our modules are $P \times_{P_+} h^*_+$-graded.

Anyhoo, the category of such modules is closed under everything you could hope it to be. To each module we associate the generating function for its dimensions. This is called the “Weyl Character” of the module. The multiplicativity and additivity say that the Weyl Character is a “monoidal exact functor” from the abelian category of modules to the “zero-category” consisting of the ring of power series.