Hilbert polynomials of graded algebras evaluated at negative numbers

Question

Let $k$ be a field and let $R$ be a commutative (standard) graded $k$-algebra, that is, $R=\bigoplus_{n=0}^\infty R_n$ with $R_0=k$ (and $R=k[R_1]$). The Hilbert function $h_R:\mathbb{N}\rightarrow \mathbb{N}$ is given by $h_R(n)=\dim_k R_n$. When $n\gg 0$, it is known that $h_R(n)$ agrees with a polnyomial $p_R(n)$ in $n$ with rational coefficients.

In many theorems in combinatorics, polynomials which count objects have combinatorially sensible meaning when evaluated at negative numbers. For example, the chromatic polynomial $\chi_G(x)$ of a graph, which counts the number of proper colorings of $G$ with $x$ colors, has the delightful property that $\chi_G(-1)$ is the number of acyclic orientations of $G$.

One other example, intimately related to the Hilbert function, is that of Ehrhart polynomials. The Ehrhart polynomial $E_P(n)$ of a convex integral polytope $P$ inside $\mathbb{R}^m$ is the number of lattice points of $\mathbb{Z}^m$ inside the $n$th dilate $nP$. It is also known that $E_P(-n)$ is the number of interior lattice points (up to sign) of $nP$. For some (all?) polytopes, the Ehrhart polynomial agrees with the Hilbert polynomial of an associated affine semigroup ring.

In general, or in the specific context of affine semigroup rings $R$, is there some sensible (algebraic or combinatorial) meaning to $p_R(-n)$, or even $p_R(-1)$?

Answer 1

This answer is essentially the same as that of Phil Tosteson, written before I saw that post. I also mention a non-Cohen-Macaulay example at the end.

If $R$ is Cohen-Macaulay (but not necessarily generated in degree one), then $R$ has associated with it a canonical module $\Omega(R)$ which can be graded so its Hilbert function agrees with $(-1)^d p_R(-n)$ for $n$ sufficiently large, where $d$ is the Krull dimension of $R$ or $\Omega(R)$. If $R$ has $n$ generators then it can be regarded as a module over the polynomial ring $A=k[x_1,\dots,x_n]$. One can then define $\Omega(R) = \mathrm{Ext}^{n-d}_A(R,A)$. If $R$ is not Cohen-Macaulay, then there are "correction terms" to the formula $p_R(-n) = (-1)^d\mathrm{HQ}(\Omega(R))$, where $\mathrm{HQ}$ denotes Hilbert quasipolynomial. Namely, $$ p_R(-n) = (-1)^d \sum_{i=0}^d (-1)^i\mathrm{HQ}\left( \mathrm{Ext}^{n-d+i}_A(R,A)\right). $$ (If $R$ is Cohen-Macaulay, then only the term indexed by $i=0$ doesn't vanish.) I don't know where this result is stated in precisely this form, but it is equivalent to Theorem 6.4 of my book Combinatorics and Commutative Algebra, second ed. Theorem 8.2 gives an example, stated in terms of Hilbert series rather than Hilbert quasipolynomials.

Answer 2

For $n \gg 0$, we have that ${\rm dim~} R_n = {\rm dim~} H^0(X, \mathcal O(n))$, where $X = {\rm Proj}(R) \subseteq \mathbb P(R_1^{\vee})$ is the projective scheme associated to $R$ and $\mathcal O(n)$ is the degree $n$ line bundle.

In fact, we have that for all integer values that $$p_R(n) = \chi(H^*(X,\mathcal O(n))) := \sum_{i} (-1)^i {\rm dim} ~ H^i(X, \mathcal O(n)).$$

This gives an interpretation of $p_R(n)$ for negative values, but it is somewhat abstract in general. However in special cases (e.g. when $X$ is Cohen-Macaulay) there are more concrete interpretations of these Euler characteristics.

Let $K$ be a dualizing complex for $X$. Then ${\rm Ext}^*_X(\mathcal O,\mathcal O(-n))$ is graded dual to $Ext^*(\mathcal O(-n),K) = H^*(X, K(n))$. Thus in particular, if $X$ is Cohen-Macaulay of dimension $d$, so that $K = \omega_X[d]$ for some dualizing sheaf $\omega_X$, then we get that for all $n \gg 0$, that $$p_R(-n) = (-1)^d \chi(H^*(X, \omega_X(n)) = (-1)^d {\rm dim~} H^0(X, \omega_X(n)).$$ (We used that $n \gg 0$ in the last equality). So in this case, the negative values of $p_R(n)$ can be interpreted as the graded dimensions of the dualizing module of $R$.

As an example, you can take $R = k[x_0, \dots, x_d]$ so that $X = \mathbb P^d$ and $\omega_X = \mathcal O(-d-1)$ and one recovers the combinatorial reciprocity $${-n + d \choose d} = (-1)^d {n -1 \choose d}.$$

Also this story is closely related to Ehrhart reciprocity, because Ehrhart functions are Hilbert functions on toric varieties.