Facard/Vortrag/3/Textabschnitt

Differential powers and differential core

We describe a second approach to the differential signature.

Definition

Let ${}R$ be a ${}K$-algebra and ${}I\subseteq R$ an ideal. Then we call

${}I^{\langle n\rangle }={\left\{f\in R\mid E(f)\in I{\text{ for all }}E\in \operatorname {Diff} ^{n-1}(R,R)\right\}}\,$

the ${}n$th differential power of ${}I$.

The differential powers are ideals, we have ${}I^{\langle 1\rangle }=I$ and ${}I^{n}\subseteq I^{\langle n\rangle }$. They from a decreasing sequence of ideals

${}I^{\langle n+1\rangle }\subseteq I^{\langle n\rangle }\,.$

Of particular importance is the family ${}{\mathfrak {m}}^{\langle n\rangle }$, ${}n\in \mathbb {N}$, for a maximal ideal in a local ring. Since these contain certain ordinary powers of the maximal ideal, they have finite colength.

If ${}(R,{\mathfrak {m}})$ is a local ${}K$-algebra with ${}R/{\mathfrak {m}}\cong K$ then there is a perfect pairing

$R/{\mathfrak {m}}^{\langle n+1\rangle }\times \operatorname {Diff} ^{n}(R,R)/\operatorname {Diff} ^{n}(R,{\mathfrak {m}})\longrightarrow R/{\mathfrak {m}},\,(f,E)\longmapsto {\overline {E(f)}}$

and the ${}K$-dimension of

${}Q_{n}=\operatorname {Diff} ^{n}(R,R)/\operatorname {Diff} ^{n}(R,{\mathfrak {m}})\,,$

which is the freerank of the module of principal parts, equals the ${}K$-dimension of ${}R/{\mathfrak {m}}^{\langle n+1\rangle }$. Hence we can also define

${}\operatorname {DS} (R)=\limsup _{n}{\frac {\operatorname {length} {\left(R/{\mathfrak {m}}^{\langle n+1\rangle }\right)}}{n^{d}/d!}}\,$

under the mentioned assumptions. This approach has the advantage that it is defined for any noetherian local ring and that one has a direct comparison with ordinary powers and so with the Hilbert-Samuel multiplicity.

Definition

Let ${}(R,{\mathfrak {m}})$ be a local noetherian ${}K$-algebra. Then the intersection

$\bigcap _{n\in \mathbb {N} }{\mathfrak {m}}^{\langle n\rangle }$

is called the differential core of ${}R$.

We have

${}\bigcap _{n\in \mathbb {N} }{\mathfrak {m}}^{\langle n\rangle }={\left\{f\in R\mid E(f)\in {\mathfrak {m}}{\text{ for all differential operators}}E\right\}}\,$

so this is the set of elements for which there exists no unitary operator at all. The differential core is a submodule of ${}R$ for the ring of differential operators.

Lemma

Let ${}R$ be a noetherian ${}K$-algebra. Then the following are equivalent.

1. ${}R$ is a simple ${}D_{R{|}K}$-module.
2. The differential core is ${}0$.
3. For all ${}f\in R$, ${}f\neq 0$, there exists a differential operator ${}E$ with ${}E(f)=1$.

Computations of differential signature

The computation of the differential signature in certain important classes of examples rests on the following strategy: realize the ring ${}R$ as a direct summand of a regular ring (polynomial ring) ${}S$. This implies that the differential signature is positive. To obtain however a precise statement one also needs that the ring extension is differentially extensible, meaning that every differential operator of ${}R$ extends to the regular ring. For example, if we look at the natural action of the symmetric group on the polynomial ring, then the invariant ring is isomorphic to the polynomial ring itself, and the extension is not differentially extensible. Therefore we need the smallness assumption in the following theorem, which implies that

$\operatorname {Spec} {\left(S\right)}\longrightarrow \operatorname {Spec} {\left(R\right)}$

is etale in codimension one.

Theorem

Let ${}G\subseteq \operatorname {GL} _{n}\!{\left(K\right)}$ be a small finite group with its natural action on the polynomial ring ${}K[X_{1},\ldots ,X_{n}]$. Suppose that the characteristic of ${}K$ does not divide the order of the group.

Then the differential signature of the invariant ring

${}R=K[X_{1},\ldots ,X_{n}]^{G}\,$

is ${}{\frac {1}{\operatorname {ord} {\left(G\right)}}}$

Here both signatures coincide. The unitary operators can also be described easily. If ${}E$ is a unitary operator on ${}S$ then ${}{\frac {1}{\operatorname {ord} {\left(G\right)}}}{\left(\sum _{g\in G}g^{-1}Eg\right)}$ is a unitary operator on ${}R$.

For a positive normal toric monoid we work with the direct summand coming from ${}M\subseteq \mathbb {N} ^{r}$. On one hand, the monoid ring

${}K[M]\subseteq K[\mathbb {N} ^{r}]\cong K[Y_{1},\ldots ,Y_{r}]\,$

is the degree ${}0$ part of the divisor class group grading, on the other hand, it inherits the standard grading from the polynomial ring. In this situation we have the following general fact. The property differentially extensible means that every differential operator of the subring extends to a differential operator in the bigger ring.

Theorem

Let ${}R\subseteq S=K[Y_{1},\ldots ,Y_{r}]$ be an ${}\mathbb {N}$-graded ${}K$-subalgebra of the standard-graded polynomial ring. Suppose that this inclusion is a direct summand which is also differentially extensible.

Then

${}R_{\geq n}={\mathfrak {m}}^{\langle n\rangle }\,$

and

${}R_{n}={\mathfrak {m}}^{\langle n\rangle }/{\mathfrak {m}}^{\langle n+1\rangle }\,,$

where ${}{\mathfrak {m}}=R_{+}$. The differential signature of ${}R$ equals its degree (as a graded ring), it is a rational number and the defining sequence converges.

This means that we have a natural grading coming from differental powers and hence from differential operators.

Theorem

Let ${}C$ be a pointed rational cone in ${}\mathbb {R} ^{d}$ defining the monoid ${}M=\mathbb {Z} ^{d}\cap C$. Let ${}F_{1},\ldots ,F_{r}$ be the facets of ${}C$ and let ${}\ell _{1},\ldots ,\ell _{r}$ be the describing integral linear forms, meaning that they are positive in the interior of the cone and that they vanish on the corresponding facets. Suppose that ${}K$ is a field of characteristic ${}0$.

Then the differential signature of the monoid ring ${}K[M]$ is

$d!\operatorname {vol} {\left\{P\in C\mid (\ell _{1}+\cdots +\ell _{r})(P)\leq 1\right\}}.$

Theorem

Let ${}K$ be an algebraically closed field of characteristic zero.

Then the differential signature of the determinantal ring of ${}m\times n$-matrices of rank ${}r$

${}R=K[X_{a,b},\,1\leq a\leq m,\,1\leq b\leq n]/({\text{ minors of seize }}r)\,$

is

${\frac {1}{2^{r(m+n-r)}}}\det {\left({\binom {m+n-i-j}{m-i}}_{1\leq i,j\leq r}\right)}.$

In this class of examples the ${}F$-signature is not known.

Necessary conditions for positive differential signature

Theorem

Let ${}R$ be a reduced noetherian local ${}K$-algebra. Suppose that the differential signature is positive.

Then ${}R$ is a simple ${}D_{R{|}K}$-module.

This rests on the fact that the core is zero, since otherwise it would be of positive dimension and that would give a lower dimensional bound for the signature function contradicting the positivity.

Theorem

Let ${}R$ be a standard-graded normal domain over an algebraically closed field ${}K$ of characteristic ${}0$. Suppose that ${}R$ is Gorenstein and that it has an isolated singularity. Suppose that the differential signature of ${}R$ is positive. Then the following hold.

1. The ${}a$-invariant of ${}R$ is negative.
2. ${}R$ has a rational singularity.
3. ${}R$ is log-terminal.
4. The Kodaira dimension of ${}\operatorname {Proj} R$ is negative.
5. ${}\operatorname {Proj} R$ is Fano.

The main idea is to look at the canonical line bundle ${}{\mathcal {O}}_{X}(a)$, where ${}a$ is the ${}a$-invariant of the ring. A result of Hsiao gives that the simplicity of ${}R$ as a module over the ring of differential operators implies that the tangent bundle ${}{\mathcal {T}}_{X}$ on ${}X$ is big. We have

${}\bigwedge ^{\operatorname {dim} (X)}{\mathcal {T}}_{X}\cong {\mathcal {O}}_{X}(-a)\,.$

Assuming

${}a\geq 0\,$

implies that ${}{\mathcal {T}}_{X}$ is semistable and then restricting to a generic complete intersetion curve gives a contradiction.

Ths theorem says in particular that a homogeneous hypersurface with an isolated singularity where the degree is larger than the dimension of the cone the differential signature is ${}0$. Such singularites are not mild in the sense of differential signature. This includes cones over elliptic curves, cones over surfaces of degree ${}4$, and so on.

Problem: Does the converse of this Theorem hold? In particular we do not know whether the cone over a cubic in four variables like ${}X^{3}+Y^{3}+Z^{3}+W^{3}$ has positive differential signature. It is known that the ${}F$-signature is positive in such a situation.

Differential signature and ${}F$-signature

Theorem

Let ${}R$ denote a local noetherian ring in positive characteristic.

If the ${}F$-signature of ${}R$ is positive, then also the differential signature of ${}R$ is positive, and we have the estimate

${}{\frac {\mu ^{d}}{d!}}\operatorname {DS} (R)\geq \operatorname {FS} (R)\,,$

where

${}\mu =\operatorname {dim} _{K^{p}}(R/{\mathfrak {m}}^{[p]})\,.$

If ${}R$ is ${}F$-pure and the differential signature is positive, then also the ${}F$-signature is positive. Moreover, if in this setting the differential signature exists as a limit, then the estimate

${}\operatorname {FS} (R)\geq {\frac {1}{d!}}\operatorname {DS} (R)\,$

holds.

The following example shows that without the ${}F$-pure hypothesis the second implication is not true.

Example

We consider in characteristic ${}p$ the ring given by the equation

${}X^{p}+Y^{p+1}+Z^{p+1}=0\,.$

This is a normal hypersurface ring. This ring is not ${}F$-pure, since ${}X\notin (Y,Z)$, but ${}X^{p}\in (Y^{p},Z^{p})$. Hence it is not strongly ${}F$-regular and so its ${}F$-signature is ${}0$. However, the differential signature is positive. This example is also an easy counterexample to the Zariski-Lipman conjecture.

The differential signature for quadrics

We want to discuss techniques which help to compute the differential signature for quadrics. The modules of principal parts are related by the short exact sequence of ${}R$-modules

$0\longrightarrow \Delta ^{n}/\Delta ^{n+1}\longrightarrow P_{R{|}K}^{n}\longrightarrow P_{R{|}K}^{n-1}\longrightarrow 0.$

Moreover, there is a surjective ${}R$-module homomorphism

$\operatorname {Sym} ^{n}(\Omega _{R{|}K})\longrightarrow \Delta ^{n}/\Delta ^{n+1}.$

Note that ${}\Omega _{R{|}K}=\Delta ^{1}/\Delta ^{2}$. The direct sums

$\bigoplus _{n}\operatorname {Sym} ^{n}(\Omega _{R{|}K})\longrightarrow \bigoplus _{n}\Delta ^{n}/\Delta ^{n+1}$

are called the tangent algebra and the graded associated ring (for the diagonal embedding). Composition and dualizing gives an exact sequence

$\operatorname {Diff} ^{n-1}(R,R)=\operatorname {Hom} _{R}(P_{R{|}K}^{n-1},R)\longrightarrow \operatorname {Diff} ^{n}(R,R)=\operatorname {Hom} _{R}(P_{R{|}K}^{n},R)\longrightarrow \operatorname {Hom} _{R}(\operatorname {Sym} ^{n}(\Omega _{R{|}K}),R).$

For an operator ${}E$ or order ${}\leq n$ and a symmetric product ${}df_{1}\cdots df_{n}$ of the generators we have

${}{\begin{aligned}E(df_{1}\cdots df_{n})&=\sum _{I\subseteq \{1,\ldots ,n\}}(-1)^{\#\left(I\right)}{\left(\prod _{i\notin I}f_{i}\right)}E{\left(\prod _{i\in I}f_{i}\right)}\\\end{aligned}}$

We have the following relation, where the symmetric product ${}\delta _{1}\cdots \delta _{n}$ of derivations is sent to the linear form, which sends a symmetric product ${}\omega _{1}\cdots \omega _{n}$ to

$\sum _{\pi \in S_{n}}\prod _{i=1}^{n}\delta _{\pi (i)}(\omega _{i}).$

Theorem

Let ${}R$ be a commutative ${}K$-Algebra over a field ${}K$ and let ${}\delta _{1},\ldots ,\delta _{n}$ be derivations.

Then the composition ${}\delta _{n}\circ \cdots \circ \delta _{1}$ is sent under the natural map

$\operatorname {Diff} _{K}^{n}(R,R)\longrightarrow \operatorname {Hom} _{R}(\operatorname {Sym} _{R}^{n}(\Omega _{R{|K}}),R)$

to the image of the symmetric product ${}\delta _{n}\cdots \delta _{1}$ under the natural map

$\operatorname {Sym} _{R}^{n}(\operatorname {Der} _{K}(R,R))\cong \operatorname {Sym} _{R}^{n}(\operatorname {Hom} _{R}(\Omega _{R{|K}},R))\longrightarrow \operatorname {Hom} _{R}(\operatorname {Sym} _{R}^{n}(\Omega _{R{|K}}),R).$

The relation between the differential operators with their action on the symmetric products is in particular helpful in the graded case. If ${}R$ is graded, then every differential operator has a decomposition into homogeneous operators. The degree of a homogeneous operator ${}E$ is given as the difference

$\operatorname {deg} (E(f))-\operatorname {deg} (f)$

for every homogeneous element of the ring. For example, the operator ${}x^{\nu }\partial ^{\lambda }$ on the polynomial ring has degree ${}\operatorname {deg} (\nu )-\operatorname {deg} (\lambda )$. The unitary condition imposes strong conditions on the degree of an operator. If a homogeneous operator is unitary and sends a homogeneous ${}f$ to ${}1$, then the degree of the operator is ${}-\operatorname {deg} (f)$. Hence the nonexistence of operators and of linear forms on symmetric powers in certain degree give bounds for the differential signature from above.

Lemma

Let ${}R$ denote a standard-graded ring over ${}K$ of dimension ${}d$ and of multiplicity ${}e$. Suppose that there exists ${}\alpha \in \mathbb {R} _{\geq 0}$ such that

${}\operatorname {Hom} _{R}(\operatorname {Sym} ^{n}(\Omega _{R{|}K}),R)_{\ell }=0\,$

for all ${}\ell <-\alpha n$.

Then the differential signature of ${}R$ is bounded from above by ${}e\alpha ^{d}$.

If ${}R$ is standard-graded and has an isolated singularity, so that its projective spectrum ${}\operatorname {Proj} R$ is smooth, then one can try to find such degree bounds for ${}\operatorname {Sym} ^{n}(\Omega _{R{|}K})$ and Homs whereof using methods from algebraic geometry.

We will apply this observation for the computation of the differential signature for quadrics.

Lemma

Let ${}K$ be an algebraically closed field of characteristic zero. Let ${}f$ be an irreducible quadric in ${}d+1$ variables, ${}d\geq 2$.

Then the symmetric powers ${}\operatorname {Sym} ^{n}(\operatorname {Syz} (x_{1},\ldots ,x_{d+1}))(m)$ on the quadric

${}Q_{d}=\operatorname {Proj} K[x_{1},\ldots ,x_{d+1}]/(f)\,$

have no nontrivial section for ${}m<{\frac {3}{2}}n$.

This is proved by induction on the dimension, the starting point is the case of a quadric curve inside the projective plane. Here the restriction of the syzygy bundle splits into line bundles.

Theorem

Let ${}K$ be an algebraically closed field of characteristic zero and ${}d\geq 2$.

Then the differential signature of the quadric hypersurface

${}R=K[X_{1},\ldots ,X_{d+1}]/{\left(X_{1}^{2}+\cdots +X_{d+1}^{2}\right)}\,$

is ${}{\left({\frac {1}{2}}\right)}^{d-1}$.

In dimension ${}2$ this ring can be realized as an invariant ring for the group ${}\mathbb {Z} /(2)$, and also as a monoid ring with the equation ${}XY=Z^{2}$. In dimension ${}3$ we have the monoid ring given by the equation ${}XY=ZW$. In general, the proof consists in two parts. On one hand, one has to show rather explicitly with the help of the Jacobi-Taylor matrices that there exist unitary differential operators of order ${}2$ on the quadric, namely of the form

${}E_{1}=(d-1)\partial _{1}+x_{1}\partial _{1}^{2}-\sum _{j\neq 1}x_{1}\partial _{j}^{2}+2\sum _{j\neq 1}x_{j}\partial _{1}\partial _{j}\,.$

A careful study compositions of these operators shows that the differential signature is ${}\geq {\left({\frac {1}{2}}\right)}^{d-1}$. Then Fakt gives the other estimate.

Compare this with the limit of the ${}F$-signatures as ${}p$ tends to infinity, which were computed by Gessel and Monsky.

${}d$	${}2$	${}3$	${}4$	${}5$	${}6$	${}7$
${}\lim _{p\rightarrow \infty }FS(R_{p})$	${}{\frac {1}{2}}$	${}{\frac {2}{3}}$	${}{\frac {19}{24}}$	${}{\frac {13}{15}}$	${}{\frac {659}{720}}$	${}{\frac {298}{315}}$

These numbers are ${}1-$the coefficient of ${}z^{d}$ in the expansion of ${}\tan z+\sec z$.