# Properties of Invariant Rings

The following functions return non-trivial structural properties of invariant rings of finite groups.

## `HilbertSeries(R): RngInvar -> FldFunUElt`

The Hilbert series of the invariant ring $R=K[V]^G$, returned as an element of the rational function field ${\mathbb{Z}}(t)$. The Molien series of $G$ will be used if possible; otherwise (the modular matrix group case) secondary invariants for $R$ will be constructed to determine the result.

## `HilbertSeriesApproximation(R, n): RngInvar, RngIntElt -> RngSerLaurElt`

The Hilbert series of the invariant ring $R=K[V]^G$, returned as a Laurent series with $n$ known terms. The conjugacy classes of $G$ will be used to compute the approximation.

## `IsCohenMacaulay(R): RngInvar -> BoolElt`

Given the invariant ring $R=K[V]^G$ of the group $G$ over the field $K$, return `true` iff $R$ is Cohen-Macaulay. This is always true in the non-modular case. Otherwise, secondary invariants for $R$ will be constructed to determine the result.

## `FreeResolution(R): RngInvar -> [ ModMPol ]`

Given the invariant ring $R=K[V]^G$ of the group $G$ over the field $K$, return a free resolution of (the module of) $R$. This is just the same as the invocation `FreeResolution(Module(R))`. The free resolution is returned as a sequence $F$ such that $F[1]$ is $M$, $F[i+1]$ is the syzygy module of $F[i]$ for $i<\#F$, and the last element of $F$ is free (its basis has no syzygies).

## `MinimalFreeResolution(R): RngInvar -> [ ModMPol ]`

Given the invariant ring $R=K[V]^G$ of the group $G$ over the field $K$, return a minimal free resolution of (the module of) $R$. This is just the same as the invocation `MinimalFreeResolution(Module(R))`.

## `HomologicalDimension(R): RngInvar -> RngInt`

Given the invariant ring $R=K[V]^G$ of the group $G$ over the field $K$, return the homological dimension of $R$. This is just the length of a minimal free resolution of $R$ minus 1 (taking account of the fact that the module $M$ of $R$ is always included in the free resolution).

## `Depth(R): RngInvar -> RngIntElt`

Given the invariant ring $R=K[V]^G$ of the group $G$ over the field $K$, return the depth of $R$. This is $n-d$ by the Auslander-Buchsbaum formula, where $n$ is the rank of $R$ and $d$ is the homological dimension of $R$.

## `Example: Depth (ex-6d79cb)`

We construct a minimal free resolution of the invariant ring of the group generated by the degree-5 Jordan block over ${\bf F}_{2}$ and verify that the depth is 3.

```magma
> K:=GF(2);
> G := MatrixGroup<5,K | [1,0,0,0,0, 1,1,0,0,0, 0,1,1,0,0,
>                         0,0,1,1,0, 0,0,0,1,1]>;
> R := InvariantRing(G);
> time F := MinimalFreeResolution(R);
Time: 0.690
> F;
Chain complex with terms of degree 3 down to -1
Dimensions of terms: 0 1 7 22 0
> Depth(R);
3
> HomologicalDimension(R);
2

```

Sections [Steenrod Operations](steenrod.md#rnginvar-steenrod) and [Minimalization and Homogeneous Module Testing](minimalize-module.md#rnginvar-minimalize-module) present functions whose scope is not limited to the context of invariant theory.