Subsection Gorenstein
Now, we pursue the injective analogues of the questions we addressed in the Section on Auslander, Buchsbaum, and Serre: What is the injective dimension of a finitely generated module when it is finite? Are there natural conditions on a ring that ensure finite injective dimension for all, or for some reasonable class, of finitely generated modules? To start, we make an observation about Bass numbers.
Definition 20.12.
Definition 4.11. The injective dimension of a finitely generated \(R\)-module \(M\text{,}\) denoted \(\operatorname{injdim}_{R}(M)\text{,}\) is the length of its minimal injective resolution. Note that any injective resolution of \(M\) has length at least equal to that of the minimal injective resoltuion, since a Bass number can be computed as an Ext, whose nonvanishing implies nonvanishing in that spot for any injective resolution.
Proposition 20.13.
Proposition 4.12. Let \(R\) be noetherian, and \(M\) finitely generated. If \(\mu_{i, \mathfrak{p}}(M) \neq 0\text{,}\) for some prime \(\mathfrak{p}\text{,}\) then \(\mu_{i+1, \mathfrak{q}}(M) \neq 0\) for all \(\mathfrak{q}\) with \(\mathrm{ht}(\mathfrak{q} / \mathfrak{p})=1\text{.}\)
Proof.
Proof. Since Bass numbers behave well under localization, we can localize at \(\mathfrak{q}\text{,}\) and assume that \(\operatorname{dim}(R / \mathfrak{p})=1\text{.}\) Pick \(x \in \mathfrak{q} \backslash \mathfrak{p}\text{.}\) From the SES
\begin{equation*}
0 \rightarrow R / \mathfrak{p} \stackrel{x}{\longrightarrow} R / \mathfrak{p} \rightarrow R /(\mathfrak{p}+(x)) \rightarrow 0
\end{equation*}
we get the LES
\begin{equation*}
\cdots \rightarrow \operatorname{Ext}_{R}^{i}(R / \mathfrak{p}, M) \stackrel{x}{\longrightarrow} \operatorname{Ext}_{R}^{i}(R / \mathfrak{p}, M) \rightarrow \operatorname{Ext}_{R}^{i+1}(R /(\mathfrak{p}+(x)), M) \rightarrow \cdots
\end{equation*}
Suppose that \(\mu_{i+1, \mathfrak{q}}(M)=0\text{.}\) Then, \(\operatorname{Ext}_{R}^{i+1}(R / \mathfrak{q}, M)=0\text{.}\) The module \(R /(\mathfrak{p}+(x))\) has finite length, and an induction on length (similar arguments we saw earlier) shows that \(\operatorname{Ext}_{R}^{i+1}(R /(\mathfrak{p}+(x)), M)=\) 0 as well. It then follows by NAK that \(\operatorname{Ext}_{R}^{i}(R / \mathfrak{p}, M)=0\text{,}\) so \(\mu_{i, \mathfrak{p}}(M)=0\text{.}\)
Corollary 20.14.
Corollary 4.13. Let \(R\) be noetherian, and \(M\) a finitely generated \(R\)-module.
\(\operatorname{injdim}_{R}(M)=\max \left\{i \mid \operatorname{Ext}_{R}^{i}(R / \mathfrak{m}, M) \neq 0, \mathfrak{m}\right.\) a maximal ideal \(\}\text{.}\)
\(\operatorname{injdim}_{R}(M) \geq \operatorname{dim}(M)\text{.}\)
Proof.
Proof. The first statement is clear. For the second, we recall that the associated primes of \(M\) are the same as those of \(E_{R}(M)\text{,}\) so the zeroth Bass numbers are nonzero for the associated primes of \(M\text{.}\) Then, we are guaranteed a chain of Bass numbers as long as \(\operatorname{dim}(M)=\max \{\operatorname{dim}(R / \mathfrak{p}) \mid \mathfrak{p} \in\) \(\operatorname{Ass}(M)\}\text{.}\)
The following is the injective analogue of Auslander-Buchsbaum.
Theorem 20.15.
Let \((R, \mathfrak{m}, k)\) be a local ring, and \(M\) a finitely generated module. If \(M\) has finite injective dimension, then \(\operatorname{injdim}_{R}(M)=\operatorname{depth}(R)\text{.}\)
Proof.
Proof. Set \(e=\operatorname{injdim}_{R}(M)\) and \(d=\operatorname{depth}(R)\text{.}\) Let \(\underline{x}=x_{1}, \ldots, x_{d}\) be a maximal \(R\)-sequence.
To see \(e \geq d\text{,}\) compute \(\operatorname{Ext}_{R}^{d}(R / \underline{x} R, M)\) by the Koszul complex: this is \(H^{d}(\underline{x} ; M)=M / \underline{x} M \neq 0\text{,}\) so the injective resolution of \(M\) is at least this long.
To see \(e \leq d\text{,}\) we can take a SES \(0 \rightarrow k \rightarrow R / \underline{x} R \rightarrow C \rightarrow 0\) and the LES
\begin{equation*}
\cdots \rightarrow \operatorname{Ext}_{R}^{e}(R / \underline{x} R, M) \rightarrow \operatorname{Ext}_{R}^{e}(k, M) \rightarrow \operatorname{Ext}_{R}^{e+1}(C, M) \rightarrow \cdots
\end{equation*}
From the definition of \(e\text{,}\) the last term vanishes. Then, by Corollary ??(1), \(\operatorname{Ext}_{R}^{e}(k, M) \neq 0\text{,}\) so \(\operatorname{Ext}_{R}^{e}(R / \underline{x} R, M) \neq 0\text{.}\) Thus, \(e \leq \operatorname{pd}(R / \underline{x} R)=d\text{.}\)
We now pursue an analogue of Auslander-Buchsbaum-Serre. Here is one such statement:
Definition 20.17.
Definition 4.16 (Gorenstein local ring). A local ring \((R, \mathfrak{m}, k)\) is Gorenstein if for every system of parameters \(x_{1}, \ldots, x_{d}\text{,}\)
\(x_{1}, \ldots, x_{d}\) is a regular sequence (so \(R\) is \(C M\) )
the ideal \(\left(x_{1}, \ldots, x_{d}\right)\) is irreducible: it cannot be written as \(I \cap J\) for \(I, J \supsetneq\left(x_{1}, \ldots, x_{d}\right)\text{.}\)
This class of rings can be characterized in many ways. Bass was the first to point this out, and to illustrate to importance of this class of rings. We will start this pursuit with artinian rings.
Theorem 20.18.
Let \((R, \mathfrak{m}, k)\) be an artinian local ring. TFAE:
\(R\) is Gorenstein: (0) is irreducible
\(\displaystyle \operatorname{dim}_{k}(\operatorname{soc}(R))=1\)
\(\displaystyle E_{R}(k) \cong R\)
\(\displaystyle \operatorname{injdim}_{R}(R)<\infty\)
4’) \(R\) is an injective \(R\)-module
Proof.
Proof. (1) \(\Rightarrow(2)\) : Given two linearly independent elements \(f, g\) in the socle, \((f) \cap(g)=0\text{.}\)
\((2) \Rightarrow(1): R\) is \(\mathfrak{m}\)-torsion, hence is an essential extension of its socle, so any ideal contains a nonzero element there. If the socle is 1-dimensional, any ideal contains the whole socle.
\((2) \Rightarrow(3): R\) is an essential extension of \(k\text{,}\) so it embeds into \(E_{R}(k)\text{.}\) Since \(\ell(R)=\ell\left(R^{\vee}\right)=\) \(\ell\left(E_{R}(k)\right)\text{,}\) this is an isomorphism.
\((3) \Rightarrow(2)\) : The socle of \(E\) is a copy of \(k\text{.}\)
\((3) \Rightarrow\left(4^{\prime}\right)\) : Trivial.
\(\left(4^{\prime}\right) \Rightarrow(3)\) : The only injectives are copies of \(E\text{,}\) and \(R\) is an indecomposable module, so this is the only option from the structure theory.
\(\left(4^{\prime}\right) \Rightarrow(4)\) : Trivial.
\((4) \Rightarrow\left(4^{\prime}\right)\) : Because injective dimension is the depth.
We note that if \(R\) is artinian, then it contains a copy of \(k\text{,}\) and we can reinterpret (3) as saying that \(R \cong \operatorname{Hom}_{k}(R, k)\text{.}\)
We now want to extend this theorem to higher dimensions. We will use another theorem of Rees on Ext.
Theorem 20.19.
Let \(M, N\) be \(R\)-modules, and \(x \in \operatorname{ann}(M)\) be \(R\)-regular and \(N\)-regular. Then \(\operatorname{Ext}_{R}^{i+1}(M, N) \cong \operatorname{Ext}_{R / x R}^{i}(M, N / x N)\) for all \(i \geq 0\text{.}\)
Proof.
We will apply the same strategy as in the Fundamental Theorem of Local Cohomology to see that for the left-exact functor \(F(-)=\operatorname{Hom}_{R / x R}(-, N / x N)\) from \(R / x R\)-modules to \(R / x R\) modules, its right derived functors are \(R^{i} F(-)=\operatorname{Ext}_{R}^{i+1}(-, N)\text{.}\) Note that \(M\) is an \(R / x R\)-module by hypothesis. By definition, \(R^{i} F(-)=\operatorname{Ext}_{R / x R}^{i}(M, N / x N)\text{,}\) so this will establish the theorem.
Step 1: The functors agree when \(i=0\text{.}\) Indeed, for the SES of \(R\)-modules \(0 \rightarrow N \stackrel{x}{\longrightarrow} N \rightarrow\) \(N / x N \rightarrow 0\text{,}\) there is an LES
\begin{equation*}
\cdots \rightarrow \operatorname{Hom}_{R}(M, N) \rightarrow \operatorname{Hom}_{R}(M, N / x N) \rightarrow \operatorname{Ext}_{R}^{1}(M, N) \stackrel{x}{\longrightarrow} \operatorname{Ext}_{R}^{1}(M, N) \rightarrow \cdots
\end{equation*}
We observe that \(\operatorname{Hom}_{R}(M, N)=0\) (any element in \(M\) has to map to something killed by \(x\) ), that \(\operatorname{Hom}_{R}(M, N / x N)=\operatorname{Hom}_{R / x R}(M, N / x N)\) (since both sides are killed by \(x\) ), and \(x\) kills \(\operatorname{Ext}_{R}^{1}(M, N)\) (since it kills \(M\) ). The specified isomorphism follows.
Step 2: \(\operatorname{Ext}_{R}^{i+1}(-, N)\) vanishes for free \(R / x R\)-modules for \(i>0\text{.}\) Indeed, a free \(R / x R\)-module has a free \(R\)-resolution of length one, so computing the Ext from this resolution shows the vanishing.
Step 3: This follows in the same way as Step 3 of the Fundamental Theorem of Local Cohomology. We leave this as an exercise.
Theorem 20.20.
Let \((R, \mathfrak{m}, k)\) be a local ring of dimension \(d\text{.}\) TFAE:
\(R\) is Gorenstein
1’) \(R\) is \(C M\) and some \(S O P\) generates an irreducible ideal
\(\displaystyle \operatorname{Ext}_{R}^{i}(k, R) \cong \begin{cases}0 & i<d \\ k & i=d\end{cases}\)
\(R\) is \(C M\) and \(\mathrm{H}_{\mathfrak{m}}^{d}(R) \cong E_{R}(k)\text{.}\)
\(\displaystyle \operatorname{injdim}_{R}(R)<\infty\)
4’) \(\operatorname{injdim}_{R}(R)=d\)
Proof.
Proof. First we want to observe that every condition implies that \(R\) is Cohen-Macaulay. For (2), this is due to Rees’s theorem on depth and Ext (the one from a while ago). For (4), and consequently for \((4 ')\text{,}\) this follows from \(\operatorname{dim}(R) \leq \operatorname{injdim}_{R}(R)=\operatorname{depth}(R)\text{.}\) For the others, this is explicit. We assume that \(R\) is Cohen-Macaulay henceforth.
\((1) \Rightarrow\left(1^{\prime}\right)\) : trivial
\(\left(1^{\prime}\right) \Rightarrow(2)\) : The Ext vanishings follow from \(\mathrm{CM}\) as noted above. Now, let \((\underline{x})\) be the given irreducible parameter ideal. The zero ideal in \(R /(\underline{x})\) is then irreducible, and by the artinian case, \(\operatorname{Hom}_{R / \underline{x} R}(k, R / \underline{x} R) \cong k\text{.}\) Applying the previous theorem \(d\) times, we obtain that \(\operatorname{Ext}_{R}^{d}(k, R) \cong k\text{.}\)
\((2) \Rightarrow(1)\) : Given any SOP \(\underline{x}\) of \(R\text{,}\) as in the previous implication (backwards), we see that \(\operatorname{Hom}_{R / \underline{x} R}(k, R / \underline{x} R) \cong k\text{,}\) and apply the artinian case to see that \((0)\) is irreducible in \(R / \underline{x} R\text{,}\) so \((\underline{x})\) is irreducible in \(R\text{.}\)
As a consequence of these implications, we see that if \(R\) is Gorenstein local, and \(\underline{x}\) a SOP, then \(R\) is Gorenstein iff \(R / \underline{x} R\) is.
\((1) \Rightarrow\left(4^{\prime}\right) \Rightarrow(4) \Rightarrow(1)\) : Applying Rees’s recent theorem again, we find that if \(\underline{x}\) is an SOP, then
\(\operatorname{injdim}_{R}(R)=\max \left\{t \mid \operatorname{Ext}_{R}^{t}(k, R) \neq 0\right\}=d+\max \left\{t \mid \operatorname{Ext}_{R / \underline{x} R}^{t}(k, R / \underline{x} R) \neq 0\right\}=d+\operatorname{injdim}_{R / \underline{x} R}(R / \underline{x} R)\text{.}\)
Then, \(R\) is Gorenstein implies \(R / \underline{x} R\) is too, and that it has injective dimension zero (by the artinian case), so \(R\) has injective dimension \(d\text{.}\) If \(R\) has finite injective dimension, so does \(R / \underline{x} R\text{,}\) so it is Gorenstein, and \(R\) is too.
\((2)+\left(4^{\prime}\right) \Rightarrow(3)\) : By \((2)\text{,}\) know that \(\mu_{i, \mathfrak{m}}(R)\) is zero for \(i<d\) and is one for \(i=d\text{;}\) by (4) it is zero for \(i>d\text{.}\) Thus, taking \(\Gamma_{\mathfrak{m}}\) of an injective resolution of \(R\) leaves just one copy of \(E_{R}(k)\) in cohomological degree \(d\text{.}\)
\((3) \Rightarrow(2)\) : Again, the Ext vanishing is a consequence of Cohen-Macaulayness. By the Lemma on ascending Bass numbers, \(\operatorname{Ext}_{R}^{d}(k, R) \neq 0\text{.}\) In the Theorem characterizing Bass numbers in terms of Ext, we saw that if \(E^{\bullet}\) is an injective resolution of \(R, \operatorname{Hom}_{R}\left(k, E^{\bullet}\right)\) has vanishing differentials; this was explicitly stated and established as a claim. This means that, in a minimal injective resolution, any socle element is in the kernel of the differential. Now consider \(\Gamma_{\mathfrak{m}}\left(E^{\bullet}\right)\text{.}\) This is zero up to cohomological dimension \(d\text{.}\) Then, we have
\begin{equation*}
0 \rightarrow E_{R}(k)^{\oplus \mu_{d, \mathfrak{m}}} \stackrel{d}{\longrightarrow} E_{R}(k)^{\oplus \mu_{d+1, \mathfrak{m}}} \rightarrow \cdots
\end{equation*}
and \(k^{\oplus \mu_{d, \mathfrak{m}}} \cong \operatorname{soc}\left(E_{R}(k)^{\oplus \mu_{d, \mathfrak{m}}}\right) \subseteq \operatorname{ker}(d)=\mathrm{H}_{\mathfrak{m}}^{d}(R)\text{.}\)
The following is now evident from the remark after the proof of Local Duality for regular rings.
Corollary 20.21.
Corollary 4.20 (Local Duality for Gorenstein rings). Let \((R, \mathfrak{m}, k)\) be a Gorenstein local ring of dimension \(d\text{.}\) Then
If \(M\) is a finitely generated \(R\)-module, then \(\mathrm{H}_{\mathfrak{m}}^{i}(M)=\operatorname{Ext}_{d-i}^{R}(M, R)^{\vee}\text{.}\)
If \(R\) is complete and \(M\) is arbitrary, then \(\mathrm{H}_{\mathfrak{m}}^{i}(M)^{\vee}=\operatorname{Ext}_{d-i}^{R}(M, R)\text{.}\)
These isomorphisms are functorial in \(M\text{.}\)
Subsection Worksheet on Gorenstein rings
Let \((R, \mathfrak{m}, k)\) be a local ring. Show \({ }^{1}\) the following:
(a) \(R\) is Gorenstein if and only if \(\widehat{R}\) is Gorenstein.
(b) If \(x_{1}, \ldots, x_{i}\) is a regular sequence, then \(R\) is Gorenstein if and only if \(R /\left(x_{1}, \ldots, x_{i}\right)\) is Gorenstein.
(c) If \(R\) is Gorenstein, and \(\mathfrak{p} \in \operatorname{Spec}(R)\text{,}\) then \(R_{\mathfrak{p}}\) is Gorenstein.
Definition 20.22.
Definition 4.21. A local ring \((R, \mathfrak{m}, k)\) is a complete intersection if there is a surjection from a complete regular local ring \(S\) onto \(\widehat{R}\) with the kernel generated by a regular sequence.
Not all of these equivalences are obvious, but we will use them freely.
Show that if \(R\) is a complete intersection, then \(R\) is Gorenstein.
Let \(K\) be a field. For each of the following rings \(R\) determine: Is \(R\) Gorenstein? Is \(R\) CohenMacaulay? Is \(R\) a complete intersection? Reuse your work from old worksheets and HW when convenient.
(a) \(R=\frac{K \llbracket x, y \rrbracket}{\left(x^{2}, x y\right)}\text{.}\)
(b) \(R=\frac{K \llbracket x, y, z \rrbracket}{(x y, x z, y z)}\text{.}\)
(c) \(R=\frac{K \llbracket x, y, z \rrbracket}{\left(x^{2}, y^{2}, z^{2}, x(y-z),(x-y) z\right)}\text{.}\)
(d) \(R=\frac{K\left[X_{2 \times 3}\right]_{\mathfrak{m}}}{I_{2}\left(X_{2 \times 3}\right)}\text{,}\) where \(\mathfrak{m}\) is the ideal generated by the entries of \(X\text{.}\)
(e) \(R=K[x, y]_{\mathfrak{m}}^{(2)}\text{,}\) where \(\mathfrak{m}\) is the ideal generated by the positive degree forms.
(f) \(R=K[x, y]_{\mathfrak{m}}^{(3)}\text{,}\) where \(\mathfrak{m}\) is the ideal generated by the positive degree forms.
(g) \(R=K[x, y, z]_{\mathfrak{m}}^{(3)}\text{,}\) where \(\mathfrak{m}\) is the ideal generated by the positive degree forms.
Let \((R, \mathfrak{m})\) and \((S, \mathfrak{n})\) be two complete Gorenstein local rings, with \(R=S / I\text{.}\)
(a) Use Local Duality to show that \(\operatorname{Ext}_{S}^{t}(R, S)= \begin{cases}0 & t<\operatorname{dim}(S)-\operatorname{dim}(R) \\ R & t=\operatorname{dim}(S)-\operatorname{dim}(R)\end{cases}\)
(b) Suppose moreover that \(S\) is regular. Let \(P_{\bullet} \rightarrow R\) be the minimal free resolution of \(R\) as an \(S\)-module. Show that, \(P_{\bullet} \cong \operatorname{Hom}_{S}\left(P_{\bullet}, S\right)\)
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(c) With the same assumptions as in the previous part, show that \(\operatorname{Tor}_{i}^{S}(R, M) \cong \operatorname{Ext}_{S}^{i}(R, M)\) for all \(S\)-modules \(M\text{.}\)
A numerical semigroup is a subsemigroup \(S\) of \(\mathbb{N}\text{;}\) our convention is that \(0 \in S\text{.}\) A numerical semigroup ring is a ring of the form \(K[S]:=K\left[\left\{x^{s} \mid s \in S\right\}\right] \subseteq K[x]\text{.}\) Assume that the GCD of the elements in \(S\) is 1 . Then, there is a largest number \(f_{S} \in \mathbb{N}\) such that \(f_{S} \notin S\text{,}\) called the Frobenius number of \(S\text{.}\)
(a) Show that \(\mathrm{H}_{\mathfrak{m}}^{1}(K[S])\) is generated as a \(K\) vector space by \(\left\{x^{a} \mid a \leq f_{S} \text { and } a \notin S\right\} !^{5}\)
(b) Show that \(K[S]_{\mathfrak{m}}\) is Gorenstein if and only if
\begin{equation*}
\left\{c \mid 0 \leq c \leq f_{S}, c \notin S\right\}=\left\{f_{S}-d \mid 0 \leq d \leq f_{S}, d \in S\right\}
\end{equation*}
(c) Check this criterion with \(K \llbracket x^{3}, x^{7} \rrbracket, K \llbracket x^{3}, x^{5}, x^{7} \rrbracket\text{,}\) and \(K \llbracket x^{4}, x^{5}, x^{6} \rrbracket\text{.}\)
Show that if \(R\) is a Gorenstein local ring, and \(M\) is a finitely generated \(R\)-module, then \(M\) has finite projective dimension if and only if \(M\) has finite injective dimension.
We summarize some implications and non-implications on the structural properties of local rings we have encountered:
\begin{equation*}
\text { regular } \Rightarrow \text { complete intersection } \Rightarrow \text { Gorenstein } \Rightarrow \text { Cohen-Macaulay }
\end{equation*}
complete intersection \(\nRightarrow\) regular: \(K[x] /\left(x^{2}\right), K\left[x^{2}, x y, y^{2}\right], \frac{K[x, y, z, u, v, w]}{\left(u^{2}+v^{2}+w^{2}, x^{2}+y^{2}+z^{2}\right)}, \ldots\)
Gorenstein \(\nRightarrow\) complete intersection: \(R=\frac{K \llbracket x, y, z \rrbracket}{\left(x^{2}, y^{2}, z^{2}, x(y-z),(x-y) z\right)}, R=K[x, y, z]_{\mathfrak{m}}^{(3)}, \ldots\)
Cohen-Macaulay \(\nRightarrow\) Gorenstein: \(R=\frac{K \llbracket x, y, z \rrbracket}{(x y, x z, y z)}, R=\frac{K\left[X_{2 \times 3}\right]_{\mathfrak{m}}}{I_{2}\left(X_{2 \times 3}\right)}, K \llbracket x^{3}, x^{5}, x^{7} \rrbracket, \ldots\)
