Ozgur Esentepe - University of Leeds
Aarhus Homological Algebra Seminar - September 6, 2023
Classical homological algebra, since around 1890s, tells us that the geometric object corresponding to $R$ is smooth if and only if $R$ has finite global dimension.
Finite global dimension $d$ means that for any two $R$-modules $M,N$, we have \[ \mathrm{Ext}_R^{d+1}(M,N) = 0 \] Equivalently, \[ \mathrm{ann}_R\mathrm{Ext}_R^{d+1}(M,N) = R \]
We define \[ \mathrm{ca}^n(R) = \bigcap_{M,N \in \mathrm{mod}(R)} \mathrm{ann}_R\mathrm{Ext}_R^{n}(M,N) \] and we put $\mathrm{ca}(R) = \bigcup_n \mathrm{ca}^n(R)$.
Since $\mathrm{Ext}^{n+1}(M,N) \cong \mathrm{Ext}^n(\Omega M,N)$, we have an ascending chain of ideals $\ldots \subseteq \mathrm{ca}^n(R) \subseteq \mathrm{ca}^{n+1}(R) \subseteq \ldots$.
And since our ring is Noetherian, this gives us that $\mathrm{ca}(R) = \mathrm{ca}^s(R)$ for some large $s$.
The main goal is to tell you stories about these annihilators.
An example: If $R = \mathbb{C}[x,y]/(x^3+y^5)$, then $\mathrm{ca}(R) = (x^2,xy,y^3)$. One way to see this is to compute stable annihilators of all indecomposable MCM modules. (This ring has finite CM-representation type).
When $R$ is a Gorenstein ring, $\underline{\mathrm{MCM}}(R) \cong \underline{\mathrm{ACP}}(R)$.
The homotopy category of acyclic complexes of projectives.
Thus, an element $r$ is a cohomology annihilator if and only if multiplication by $r$ is nullhomotopic on every acyclic complex of projective modules.
Assume that $R = \mathbb{C}[x_0, \ldots x_d]/(f)$ is a hypersurface ring. Any object in $\underline{\mathrm{ACP}}(R)$ is isomorphic to a 2-periodic complex, say given by two linear maps $A, B$.
Leibniz rule:
\[
0 = \frac{\partial}{\partial x_n}(AB) = \frac{\partial A}{\partial x_n} B + A \frac{\partial B}{\partial x_n}
\]
Assume that $R = \mathbb{C}[x_0, \ldots x_d]/(f)$ is a hypersurface ring. Any object in $\underline{\mathrm{ACP}}(R)$ is isomorphic to a 2-periodic complex, say given by two linear maps $A, B$.
Leibniz rule:
\[
0 = \frac{\partial}{\partial x_n}(AB) = \frac{\partial A}{\partial x_n} B + A \frac{\partial B}{\partial x_n}.
\]
Therefore, the ideal generated by the partial derivatives of $f$ is contained in the cohomology annihilator ideal of $R$.
More generally, the Jacobian ideal is contained in the cohomology annihilator ideal for complete intersections. (B, also IT).
In general, it is difficult to give a precise description of $\mathrm{ca}(R)$.
In my thesis, I showed that for reduced plane curves singularities, it coincides with
the conductor ideal.
\[ \mathrm{co}(R) = \{r \in R \colon r \overline{R} \subseteq R \} \] where $\overline{R}$ is the normalization of $R$.
Consider the semigroup $S$ generated by 3 and 5
i.e. $S = \{0,3,5,6,8,9,10,11,...\}$
and the semigroup ring $T = \mathbb{C}[t^3,t^5]$.
Joint with
Back to the theorem in my thesis:
Question: In general, can you find a special MCM module with this property?
Rephrase: let's put \[\mathrm{cl}(M) = \{L \in \mathrm{MCM}(R) \colon \underline{\mathrm{ann}}_R(L) \subseteq \underline{\mathrm{ann}}_R(M)\}.\]
Can you find an $X$ which belongs to $\mathrm{cl}(M)$ for every $M$?
Consider the set of isomorphism classes of MCM modules over $R$ and $\mathcal{S}$ be a subset. Define \[ \mathrm{cl}(\mathcal{S}) = \bigcup_{M \in \mathcal{S}} \mathrm{cl}(M) =\bigcup_{M \in \mathcal{S}} \{L\colon \underline{\mathrm{ann}}_R(L) \subseteq \underline{\mathrm{ann}}_R(M)\}. \]
Then, $\mathrm{cl}$ is a closure operator and defines a topology.
Original question: In general (over a Gorenstein ring), can you find an MCM module $M$ such that $\mathrm{ca}(R) = \underline{\mathrm{ann}}_R(M)$?
Theorem (AACE): The answer is affirmative if and only if the topological space $\mathrm{MCM}(R)$ is compact.
Some properties of $\mathrm{cl}$:
Note: You can do this for any $R$-linear small Krull-Schmidt category.
Joint with
The dimension of a triangulated category measures how much it costs to build it from a single object using
So far, we have been interested in ring elements which stably annihilate ALL maximal Cohen-Macaulay modules.
Now, let us pick an arbitrary element $r$ (assume it is a nzd) and define
\[
\Sigma(r)=\{M \in \mathrm{MCM}(R) \colon r \in \underline{\mathrm{ann}}_R(M) \}.
\]
Note that
\[ \Sigma(r)=\{M \in \mathrm{MCM}(R) \colon r \in \underline{\mathrm{ann}}_R(M) \}. \]
Lemma: Say we have a triangle \[ M \to L \to N \to \Omega^{-1}M \] and $M \in \Sigma(r)$ and $N \in \Sigma(s)$, then $L \in \Sigma(rs)$.
Proof: Use the long exact sequence you get from this triangle.
\[ \Sigma(r)=\{M \in \mathrm{MCM}(R) \colon r \in \underline{\mathrm{ann}}_R(M) \}. \]
An improved version of the Lemma: We have \[ \Sigma(rs) = \Sigma(r) \diamond \Sigma(s). \]
In particular, \[ \Sigma(r^n) = [\Sigma(r)]_n \]
Corollary: If $r^n \in \mathrm{ca}(R)$, then \[\underline{\mathrm{MCM}}(R) = [\Sigma(r)]_n \]
\[ \Sigma(r)=\{M \in \mathrm{MCM}(R) \colon r \in \underline{\mathrm{ann}}_R(M) \}. \]
Corollary: If $r^n \in \mathrm{ca}(R)$, then $\underline{\mathrm{MCM}}(R) = [\Sigma(r)]_n$.
So, if we can say that $\Sigma(r) = \mathrm{add}(M)$ for some $M$, we can deduce that \[\dim \underline{\mathrm{MCM}}(R) \leq n -1\]
provided that $r^n \in \mathrm{ca}(R)$.
Definition: $ \Sigma(r)=\{M \in \mathrm{MCM}(R) \colon r \in \underline{\mathrm{ann}}_R(M) \}. $
Fact: $\Sigma(r) = \{ \Omega_R N \colon N \in \mathrm{MCM}(R/rR) \}$.
Example: Let $R = \mathbb{C}[x,y]/(x^3+y^n)$ with $n \geq 6$. Then, $x^2 \in \mathrm{ca}(R)$ and so $\underline{\mathrm{MCM}}(R) = [\Sigma(x)]_2$. Moreover, \[\Sigma(x) = \{\Omega_R N \colon N \in \mathrm{MCM}(\mathbb{C}[y]/y^n ) \} \]We conclude that $\dim \underline{\mathrm{MCM}}(R) = 1$.
joint with
I am just going to leave this here.
So, we have a short exact sequence
\[
0 \to \Omega M \to \Omega(M/rM) \to M \to 0
\]
and that
\[
\Omega_R(M/rM) \cong M \oplus \Omega M
\]
if and only if $r \mathrm{Ext}_R^1(M, \Omega M) = 0$
and when $R$ is Gorenstein and $M$ is MCM
iff $r \in \underline{\mathrm{ann}_R}(M)$
iff $M \in \Sigma(r)$.
From now on, assume $R$ is Gorenstein and $M$ is MCM. \[ \Sigma(r)=\{M \in \mathrm{MCM}(R) \colon r \in \underline{\mathrm{ann}}_R(M) \} = \{ \Omega_R N \colon N \in \mathrm{MCM}(R/rR)\} \] and we have a short exact sequence \[ 0 \to \Omega M \to \Omega(M/rM) \to M \to 0 \quad (*) \]
$\mathrm{MCM}(R)$ is an exact category.
We can consider a new exact structure on it given by short exact sequences divisible by $r$. i.e. We consider the exact structure given by the subbifunctor $r\mathrm{Ext}(-,-)$.
$M$ is projective in this exact structure if and only if $M \in \Sigma(r)$.
$M$ is injective in this exact structure if and only if $M \in \Sigma(r)$.
$\Sigma(r)$ is functorially finite, so there is enough projectives (and injectives).
We have a new Frobenius structure!
We have a new Frobenius structure on $\mathrm{MCM}(R)$ whose projective-injectives are $\Sigma(r)$.
So, we have a new triangulated category $\mathsf{T}(r)$ defined as the stable category \[ \mathsf{T}(r) = \frac{\mathrm{MCM}(R)}{[\Sigma(r)]}. \]
Theorem (BE): Let $(R, \mathfrak{m})$ be a Gorenstein local ring of Krull dimension $d$ and $x \in \mathfrak{m}$ be a nonzerodivisor such that \[x \in \underline{\mathrm{ann}}_{R_\mathfrak{p}}M_\mathfrak{p} \] for all $M \in \mathrm{MCM}(R)$. Then, for any $M,N \in \mathrm{MCM}(R)$, we have \[ D \mathrm{Hom}_{\mathsf{T(r)}}(M,N) \cong \mathrm{Hom}_{\mathsf{T(r)}}(N, \Omega^{d-1}M). \] In particular, $\mathsf{T}(r)$ is $(d-1)$-Calabi-Yau.