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.

- We have $\mathrm{ca}(R) = R$ if and only if $\mathrm{gldim} R < \infty$.
- (IT) If $R$ is a localization of a finitely generated algebra over a perfect field of finite Krull dimension, then the vanishing locus of $\mathrm{ca}(R)$ is the singular locus of $R$.

**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 $T = \mathbb{C}[t^3,t^5]$.

- Then, the conductor ideal of $T$ is $(t^8, t^9, t^{10})$,
- We have a ring isomorphism $ \mathbb{C}[t^3,t^5]\cong \mathbb{C}[x,y]/(x^3+y^5)$,
- Under this isomorphism, $(t^8, t^9, t^{10})$ maps to $(x^2,xy,y^3)$.

Joint with

Back to the theorem in my thesis:

- We are in the 1 dimensional Gorenstein setting. An element $r \in R$ is in $\mathrm{ca}(R)$ iff $r \in \underline{\mathrm{ann}}_R(M) := \mathrm{ann}_R \underline{\mathrm{End}}_R(M)$ for all $M \in \mathrm{MCM}(R)$.
- The normalization $\overline{R}$ is maximal Cohen-Macaulay as an $R$-module.
- We have $\mathrm{co}(R) = \underline{\mathrm{ann}}_R(M)$.
- The main theorem statement: if an element stably annihilates a single MCM module (in this case $\overline{R}$), then it annihilates the entire singularity category.

**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 $\mathrm{cl}$:

- For all $M, N \in \mathrm{MCM}(R)$, we have $\mathrm{cl}(M\oplus N) = \mathrm{cl}(M) \oplus \mathrm{cl}(N)$.
- If $N \in \mathrm{add}(M)$, then we have $M \in \mathrm{cl}(N)$.
- For any integer $n$, we have that $\mathrm{cl}(M) = \mathrm{cl}(\Omega^n M)$.

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

- Free operations: finite direct sums, direct summands and shifts,
- The cone operation: 1 DK.

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

- $\Sigma(1) = \mathrm{proj}(R)$
- $\Sigma(r) = \mathrm{MCM}(R)$ if and only if $r \in \mathrm{ca}(R)$.

\[ \Sigma(r)=\{M \in \mathrm{MCM}(R) \colon r \in \underline{\mathrm{ann}}_R(M) \}. \]

**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 $\dim \underline{\mathrm{MCM}}(R) = 1$.

- This work was inspired by the work of Dumas-Leuschke who was inspired by the work of Knorrer and Herzog-Popescu.
- We can actually do this in much more generality. (ET) Let $x_1, \ldots, x_n$ be elements of a commutative Noetherian ring $R$ such that the product $x_1 \cdots x_n$ belongs to $\mathrm{ann}_R \mathsf{D}_\mathrm{sg}(R)$. If no $x_i$ is a unit or a zerodivisor, then we have \[\dim \mathsf{D}_\mathrm{sg}(R) \leq \sum_{i = 1}^n \dim \mathsf{D}_\mathrm{sg}(R/x_i R) + n - 1 \]

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 \[ 0 \to \Omega M \to \Omega(M/rM) \to M \to 0 \quad (*) \]

- We have that $\Omega(M/rM) \in \Sigma(r)$.
- (*) is a right $\Sigma(r)$-approximation of $M$.
- (*) is a left $\Sigma(r)$-approximation of $\Omega M$.
- $\Sigma(r)$ is functorially finite in $\mathrm{MCM}(R)$.

$\mathrm{MCM}(R)$ 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 \[ \mathsf{T}(r) = \frac{\mathrm{MCM}(R)}{[\Sigma(r)]}. \]

- $\mathsf{T}(r)$ is the usual singularity category if $r=1$.
- $\mathsf{T}(r) = 0$ if and only if $r \in \mathrm{ca}(R)$.

**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.