Annihilation of cohomology and applications

Ozgur Esentepe - University of Leeds
Aarhus Homological Algebra Seminar - September 6, 2023

Let $R$ be a commutative Noetherian ring of Krull dimension $d$.

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 $s$.

Today's Goal

The main goal is to tell you stories about these annihilators.

Some Facts

  • 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 $R$.
  • When $R$ is a Gorenstein ring, we have \[ \mathrm{ca}(R) = \mathrm{ann}_R D_{\mathrm{sg}}(R)= \mathrm{ann}_R \underline{\mathrm{MCM}}(R) \] This means $r \in \mathrm{ca}(R)$ iff $r \underline{\mathrm{Hom}}_R(M,N) = 0$ for all $M, N \in \mathrm{MCM}(R)$ iff $r \underline{\mathrm{End}}_R(M) = 0$ for all $M \in \mathrm{MCM}(R)$.
  • 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 $(x^2,xy,y^3)$.

    A topological space

    Joint with

    Akdenizli, Aytekin, Cetin

    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.

    Dimension of the singularity category

    Joint with

    Ryo Takahashi

    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 \[\dim \mathsf{D}_\mathrm{sg}(R) \leq \sum_{i = 1}^n \dim \mathsf{D}_\mathrm{sg}(R/x_i R) + n - 1 \]

    A functorially finite subcategory

    joint with

    Benjamin Briggs

    \[ \xymatrix{ & & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & \Omega M \ar[r] \ar@{=}[d] & Q \ar[r] \ar[d] & M \ar[r] \ar[d]^{x} & 0 \\ 0 \ar[r] & \Omega M \ar[r] & P \ar[r]^{\varepsilon} \ar[d] & M \ar[r] \ar[d] & 0\\ & & M/xM \ar@{=}[r] \ar[d] & M/xM \ar[d] \\ & & 0 & 0 } \]

    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.