Zero loci of Bernstein–Sato ideals

Abstract

We prove a conjecture of the first author relating the Bernstein–Sato ideal of a finite collection of multivariate polynomials with cohomology support loci of rank one complex local systems. This generalizes a classical theorem of Malgrange and Kashiwara relating the b-function of a multivariate polynomial with the monodromy eigenvalues on the Milnor fibers cohomology.

Appendix

We recall some facts for not-necessarily commutative rings from [7, A.III and A.IV] that we use in the proof of the main theorem.

4.1.

Let A be a ring, by which we mean an associative ring with a unit element. Let ${\text{ Mod }}_{f}(A)$ be the abelian category of finitely generated left A-modules.

We say that A is a positively filtered ring if A is endowed with a ${\mathbb {Z}}$-indexed increasing exhaustive filtration $\{F_iA\}_{i\in {\mathbb {Z}}}$ of additive subgroups such that $F_iA\cdot F_jA\subset F_{i+j}A$ for all i, j in ${\mathbb {Z}}$, and $F_{-1}A=0$. The associated graded object ${\text{ gr }}^FA=\oplus _i (F_iA/F_{i-1}A)$ has a natural ring structure. When we do not need to specify the filtration, we write ${\text{ gr }}\,A$ for ${\text{ gr }}^FA$.

If A is a positively filtered ring such that ${\text{ gr }}\,A$ is noetherian, then A is noetherian, [7, A.III 1.27]. Here, noetherian means both left and right noetherian.

4.2.

Let A be a noetherian ring, positively filtered. A good filtration on $M\in {\text{ Mod }}_f(A)$ is an increasing exhaustive filtration $F_\bullet M$ of additive subgroups such that $F_iA\cdot F_jM\subset F_{i+j}M$ for all i, j in ${\mathbb {Z}}$, and such that its associated graded object ${\text{ gr }}\,M$ is a finitely generated graded module over ${\text{ gr }}\,A$, cf. [7, A.III 1.29].

Proposition 4.2.1

([7, A.III 3.20–3.23]) Let A be a noetherian ring, positively filtered.

• (1) Let M be in ${\text{ Mod }}_f(A)$ with a good filtration. Then the radical of the annihilator ideal in ${\text{ gr }}\,A$

  $$\begin{aligned} J(M):=\sqrt{{\text{ Ann }}_{{\text{ gr }}\,A}({\text{ gr }}\,M)} \end{aligned}$$

  and the multiplicities $m_{\mathfrak {p}}(M)$ of ${\text{ gr }}\,M$ at minimal primes ${\mathfrak {p}}$ of J(M) do not depend on the choice of a good filtration.

• (2) If

  $$\begin{aligned} 0\rightarrow M'\rightarrow M\rightarrow M''\rightarrow 0 \end{aligned}$$

  is an exact sequence in ${\text{ Mod }}_f(A)$ then

  $$\begin{aligned} J(M)=J(M')\cap J(M'') \end{aligned}$$

  and if ${\mathfrak {p}}$ is a minimal prime of J(M) then

  $$\begin{aligned} m_{\mathfrak {p}}(M)=m_{\mathfrak {p}}( M')+m_{\mathfrak {p}}( M''). \end{aligned}$$

Note that the last assertion is equivalent to the existence of a ${\mathbb {Z}}$-valued additive map $m_{{\mathfrak {p}}}$ on the Grothendieck group generated by the finitely generated modules N over ${\text{ gr }}\,A$ with $J(M)\subset \sqrt{{\text{ Ann }}_{{\text{ gr }}\,A} N}$, as it is phrased in loc. cit.

Proposition 4.2.2

([7, A.IV 4.5]) Let A be a noetherian ring, positively filtered. Let M be in ${\text{ Mod }}_f(A)$ with a good filtration. For every $k\ge 0$, there exists a good filtration on the right A-module ${\text{ Ext }}^k_A(M,A)$ such that ${\text{ gr }}\,({\text{ Ext }}^k_A(M,A))$ is a subquotient of ${\text{ Ext }}^k_{{\text{ gr }}\,A}({\text{ gr }}\,M,{\text{ gr }}\,A)$.

4.3.

Let A be a noetherian ring. The smallest $k\ge 0$ for which every M in ${\text{ Mod }}_f(A)$ has a projective resolution of length $\le k$ is called the homological dimension of A and it is denoted by $\text{ gl.dim }(A).$

Definition 4.3.1

For a nonzero M in ${\text{ Mod }}_f(A)$, the smallest integer $k\ge 0$ such that ${\text{ Ext }}^k_A(M,A)\ne 0$ is denoted

$$\begin{aligned} j_A(M) \end{aligned}$$

and it is called the grade number of M. If $M=0$ the grade number is taken to be $\infty$.

The ring A is Auslander regular if it has finite homological dimension and, for every M in ${\text{ Mod }}_f(A)$, every $k\ge 0$, and every nonzero right submodule N of ${\text{ Ext }}^k_A(M,A)$, one has $j_A(N)\ge k$. This implies the similar condition phrased for right A-modules M, see [7, A.IV 1.10] and the comment thereafter.

Theorem 4.3.2

([7, A.IV 5.1]) If A is a positively filtered ring such that ${\text{ gr }}\,A$ is a regular commutative ring, then A is an Auslander regular ring.

Proposition 4.3.3

([7, A.IV 1.11]) Let A be an Auslander regular ring. Then

$$\begin{aligned} \text{ gl.dim }(A)=\sup \{j_A(M)\mid 0\ne M\in {\text{ Mod }}_f(A)\}. \end{aligned}$$

Definition 4.3.4

A nonzero module M in ${\text{ Mod }}_f(A)$ is j-pure (or simply, pure) if $j_A(N)=j_A(M)=j$ for every nonzero submodule N.

Lemma 4.3.5

([7, A.IV 2.6]) Let A be an Auslander regular ring, M nonzero in ${\text{ Mod }}_f(A)$, and $j=j_A(M)$. Then:

• (1) ${\text{ Ext }}^j_A(M,A)$ is a j-pure right A-module;

• (2) M is pure if and only if ${\text{ Ext }}^k_A({\text{ Ext }}^k_A(M,A),A)=0$ for every $k\ne j$.

4.4.

We assume now that A is a positively filtered ring such that ${\text{ gr }}\,A$ is a regular commutative ring. Then A is also Auslander regular by Theorem 4.3.2. Moreover, with these assumptions one has the following two results.

Proposition 4.4.1

([7, A.IV 4.10 and 4.11]) If M in ${\text{ Mod }}_f(A)$ is j-pure, there exists a good filtration on M such that ${\text{ gr }}\,M$ is a j-pure ${\text{ gr }}\,A$-module.

Proposition 4.4.2

([7, A.IV 4.15]) For any M in ${\text{ Mod }}_f(A)$ and any good filtration on M,

$$\begin{aligned} j_A(M)=j_{{\text{ gr }}\,A}({\text{ gr }}\,M). \end{aligned}$$

4.5.

Lastly, we consider a regular commutative ring A. Then $\mathrm{gl.dim.}(A)=\sup \{\mathrm{gl.dim.}(A_{\mathfrak {m}})\mid {\mathfrak {m}}\subset A\text { maximal ideal}\,\}$, cf. [6, Ch. 2, 5.20]. We let $\dim (A)$ denote the Krull dimension. For a module $M\in {\text{ Mod }}_f(A)$, $\dim _A(M)$ denotes $\dim (A/ {\text{ Ann }}_A(M))$. If A is a regular local commutative ring, then $\dim (A)=\mathrm{gl.dim.}(A)$, cf. [7, A.IV 3.5].

Proposition 4.5.1

Let A be a regular commutative ring and M nonzero in ${\text{ Mod }}_f(A)$. Then:

• (i) ([7, A.IV 3.4]) A is Auslander regular;

• (ii) ([6, Ch. 2, Thm. 7.1]) if $\dim (A_{\mathfrak {m}})=m$ for every maximal ideal ${\mathfrak {m}}$ of A,

  $$\begin{aligned} j_A(M)+\dim _A(M) = m\, ; \end{aligned}$$

• (iii) ([7, A.IV 3.7 and 3.8]) M is a pure A-module if and only if every associated prime of M is a minimal prime of M and $j_A(M)=\dim (A_{{\mathfrak {p}}})$ for every minimal prime ${\mathfrak {p}}$ of M.