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.

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Notes

  1. 1.

    This is called “le complexe d’Alexander” in [24].

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Acknowledgements

We would like to thank L. Ma, P. Maisonobe, C. Sabbah for some discussions, to M. Mustaţă for drawing our attention to a mistake in the first version of this article, and to the referees for comments that helped improve the article.

The first author was partly supported by the grants STRT/13/005 and Methusalem METH/15/026 from KU Leuven, G097819N and G0F4216N from the Research Foundation - Flanders. The second author is supported by a PhD Fellowship of the Research Foundation - Flanders. The fourth author is supported by the Simons Postdoctoral Fellowship as part of the Simons Collaboration on HMS.

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Appendix

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 ij 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 ij 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. (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. (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. (1)

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

  2. (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:

  1. (i)

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

  2. (ii)

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

    $$\begin{aligned} j_A(M)+\dim _A(M) = m\, ; \end{aligned}$$
  3. (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.

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Budur, N., van der Veer, R., Wu, L. et al. Zero loci of Bernstein–Sato ideals. Invent. math. (2021). https://doi.org/10.1007/s00222-020-01025-x

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