[Paper Review] Ulrich ideals and almost Gorenstein rings

Shirô Gotô, Ryo Takahashi+1 more|arXiv (Cornell University)|Jul 16, 2015

Commutative Algebra and Its Applications15 references3 citations

TL;DR

This paper investigates Ulrich ideals in one-dimensional almost Gorenstein local rings, establishing a structural theorem for the derived Hom complex $\operatorname{\mathbf{R}Hom}_{R}(R/I,R)$ when $I$ is an Ulrich ideal. It proves that in a one-dimensional non-Gorenstein almost Gorenstein local ring, the only non-parameter Ulrich ideal is the maximal ideal, resolving a conjecture of Watanabe and showing that Ulrich ideals in such rings have a unique minimal number of generators.

ABSTRACT

The structure of the complex $\operatorname{\mathbf{R}Hom}_R(R/I,R)$ is explored for an Ulrich ideal $I$ in a Cohen-Macaulay local ring $R$. As a consequence, it is proved that in a one-dimensional almost Gorenstein but non-Gorenstein local ring, the only possible Ulrich ideal is the maximal ideal. It is also studied when Ulrich ideals have the same minimal number of generators.

Motivation & Objective

• To clarify the structure of Ulrich ideals in almost Gorenstein local rings, particularly in one-dimensional settings.
• To resolve the conjecture that in one-dimensional non-Gorenstein almost Gorenstein rings, the only Ulrich ideal is the maximal ideal.
• To investigate whether all Ulrich ideals in such rings have the same minimal number of generators.
• To provide a method for constructing Ulrich ideals with different numbers of generators in non-almost-Gorenstein rings.

Proposed method

• Deriving a structure theorem for the derived complex $\operatorname{\mathbf{R}Hom}_{R}(R/I,R)$ for Ulrich ideals in Cohen–Macaulay local rings.
• Using the isomorphism $\operatorname{\mathbf{R}Hom}_{R}(R/I,R) \cong \bigoplus_{i\in\mathbb{Z}}(R/I)^{\oplus u_i}[-i]$ in the derived category, with explicit $u_i$ depending on $\nu(I) - d$.
• Applying this structure to characterize finitess of G-dimension and Bass numbers in terms of $\nu(I)$.
• Constructing Ulrich ideals with varying numbers of generators via a quotient of a polynomial ring $S = A[X_1,\dots,X_\ell]$ modulo an ideal $\mathfrak{c}$ generated by $X_i^2 - a_i$ and $X_iX_j$ for $i \neq j$, with $a_i \in \mathfrak{q}^2$.
• Verifying that the constructed ideals are Ulrich by showing $I^2 = QI$ and $\ell_{A}(I/Q) = \ell \cdot \ell_{A}(A/\mathfrak{q})$, ensuring $I$ is Ulrich.
• Using the construction to produce examples where $\nu(I) = d + \ell$ and $\nu(J) = n > d + \ell$ for different ideals $I,J$ in the same ring $R$.

Experimental results

Research questions

• RQ1 In a one-dimensional non-Gorenstein almost Gorenstein local ring, what are the possible Ulrich ideals?
• RQ2 Do all Ulrich ideals in a non-Gorenstein almost Gorenstein local ring have the same minimal number of generators?
• RQ3 Can Ulrich ideals with different numbers of generators exist in non-almost-Gorenstein rings?
• RQ4 How does the derived complex $\operatorname{\mathbf{R}Hom}_{R}(R/I,R)$ behave for Ulrich ideals in Cohen–Macaulay rings?
• RQ5 What conditions ensure that an ideal $I$ is Ulrich in a ring $R$ constructed as a quotient of a polynomial ring over a local ring?

Key findings

• In a one-dimensional non-Gorenstein almost Gorenstein local ring, the only non-parameter Ulrich ideal is the maximal ideal.
• For a non-Gorenstein almost Gorenstein ring of prime Cohen–Macaulay type, all Ulrich ideals have the same minimal number of generators.
• If $R$ is G-regular (e.g., non-Gorenstein with minimal multiplicity or non-Gorenstein almost Gorenstein), then $\nu(I) \geq d + 2$ for any Ulrich ideal $I$.
• The derived complex $\operatorname{\mathbf{R}Hom}_{R}(R/I,R)$ decomposes as a direct sum of shifts of $R/I$, with $u_i$ explicitly computed in terms of $\nu(I) - d$.
• In non-almost-Gorenstein rings, one can construct Ulrich ideals with different numbers of generators, such as $\nu(I) = 2$ and \nu(J) = n > 2$ in the same ring $R$.
• The construction via $R = S/\mathfrak{c}$ with $S = A[X_1,\dots,X_\ell]$ and $\mathfrak{c} = (X_i^2 - a_i, X_iX_j \text{ for } i \neq j)$ yields Ulrich ideals with $\nu(I) = d + \ell$ when $a_i \in \mathfrak{q}^2$.