[Paper Review] The resolution of the universal ring for finite length modules of projective dimension two

Andrew R. Kustin|ArXiv.org|Jul 25, 2006

Commutative Algebra and Its Applications15 references3 citations

TL;DR

This paper constructs a coordinate-free, characteristic-independent resolution of the universal ring $\mathcal{R}$ for finite-length modules of projective dimension two, using a mapping cone of Koszul complexes over a polynomial ring $\mathcal{P}$. The key result shows that $\operatorname{Tor}^\mathcal{P}_\bullet(\mathcal{R},\mathbb{Z})$ is not a free abelian group when $e,g \geq 5$, implying that the graded Betti numbers of $\mathcal{K} \otimes_\mathbb{Z} \mathcal{R}$ depend on the characteristic of the field $\mathcal{K}$.

ABSTRACT

Hochster established the existence of a commutative noetherian ring $\Cal R$ and a universal resolution $\Bbb U$ of the form $0 o \Cal R^{e} o \Cal R^{f} o \Cal R^{g} o 0$ such that for any commutative noetherian ring $S$ and any resolution $\Bbb V$ equal to $0 o S^{e} o S^{f} o S^{g} o 0$, there exists a unique ring homomorphism $\Cal R o S$ with $\Bbb V=\Bbb U\otimes_{\Cal R} S$. In the present paper we assume that $f=e+g$ and we find a resolution $\Bbb F$ of $\Cal R$ by free $\Cal P$-modules, where $\Cal P$ is a polynomial ring over the ring of integers. The resolution $\Bbb F$ is not minimal; but it is straightforward, coordinate free, and independent of characteristic. Furthermore, one can use $\Bbb F$ to calculate $\operatorname{Tor}^{\Cal P}_{\bullet}(\Cal R, \Bbb Z)$. If $e$ and $g$ both at least 5, then $\operatorname{Tor}^{\Cal P}_{\bullet}(\Cal R, \Bbb Z)$ is not a free abelian group; and therefore, the graded betti numbers in the minimal resolution of $\pmb K\otimes_{\Bbb Z} \Cal R$ by free $\pmb K\otimes_{\Bbb Z} \Cal P$-modules depend on the characteristic of the field $\pmb K$. We record the modules in the minimal $\pmb K\otimes_{\Bbb Z} \Cal P$ resolution of $\pmb K\otimes_{\Bbb Z} \Cal R$ in terms of the modules which appear when one resolves divisors over the determinantal ring defined by the $2 imes 2$ minors of an $e imes g$ matrix.

Motivation & Objective

• To provide a coordinate-free, characteristic-independent resolution of the universal ring $\mathcal{R}$ for finite-length modules of projective dimension two.
• To establish a resolution $\mathbb{F}$ of $\mathcal{R}$ over a polynomial ring $\mathcal{P}$, enabling computation of $\operatorname{Tor}^\mathcal{P}_\bullet(\mathcal{R},\mathbb{Z})$.
• To demonstrate that the graded Betti numbers in the minimal resolution of $\mathcal{K} \otimes_\mathbb{Z} \mathcal{R}$ depend on the characteristic of the field $\mathcal{K}$ when $e$ and $g$ are both at least 5.
• To express the minimal resolution of $\mathcal{K} \otimes_\mathbb{Z} \mathcal{R}$ in terms of modules arising from resolving divisors over the determinantal ring defined by $2 \times 2$ minors of an $e \times g$ matrix.

Proposed method

• The resolution $\mathbb{F}$ is constructed as the mapping cone of a map between complexes involving Schur functors and exterior powers of $E \otimes G^*$, $F^*$, and $G^*$.
• The construction uses the Koszul complex associated with the composition $X \circ V: E \to G$, which induces a map $E \otimes G^* \to \mathcal{P}$.
• The resolution is built from tensor products of symmetric powers of $G^*$, exterior powers of $F^*$, and divided powers of $E$, indexed by triples $(a,c,d)$.
• The resolution is infinite but straightforward and independent of the characteristic of the base ring.
• The structure of the resolution is analyzed via the homology of complexes $\mathbb{M}(p,q)$, which are used to compute $\operatorname{Tor}$-modules.
• The minimal resolution of $\mathcal{K} \otimes_\mathbb{Z} \mathcal{R}$ is derived from the $\mathcal{P}$-resolution by base change, with modules expressed in terms of Schur modules and their twists.

Experimental results

Research questions

• RQ1 Does the resolution of the universal ring $\mathcal{R}$ for finite-length modules of projective dimension two admit a characteristic-independent, coordinate-free construction?
• RQ2 How does the structure of $\operatorname{Tor}^\mathcal{P}_\bullet(\mathcal{R},\mathbb{Z})$ relate to the Betti numbers of $\mathcal{K} \otimes_\mathbb{Z} \mathcal{R}$?
• RQ3 What is the dependence of the graded Betti numbers in the minimal resolution of $\mathcal{K} \otimes_\mathbb{Z} \mathcal{R}$ on the characteristic of the field $\mathcal{K}$?
• RQ4 Can the minimal resolution of $\mathcal{K} \otimes_\mathbb{Z} \mathcal{R}$ be described in terms of modules from the resolution of divisors over the determinantal ring of $2 \times 2$ minors of an $e \times g$ matrix?
• RQ5 What is the role of the Buchsbaum-Eisenbud multipliers and the factorization theorem in defining the universal ring $\mathcal{R}$?

Key findings

• The resolution $\mathbb{F}$ of $\mathcal{R}$ over $\mathcal{P}$ is infinite, coordinate-free, and independent of the characteristic of the base ring.
• The resolution $\mathbb{F}$ is constructed as the mapping cone of a map between complexes built from Koszul complexes and Schur functors.
• When $e$ and $g$ are both at least 5, $\operatorname{Tor}^\mathcal{P}_\bullet(\mathcal{R},\mathbb{Z})$ is not a free abelian group, indicating that the Betti numbers of $\mathcal{K} \otimes_\mathbb{Z} \mathcal{R}$ depend on the characteristic of $\mathcal{K}$.
• The minimal resolution of $\mathcal{K} \otimes_\mathbb{Z} \mathcal{R}$ is expressed in terms of Schur modules $T(\alpha,\beta,\gamma)$, with specific ranks and twists determined by combinatorial data.
• For $e=3$, $g=4$, the resolution contains summands like $\mathcal{P}'[-3,-8]^{210}$ and $\mathcal{P}'[-6,-8]^{420}$, which cannot be predicted from the Hilbert function alone.
• The dimension of $\operatorname{Tor}^\mathfrak{P}_{3,5}(M_0,\boldsymbol{K})$ depends on the characteristic of $\boldsymbol{K}$ when $e,g \geq 5$, confirming the characteristic dependence of Betti numbers.