# Resolving Decompositions for Polynomial Modules

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Resolving Decompositions

**Definition**

**1.**

- (i)
- $U=\langle \mathcal{B}\rangle $.
- (ii)
- Let $\mathbf{h}\in \mathcal{B}$ be an arbitrary generator. Each module term ${x}^{\mu}{\mathbf{e}}_{k}^{(0)}\in supp(\mathbf{h})\backslash \{hm(\mathbf{h})\}$ must satisfy ${x}^{\mu}{\mathbf{e}}_{k}^{(0)}\notin hm(U)$.
- (iii)
- The assigned multiplicative variables induce direct sum decompositions of both the head module$$hm(U)=\underset{\mathbf{h}\in \mathcal{B}}{\u2a01}\mathbb{k}\left[{X}_{\mathcal{B}}(\mathbf{h})\right]\xb7hm(\mathbf{h})$$$$U=\underset{\mathbf{h}\in \mathcal{B}}{\u2a01}\mathbb{k}\left[{X}_{\mathcal{B}}(\mathbf{h})\right]\xb7\mathbf{h}\phantom{\rule{0.166667em}{0ex}}.$$
- (iv)
- We have a direct sum decomposition ${({\mathcal{P}}_{\mathbf{d}}^{m})}_{r}={U}_{r}\oplus {\langle \mathcal{N}{(hm(U))}_{r}\rangle}_{\mathbb{k}}$ for all degrees $r\ge 0$.
- (v)
- Let $\{{\mathbf{e}}_{1}^{(1)},\dots ,{\mathbf{e}}_{s}^{(1)}\}$ denote the canonical basis of the free module ${\mathcal{P}}^{s}$. Given an arbitrary term ${x}^{\delta}\in \mathbb{T}$ and an arbitrary generator ${\mathbf{h}}_{\alpha}\in \mathcal{B}$, we find for every term ${x}^{\u03f5}{\mathbf{e}}_{i}^{(0)}\in supp({x}^{\delta}{\mathbf{h}}_{\alpha})\cap hm(U)$ a unique ${\mathbf{h}}_{\beta}\in hm(\mathcal{B})$ such that ${x}^{\u03f5}{\mathbf{e}}_{i}^{(0)}={x}^{{\delta}^{\prime}}hm({\mathbf{h}}_{\beta})$ with ${x}^{{\delta}^{\prime}}\in \mathbb{k}\left[{X}_{\mathcal{B}}({\mathbf{h}}_{\beta})\right]$ by (iii). Then the inequality ${x}^{\delta}{\mathbf{e}}_{\alpha}^{(1)}{\u2ab0}_{\mathcal{B}}{x}^{{\delta}^{\prime}}{\mathbf{e}}_{\beta}^{(1)}$ holds with respect to the term order ${\prec}_{\mathcal{B}}$.

**Remark**

**1.**

**Lemma**

**1.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

**Proposition**

**1.**

**Proof.**

**Lemma**

**2.**

**Proof.**

**Lemma**

**3.**

**Proof.**

**Example**

**3.**

**Example**

**4.**

## 3. The Syzygy Resolutions Induced by a Resolving Decomposition

**Lemma**

**4.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Remark**

**2.**

## 4. Explicitly Determining the Differentials

**Definition**

**2.**

**Remark**

**3.**

**Lemma**

**5.**

**Type****1:**- This is the case where $\mathbf{l}=(\mathbf{k}\cup i)\backslash j$, ${x}^{\nu}=\frac{{x}^{\mu}}{{x}_{i}}$ and $\beta =\alpha $. Note that it is allowed that $i=j$. We define $\u03f5(i;\mathbf{k})={(-1)}^{|\{j\in \mathbf{k}\mid j>i\}|}$. Then the corresponding reduction is$$\rho ({\mathbf{v}}_{\mathbf{k}}{x}^{\mu}{\mathbf{h}}_{\alpha})=\u03f5(i;\mathbf{k}\cup i)\u03f5(j;\mathbf{k}\cup i){x}_{j}{\mathbf{v}}_{(\mathbf{k}\cup i)\backslash j}\left(\frac{{x}^{\mu}}{{x}_{i}}{\mathbf{h}}_{\alpha}\right)\phantom{\rule{0.166667em}{0ex}}.$$
**Type****2:**- In this case $\mathbf{l}=(\mathbf{k}\cup i)\backslash j$ and the term ${x}^{\nu}{\mathbf{h}}_{\beta}$ appears in the involutive standard representation of the product $\frac{{x}^{\mu}{x}_{j}}{{x}_{i}}{\mathbf{h}}_{\alpha}$ with the coefficient ${\lambda}_{j,i,\alpha ,\mu ,\nu ,\beta}\in \mathbb{k}$. By the construction of the Morse matching, we now always find $i\ne j$. The corresponding reduction is$$\rho ({\mathbf{v}}_{\mathbf{k}}{x}^{\mu}{\mathbf{h}}_{\alpha})=-\u03f5(i;\mathbf{k}\cup i)\u03f5(j;\mathbf{k}\cup i){\lambda}_{j,i,\alpha ,\mu ,\nu ,\beta}{\mathbf{v}}_{(\mathbf{k}\cup i)\backslash j}({x}^{\nu}{\mathbf{h}}_{\beta})\phantom{\rule{0.166667em}{0ex}}.$$

**Theorem**

**3.**

**Theorem**

**4.**

**Proof.**

**Lemma**

**6.**

**Lemma**

**7.**

**Corollary**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

^{2}(712689).

## Conflicts of Interest

## References

- Albert, M.; Fetzer, M.; Sáenz-de Cabezón, E.; Seiler, W. On the free resolution induced by a Pommaret basis. J. Symb. Comp.
**2015**, 68, 4–26. [Google Scholar] [CrossRef][Green Version] - Seiler, W. A Combinatorial Approach to Involution and δ-Regularity I: Involutive Bases in Polynomial Algebras of Solvable Type. Appl. Algebr. Eng. Commun. Comput.
**2009**, 20, 207–259. [Google Scholar] [CrossRef] - Jöllenbeck, M.; Welker, V. Minimal Resolutions via Algebraic Discrete Morse Theory; Memoirs American Mathematical Society (AMS): Providence, RI, USA, 2009; Volume 197. [Google Scholar]
- Sköldberg, E. Morse Theory from an Algebraic Viewpoint. Trans. Am. Math. Soc.
**2006**, 358, 115–129. [Google Scholar] [CrossRef] - Abbott, J.; Bigatti, M. CoCoALib: A C++ Library for Doing Computations in Commutative Algebra. Available online: http://cocoa.dima.unige.it/cocoalib (accessed on 9 August 2018).
- Albert, M.; Fetzer, M.; Seiler, W.M. Janet Bases and Resolutions in CoCoALib. In Computer Algebra in Scientific Computing, Proceedings of the 17th International Workshop on Computer Algebra in Scientific Computing (CASC 2015), Aachen, Germany, 14–18 September 2015; Gerdt, V.P., Koepf, W., Seiler, W.M., Vorozhtsov, E.V., Eds.; Springer International Publishing: Cham, Switzerland, 2015; pp. 15–29. [Google Scholar]
- Albert, M. Computing Quot Schemes. Ph.D. Thesis, Institut für Mathematik, Universität Kassel, Kassel, Germany, 2017. [Google Scholar]
- Albert, M.; Bertone, C.; Roggero, M.; Seiler, W.M. Computing Quot Schemes via Marked Bases over Quasi-Stable Modules. arXiv, 2018; arXiv:1511.03547v2. [Google Scholar]
- Albert, M.; Seiler, W. Resolving Decompositions for Polynomial Modules. In Computer Algebra in Scientific Computing, Proceedings of the 18th International Workshop on Computer Algebra in Scientific Computing—CASC 2016, Bucharest, Romania, 19–23 September 2016; Lecture Notes in Computer Science; Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E., Eds.; Springer: Cham, Switzerland, 2016; Volume 9890, pp. 13–27. [Google Scholar]
- Sköldberg, E. Resolutions of Modules with Initially Linear Syzygies. arXiv, 2011; arXiv:1106.1913. [Google Scholar]
- Stanley, R. Hilbert Functions of Graded Algebras. Adv. Math.
**1978**, 28, 57–83. [Google Scholar] [CrossRef] - Cox, D.; Little, J.; O’Shea, D. Ideals, Varieties, and Algorithms; Undergraduate Texts in Mathematics; Springer: New York, NY, USA, 1992. [Google Scholar]
- Cox, D.; Little, J.; O’Shea, D. Using Algebraic Geometry; Graduate Texts in Mathematics 185; Springer: New York, NY, USA, 1998. [Google Scholar]
- Seiler, W. A Combinatorial Approach to Involution and δ-Regularity II: Structure Analysis of Polynomial Modules with Pommaret Bases. Appl. Algebr. Eng. Commun. Comput.
**2009**, 20, 261–338. [Google Scholar] [CrossRef] - Given a term order ≺ on $\mathcal{P}$, its term over position (TOP) lifts to a module term order ${\prec}_{TOP}$ on ${\mathcal{P}}_{\mathbf{d}}^{m}$ is defined as follows: let ${x}^{\mu}{\mathbf{e}}_{k}^{\left(0\right)},{x}^{\nu}{\mathbf{e}}_{l}^{\left(0\right)}\in {\mathcal{P}}_{\mathbf{d}}^{m}$, then ${x}^{\mu}{\mathbf{e}}_{k}^{\left(0\right)}{\succ}_{TOP}{x}^{\nu}{\mathbf{e}}_{l}^{\left(0\right)}$ if ${x}^{\mu}\succ {x}^{\nu}$ or if ${x}^{\mu}={x}^{\nu}$ and $k<l$.
- Given a term order ≺ on $\mathcal{P}$, its position over term (POT) lift to a module term order ${\prec}_{POT}$ on ${\mathcal{P}}_{\mathbf{d}}^{m}$ is defined as follows: let ${x}^{\mu}{\mathbf{e}}_{k}^{\left(0\right)},{x}^{\nu}{\mathbf{e}}_{l}^{\left(0\right)}\in {\mathcal{P}}_{\mathbf{d}}^{m}$, then ${x}^{\mu}{\mathbf{e}}_{k}^{\left(0\right)}{\succ}_{POT}{x}^{\nu}{\mathbf{e}}_{l}^{\left(0\right)}$ if $k<l$ or if $k=l$ and ${x}^{\mu}\succ {x}^{\nu}$.
- Herzog, J. A Survey on Stanley Depth. In Monomial Ideals, Computations and Applications; Bigatti, A., Gimenez, P., Sáenz-de Cabezón, E., Eds.; Lecture Notes in Mathemmatics; Springer: Berlin/Heidelberg, Germany, 2013; Volume 2083, pp. 3–45. [Google Scholar]
- Eisenbud, D. Commutative Algebra with a View Toward Algebraic Geometry; Graduate Texts in Mathematics 150; Springer: New York, NY, USA, 1995. [Google Scholar]
- Gerdt, V.P.; Blinkov, Y.A. Involutive Division Generated by an Antigraded Monomial Ordering. In Computer Algebra in Scientific Computing, Proceedings of the 13th International Workshop on Computer Algebra in Scientific Computing (CASC 2011), Kassel, Germany, 5–9 September 2011; Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V., Eds.; Springer: Berlin/Heidelberg, Germany, 2011; pp. 158–174. [Google Scholar]
- Eisenbud, D. The Geometry of Syzygies: A Second Course in Algebraic Geometry and Commutative Algebra (Graduate Texts in Mathematics); Springer: Berlin/Heidelberg, Germany, 2005. [Google Scholar]

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Albert, M.; Seiler, W.M. Resolving Decompositions for Polynomial Modules. *Mathematics* **2018**, *6*, 161.
https://doi.org/10.3390/math6090161

**AMA Style**

Albert M, Seiler WM. Resolving Decompositions for Polynomial Modules. *Mathematics*. 2018; 6(9):161.
https://doi.org/10.3390/math6090161

**Chicago/Turabian Style**

Albert, Mario, and Werner M. Seiler. 2018. "Resolving Decompositions for Polynomial Modules" *Mathematics* 6, no. 9: 161.
https://doi.org/10.3390/math6090161