The commutative algebra group of the Institute for Research in Fundamental Sciences (IPM) is organizing the 15th seminar on Commutative Algebra and Related Topics. The goal is to bring together people who are working on this area and to introduce recent activities to the students and young researchers.

In addition, a workshop will be provided on January 12-14, 2019. The aim is to acquaint the young researchers with the new trends in the subject.

All commutative algebra researchers, specially graduate students, are welcome to attend the both events and also give a lecture at the seminar.

Abdolnaser Bahlekeh
(Gonbade-Kavous University, Iran)

Fully Decomposability of Balanced Big Cohen-Macaulay Modules

Fully Decomposability of Balanced Big Cohen-Macaulay Modules

[1] M. Auslander, A functorial approach to representation theory, in Representatios of Algebra, Workshop Notes of the Third Inter. Confer., Lecture Notes Math. 944, 105-179, Springer-Verlag, 1982.

[2] M. Auslander and I. Reiten, Applications of contravariantly nite subcategories, Adv. Math. 86 (1991), no. 1, 111-152.

[3] A. Beligiannis, On algebras of nite Cohen-Macaulay type, Adv. Math. 226 (2011), no. 2, 1973-2019.

[4] X. W. Chen, An Auslander-type result for Gorenstein projective modules, Adv. Math. 218 (2008), 2043-2050.

[5] C. M. Ringel and H. Tachikawa, QF-3 rings, J. Reine Angew. Math. 272 (1975), 49-72.

Emanuela De Negri
(Università di Genova, Italy)

Cartwright-Sturmfels Ideals of Graphs and Linear Spaces

Cartwright-Sturmfels Ideals of Graphs and Linear Spaces

These results have been obtained jointly with Aldo Conca and Elisa Gorla.

Amir Mafi
(Univerisity of Kurdistan, Iran)

Results on Linear Resolution and Polymatroidal Ideals

Results on Linear Resolution and Polymatroidal Ideals

[1] S. Bandari and J. Herzog, Monomial localizations and polymatroidal ideals, Eur. J. Comb., 34(2013),752-763.

[2] A. Conca Regularity jumps for powers of ideals, Lect. Notes Pure Appl. Math., 244(2006), 21-32.

[3] A. Conca and J. Herzog, Castelnuovo-Mumford regularity of products of ideals, Collect. Math., 54(2003),137-152.

[4] D. Eisenbud, Commutative Algebra with a View Towards Algebraic Geometry, GTM., vol.150, Springer, Berlin, (1995).

[5] D. R. Grayson and M. E. Stillman, Macaulay 2, a software system for research in algebraic geometry, Available at http://www.math.uiuc.edu/Macaulay2/.

[6] J. Herzog and T. Hibi, Discrete polymatroids, J. Algebraic Combin., 16(2002), 239-268.

[7] J. Herzog and T. Hibi, Monomial ideals, GTM., vol.260, Springer, Berlin, (2011).

[8] J. Herzog, T. Hibi and X. Zheng, Monomial ideals whose powers have a linear resolution, Math. Scand., 95(2004), 23-32.

[9] J. Herzog, A. Rauf and M. Vladoiu, The stable set of associated prime ideals of a polymatroidal ideal, J. Algebraic Combin., 37(2013), 289-312. (2012).

[10] J. Herzog and Y. Takayama, Resolutions by mapping cones, Homology Homotopy Appl., 4(2002), 277-294.

[11] J. Herzog and M. Vladoiu, Monomial ideals with primary components given by powers of monomiali prime ideals, Electron. J. Combin., 21 (2014), P1.69.

[12] Sh. Karimi and A. Mafi, On stability properties of powers of polymatroidal ideals, to appear in Collect. Math.

[13] B. Sturmfels, Four counterexamples in cobinatorial algebraic geometry, J. Algebra, 230(2000), 282-294.

[14] T. N. Trung, Stability of associated primes of integral closures of monomial ideals, J. Comb. Theory Ser. A 116(2009), 44-54.

Zahra Nazemian
(IPM, Iran)

Some New Classes of Modules to Tackly Enochs' Conjecture

Some New Classes of Modules to Tackly Enochs' Conjecture

This is a joint work with Alberto Facchini.

Enrico Sbarra
(Università degli Studi di Pisa, Italy)

Jet Schemes Ideals of Pfaffians

Jet Schemes Ideals of Pfaffians

This is a joint work with E. De Negri.

Russ Woodroofe
(University of Primorska, FAMNIT, Slovenia)

Arrangements and the Independence Polynomial

Arrangements and the Independence Polynomial

Rasoul Ahangari Maleki
(IPM, Iran)

The Absolutely Koszul and Backelin-Roos Properties for Spaces of Quadrics of Small Codimension

The Absolutely Koszul and Backelin-Roos Properties for Spaces of Quadrics of Small Codimension

This is a joint work with Liana M. Sega.

[1] J. Backelin, A ditributiveness property of augmented algebras and some related homological results, PhD thesis, Stockholm University, 1982.

[2] A. Conca, Groebner Bases for Spaces of Quadrics of Low Codimension, Adv. Appl. Math. 24 (2000), 111-124.

[3] A. Conca, Groebner Bases for Spaces of Quadrics of Codimension 3, J. Pure Appl. Algebra 213 (2009), 1564-1568.

[4] A. Conca, S. B. Iyengar, H. D. Nguyen, T. Roemer, Absolutely Koszul algebras and the Backelin-Roos property, Acta Math. Vietnam. 40 (2015), 353-374.

[5] A. D'Ali, The Koszul property for spaces of quadrics of codimension three, J. Algebra 490 (2017), 256-282.

[6] J. Herzog, S. Iyengar, Koszul modules, J. Pure Appl. Algebra 201 (2005), 154-188.

[7] S. Iyengar, T. Roemer, Linearity defects of modules over commutative rings, J. Algebra 322 (2009), 3212-3237.

[8] J.-E. Roos, Good and bad Koszul algebras and their Hochschild homology, J. Pure Appl. Algebra 201(2005), 295-3327.

Shamila Bayati
(Amirkabir University of Technology, Iran)

Multigraded Shifts of Monomial Ideals

Multigraded Shifts of Monomial Ideals

More precisely, let $S=k[x_1,\ldots, x_n]$ be the polynomial ring over a field $k$. We consider this ring with its natural multigrading. Suppose that $I\subseteq S$ is a monomial ideal and consider the ideal $J_k(I)=(\{\mathbf{x}^\mathbf{a}| \,\,\beta_{k,\mathbf{a}}(I)\neq 0 \, \})$ generated by the $k$-th multigraded shifts of $I$. It is under question that transferring from $I$ to $J_k(I)$ which properties are preserved.

We first investigate that the property of being (poly)matroidal is inherited by the ideals generated by multigraded shifts.

Regarding this question, we also study Borel ideals and squarefree Borel ideals. A result by Miller and Strumfels shows that the ideal generated by the first multigraded shifts of an equigenerated Borel ideal inherited the property of having linear resolution. We will show this is also the case if we consider the linear quotients property. Moreover, it is shown that if $I$ is a principal Borel ideal or a squarefree Borel ideal, then $J_k(I)$ has linear quotients for each $k=0,\ldots,pd(I)$. Furthermore, it is shown that the property of being squarefree Borel is inherited by $J_k(I)$ whenever $I$ is equigenerated.

Some results are based on a joint work with Iman Jahani and Nadiya Taghipour.

Majid Eghbali
(Tafresh University, Iran)

A View to Set-theoretically Cohen-Macaulay Ideals

A View to Set-theoretically Cohen-Macaulay Ideals

Fahimeh Sadat Fotouhi
(University of Isfahan, Iran)

Using Gabriel-Roiter (co)measure for Maximal Cohen-Macaulay Modules

Using Gabriel-Roiter (co)measure for Maximal Cohen-Macaulay Modules

Abbas Nasrollah Nejad
(Institute for Advanced Studies in Basic Sciences (IASBS), Iran)

The Gauss Algebra of Toric Algebras

The Gauss Algebra of Toric Algebras

This talk is based on joint work with Juergen Herzog and Raheleh Jafari.

Amin Nematbakhsh
(IPM, Iran)

Linear Strands of Edge Ideals of Multipartite Uniform Clutters

Linear Strands of Edge Ideals of Multipartite Uniform Clutters

[1] Joseph Alvarez Montaner and Alireza Vahidi. Lyubeznik numbers of monomial ideals. Tran- sactions of the American Mathematical Society, 366(4):18291855, 2014.

[2] Dave Bayer and Bernd Sturmfels. Cellular resolutions of monomial modules. Journal fur die Reine und Angewandte Mathematik, 502:123-140, 1998.

[3] Alessio D'Ali, Gunnar Floystad, and Amin Nematbakhsh. Resolutions of co-letterplace ideals and generalizations of Bier spheres. Transactions of the American Mathematical Society, to appear.

[4] Giuseppe Favacchio, Elena Guardo, and Juan Migliore. On the arithmetically CohenMacaulay property for sets of points in multiprojective spaces. Proceedings of the American Mathematical Society, 146:2811-2825, 2018.

[5] Juergen Herzog and Yukihide Takayama. Resolutions by mapping cones. Homology, Homotopy and Applications, 4(2):277-294, 2002.

[6] Amin Nematbakhsh. Linear strands of edge ideals of multipartite uniform clutters. arXiv:1805.11432.

Mohammad Rouzbahani Malayeri
(Amirkabir University of Technology, Iran)

On a Conjecture About Castelnuovo-Mumford Regularity of Binomial Edge Ideals

On a Conjecture About Castelnuovo-Mumford Regularity of Binomial Edge Ideals

The aforementioned result by Matsuda and Murai proves this conjecture in the case of trees. In [1], Ene and Zarojanu verified the above conjecture for a class of chordal graphs for which any two maximal cliques intersect in at most one vertex. Later, this result was improved for a bigger subclass of chordal graphs called

This talk is based on a joint work with Dariush Kiani and Sara Saeedi Madani.

[1] V. Ene, A. Zarojanu, On the regularity of binomial edge ideals, Math. Nachr. 288, No. 1 (2015), 19-24.

[2] J. Herzog, T. Hibi, F. Hreinsdottir, T. Kahle, J. Rauh, Binomial edge ideals and conditional independence statements, Adv. Appl. Math. 45 (2010), 317-333.

[3] D. Kiani, S. Saeedi Madani The regularity of binomial edge ideals of graphs, (2013) arXiv:1310.6126v2.

[4] D. Kiani, S. Saeedi Madani, The Castelnuovo-Mumford regularity of binomial edge ideals, J. Combin. Theory Ser. A. 139 (2016), 80-86.

[5] K. Matsuda, S. Murai, Regularity bounds for binomial edge ideals, J. Commutative Algebra. 5(1) (2013), 141-149.

[6] M. Ohtani, Graphs and ideals generated by some 2-minors, Comm. Algebra. 39 (2011), 905-917.

[7] S. Saeedi Madani, D. Kiani, Binomial edge ideals of graphs, Electronic J. Combin. 19(2) (2012), ♯ P44.

[8] S. Saeedi Madani, D. Kiani, On the binomial edge ideal of a pair of graphs, Electronic J. of Combinatorics. 20(1) (2013), ♯ P48.

Sara Saeedi Madani
(Amirkabir University of Technology, Iran)

Toric Algebras Arising from Cuts in Graphs

Toric Algebras Arising from Cuts in Graphs

This talk is based on a joint work with Tim Roemer.

Amir Saki
(Amirkabir University of Technology, Iran)

Lattice Theoretical Approachers for Racks

Lattice Theoretical Approachers for Racks

This talk is based on a joint work with Dariush Kiani.

[1] I. Heckenberger, J. Shareshian, V. Welker, On the lattice of subracks of the rack of a finite group, Trans. Amer. Math. Soc. (2018), https://doi.org/10.1090/tran/7644, in press.

[2] D. Kiani and A. Saki, The lattice of subracks is atomic, J. Combin. Theory Ser. A, vol. 162 (2019), 55-64.

Zahra Shahidi
(Institute for Advanced Studies in Basic Sciences (IASBS), Iran)

Torsion-free Aluffi Algebras

Torsion-free Aluffi Algebras

This talk is based on the joint work with A.Nasrollah Nejad and Rashid Zaare-Nahandi.

Ali Soleyman Jahan
(University of Kurdistan, Iran)

Linear Syzygy Graph, Linear Resolution, Linear Quotients and Variable Decomposability

Linear Syzygy Graph, Linear Resolution, Linear Quotients and Variable Decomposability

$I$ is variable-decomposable $\Longrightarrow \; I$ has linear quotients $\Longrightarrow \; I$ has a linear resolution.

Let $G_I$ be the graph which its nodes are the generators of $I$, and two vertices $u_i$ and $u_j$ are adjacent if there exist variables $x, y$ such that $xu_i = yu_j$. Let $I\subset S$ be a squarefree monomial ideal generated in degree $d$. we show that if $d=2$, $d=n-2$, $n\leq 5$, $G_I$ is cycle and $G_I$ is a tree, then these three concepts are equivalent. As applications of our results, we characterize all Cohen-Macaulay monomial ideals of codimension $2$ with a linear resolution. Let $\Delta=\langle F_1,\ldots, F_m\rangle$ be a simplicial complex. It is shown that $\Delta$ is connected in codimension one if and only if $G_{I_{\Delta^{\vee}}}$ is a connected graph. We show that $I_{\Delta^{\vee}}$ has linear relations if and only if $\Delta^{(F,G)}$ is connected in codimension one for all facets $F$ and $G$ of $\Delta$. Also, we introduce a simple graph $G_{\Delta}$ on vertex set $\{F_1,\ldots,F_m\}$ which is isomorphic to $G_{I_{\Delta^{\vee}}}$. As Corollaries of our results, we show that if $G_{\Delta}$ is a cycle or a tree, then the following are equivalent:

(a) $\Delta$ is Cohen-Macaulay;

(b) $\Delta$ is pure shellable;

(c) $ \Delta$ is pure vertex-decomposable.

This is a joint work with E. Manouchehri.

To register for the seminar, please fill out the Registration Form. Your request will be considered by the organizing committee. Upon acceptance you will receive an email. After the acceptance, you may complete the registration by doing the payment process.

**Registration fee for the seminar:**

1) Registration fee for Iranian participants:

You can get more information about the registration fee here.

2) Registration fee for international participants: The registration fee is 100 Euro. The registration fee for international participants will be due in cash at the time of registration on the first day of the meeting. Please note that standard credit cards; e.g., Visa, Master or AmEXP, cannot be used in Iran.

Registration fee includes: Documentation package, participation in the lectures, lunches coupons, and coffee breaks during the meeting.

Residence fee at IPM guest house is 25 Euro per night.

1) Registration fee for Iranian participants:

You can get more information about the registration fee here.

2) Registration fee for international participants: The registration fee is 100 Euro. The registration fee for international participants will be due in cash at the time of registration on the first day of the meeting. Please note that standard credit cards; e.g., Visa, Master or AmEXP, cannot be used in Iran.

Registration fee includes: Documentation package, participation in the lectures, lunches coupons, and coffee breaks during the meeting.

Residence fee at IPM guest house is 25 Euro per night.

Decision on acceptance of attendance and talks:

Deadline for registration confirmation:

All commutative algebraists are welcome to participate in this event. Please note that attending in the workshop is free of charge and there is no need for registration.

Emanuela De Negri
(Università di Genova, Italy)

Determinantal Ideals and Associated Simplical Complexes

Determinantal Ideals and Associated Simplical Complexes

Classical determinantal rings are defined by minors of a generic matrix, by minors of a generic symmetric matrix or by Pfaffians of a generic skew-symmetric matrix. A classical tool in the study of determinantal rings is the notion of Algebra with Straightening Law (ASL), which was born in the invariant theory and stands in the intersection among geometry, algebra and combinatorics. More recently Groebner bases theory has been successfully used to study classical and non-classical determinantal rings, allowing to deduce properties of the determinantal ideals by studying simplical complexes associates to their initial ideals. Aim of the course is to explain this tecniques and to show two applications to the study of cogenerated ideals of Pfaffians and of ideals of minors of symmetric matrices.

Arash Sadeghi
(IPM, Iran)

Vanishing of (co)homology and Depth Formula

Vanishing of (co)homology and Depth Formula

References:

[1] M. Auslander, Modules over unramified regular local rings, Illinois J. Math. 5 (1961), 631647.

[2] C. Huneke and R. Wiegand, Tensor products of modules and the rigidity of Tor, Math. Ann. 299 (1994),449476.

Enrico Sbarra
(Università degli Studi di Pisa, Italy)

Generic Initial Ideals and Local Cohomology Tables

Generic Initial Ideals and Local Cohomology Tables

After discussing the notion of genericity and the definition of generic initial ideals, we recall some of their basic properties. We shall then focus on the Betti tables and local cohomology tables of generic initial ideals, and on some generalizations of Bayer-Stillman results. We shall also survey some questions which are still open in the field, and see how the theory of generic initial ideals contributed and may contribute to their solutions.

Some of the results we present are in collaboration with G. Caviglia (Univ. Purdue) and F. Strazzanti (Univ. Barcelona).

Russ Woodroofe
(University of Primorska, FAMNIT, Slovenia)

Commutative Algebra via Simplicial Combinatorics

Commutative Algebra via Simplicial Combinatorics

Notes