RTG Learning Seminar, Spring 2026: Koszul Algebras

Mondays, 2:30–3:20, in Avery 351

During the Spring 2026 semester, Adam LaClair, with the help of Eloísa Grifo and Mark Walker, will be leading the first RTG Learning Seminar. In this seminar, graduate students, post-docs and faculty will take turns presenting material covering various topics centering on a general theme. It will be similar to the CMinar we ran in the Spring of 2025 (but it will last the entire semester, and will be on a different topic.) The goal is to learn a focused topic in great depth, with the aspiration of getting the participants up to speed on the cutting edge status of a research topic, and perhaps even sparking some new research projects. The topic of this semester's Learning Seminar will be Koszul Algebras. Here is a tentative list of specific topics we plan to discuss:

Date	Title	Presenter
January 12, 2026	Introduction	Adam LaClair
January 19, 2026	MLK Holiday
January 26, 2026	The pretty pretty pretty Priddy good complex	Mark Walker
February 2, 2026	Löfwall's Characterization : P*H = 1	Eloísa Grifo
February 9, 2026	Fröberg's Proof	Stephen Stern
February 16, 2026	Tate's Resolution	Julianne Faur
February 23, 2026	Pre-requisites to Irrational Poincaré Series	Lauren Cranton Heller
March 2, 2026	Irrational Poincaré Series	Jack Jeffries
March 9, 2026	Avramov–Eisenbud Characterization	Kesavan Mohana Sundaram
March 16, 2026	Spring Break
March 23, 2026	Avramov–Peeva Characterization	Ryan Watson
March 30, 2026	G-quadratic, LG-quadratic, Koszul filtrations
April 6, 2026	Application of Koszul filtrations
April 13, 2026	Transfer of Koszulness
April 20, 2026	Roos's Example	Em Stephen
April 27, 2026	Roos's Example for Quadratic Gorenstein Algebras

Resources

January 26: In McCullough-Peeva, p.243-4 they introduce the quadratic dual and have some examples, and p.244-5, they use this to define the Priddy resolution and present an example. In Beilinson, Ginzburg, Soergel, p. 483, 2.6 and 2.6.1, they introduce the Priddy complex and prove that it is a resolution when the ring is Koszul. Here are draft lecture notes.
February 2: In Beilinson, Ginzburg, Soergel, Section 2.11, they give a proof of this result. A proof that Koszul implies the stated equality is given in Löfwall, ``Hilbert series, Poincaré series and homotopy Lie algebras of graded algebras — a seminar," Section 1.2. Another proof that the stated equality implies that the ring is Koszul is given in Löfwall, "On the subalgebra generated by the one dimensional elements in the Yoneda Ext algebra," Theorem 1.2, p.305.
February 9: The relevant paper is Fröberg's paper "Determination of a Class of Poincaré Series."
February 16: The relevant paper is Tate's paper "Homology of Noetherian Local Rings."
February 23: The relevant paper is Roos-Sturmfels' paper, "A toric ring with irrational Poincaré series." We would like to understand the reduction steps in Section 3. Specifically, we would like to see large homomorphisms in the sense of Levin, "Large homomorphisms of local rings." Golod maps and implication on Poincaré series. In Löfwall, "Hilbert series, Poincaré series and homotopy Lie algebras of graded algebras — a seminar," Section 2.3, an introduction to the Poincaré-Birkhoff-Witt is given.
March 2: The relevant paper is Roos-Sturmfels' paper, "A toric ring with irrational Poincaré series."
March 9: The relevant paper is Avramov-Eisenbud, "Regularity of Modules over a Koszul Algebra." Given sufficient time it would also be nice to cover the proof of Corollary 2, whose proof is in Eisenbud-Goto, "Linear Free Resolutions and Minimal Multiplicity".
March 16: No meeting.
March 23: The relevant paper is Avramov-Peeva, "Finite Regularity and Koszul Algebras".
March 30: The relevant paper is Conca, "Koszul algebras and their syzygies," p. 8 - 10 and Definition 3.6.
April 6: The relevant paper is Conca, Trung, Valla, "Koszul Property for Points in Projective Spaces," p. 201-6.
April 13: The relevant paper is Conca, "Koszul algebras and their syzygies," Section 3.1, p. 18 - 20.
April 20: The relevant paper is Roos, "Commutative Non Koszul Algebras Having a Linear Resolution of Arbitrary High Order," p.1123-6.
April 27: The relevant paper is McCullough-Seceleanu, "Quadratic Gorenstein Algebras," Section 5.

Papers

These papers, and many others, are also available in a dropbox folder. Click here to access this folder, or scan the following QR code:

Topics:

Introduction to Koszul algebras and properties: Priddy resolution, Avramov-Eisenbud-Peeva regularity characterization, rational Poincaré series
Proving Koszulness: Quadratic Grobner bases, Froberg's theorem, LG quadratic, Koszul filtrations
Examples of Koszul algebras: Quadratic complete intersections, high Veronese algebras, pinched Veroneses (sometimes), quadratic Gorenstein algebras (sometimes), strongly chordal and claw-free binomial edge ideals
Other interesting examples: Anick's examples of irrational Poincaré series, Conca's strictness-of-implication examples, Roos's nonlinear syzygies examples, D'Alì-Venturello examples of dependence on characteristic
Conjectures: Bounds on Betti numbers, Bogvad's conjecture, Veroneses of non-standard graded polynomial rings

Some general references on Koszul algebras and related topics:

Here are some links to papers of relevance:

Conca, de Negri, and Gorla: Koszul Algebras and Regularity
Löfwall: Hilbert series, Poincaré series and homotopy Lie algebras of graded algebras — a seminar
Peeva and McCullough: Infinite Graded Free Resolutions

Photos

Scene from the organizational meeting.