Jürgen Herzog was born on December 21, 1941 in Heidelberg, Germany. He grew up in the nearby town of Eberbach, where he graduated from the local gymnasium. Following his army service he started studying mathematics and physics at Kiel in 1963, but soon returned to his hometown, transferring in 1964 to the University of Heidelberg.

At Heidelberg he became a student of Ernst Kunz. In the sixties, commutative algebra had spread quite rapidly and, because of their importance in number theory, field theory, algebraic geometry and complex analysis, the study of modules of differentials and derivations was a hot topic. Volume 38 of the Springer Lecture Notes, Differentialrechnung in der analytischen Geometrie (1967) by Berger, Kiehl, Kunz and Nastold reflects this trend.

Under the influence of Abhyankar and Lipman, Purdue University was one of the centers of local algebra and geometry at the time, and Kunz, like other German mathematicians, spent a year at Purdue, together with Jürgen who had finished his Diplom thesis in Heidelberg (1967). The year at Purdue was followed by a year at Louisiana State University. In the US, Jürgen earned an MSc from Purdue (1968) and the PhD from Louisiana State (1969).

Jürgen returned to Germany to complete his Diplom examination at Heidelberg, and then became Wissenschaftlicher Assistent at Regensburg, finishing his Habilitation in 1974 with the thesis *Komplexe, Auflösungen und Dualität in der lokalen Algebra*. In 1975 he was appointed professor of mathematics at the University of Essen, where he remained until his retirement in March 2009.

Beginning with his graduate studies, Jürgen has worked in commutative algebra with great enthusiasm and endurance. He has left deep footprints in almost every subfield that touches homological or combinatorial problems. At this moment MathSciNet lists 263 publications by Jürgen. So far they have earned him more than 8700 citations. Jürgen’s infectious appetite for interesting mathematics and his communication skills are witnessed by more than 110 coauthors. At the occasion of his 65th birthday a conference in 2007 at Cortona honored his work, and a conference for his 80th was held at Osnabrück in 2022. Two collections of papers were dedicated to Jürgen: volume 5 of the Journal of Commutative Algebra (2013) and a special volume of Research in the Mathematical Sciences (2020).

The book *Der kanonische Moduleines Cohen–Macaulay Rings* (1971) edited by Herzog and Kunz played an important role in the popularization of Grothendieck’s local duality. The structure of the canonical module and of Gorenstein rings has been a recurring theme in Jürgen’s mathematical work. Another leitmotif, originating from his mathematical youth and often pursued in collaboration with Rolf Waldi, is the investigation of differentials, derivations and deformations.

Serre asked if the Poincaré-Betti series of local rings are rational. This question was a driving force in local algebra during the sixties and seventies, and in particular stimulated the investigation of Koszul algebras and Golod rings. These types of algebras have played a central role in Jürgen’s work through the years, in particular in cooperation with Luchezar Avramov and Srikanth Iyengar.

In the early eighties, Jürgen, Aron Simis and Wolmer Vasconcelos developed the theory of approximation complexes, an important tool for the study of Rees algebras. Since then, he has been interested in the asymptotics of powers of ideals, above all in their homological and arithmetical invariants and especially their Castelnuovo–Mumford regularity, whose asymptotic linearity he proved in collaboration with Dale Cutkosky and Ngô Việt Trung. Giuseppe Valla was another important collaborator in the area of Rees and symmetric algebras.

Jürgen’s 1978 article in Math. Ann. on rings of finite Cohen–Macaulay representation type marks the start of an important trend connecting representation theory and commutative algebra. A key aspect are matrix factorizations of polynomials defining Cohen–Macaulay modules over hypersurface rings. Matrix factorizations, introduced by David Eisenbud, have found by now their way in to knot homology and string theory. Ten years after his first important contribution to finite representation type, Jürgen completed the work on *The classification of homogeneous Cohen–Macaulay rings of finite representation type* (with David Eisenbud, Math. Amn. 1988). A special class of maximal Cohen–Macaulay modules are those with a maximal number of generators, introduced in Jürgen’s 1987 paper with Joseph Brennan and Bernd Ulrich.

The fundamental importance of Cohen–Macaulay rings was highlighted in the seventies by the work of Mel Hochster on determinantal rings and rings of invariants. When the topic of Cohen–Macaulay modules was suggested to Jürgen by Cambridge University Press, he and Winfried Bruns responded with the book Cohen–Macaulay rings. Published in 1993, it has been extremely successful, quickly becoming the standard reference in the area. In fact, it is quoted in Wiles’ 1994 paper on Fermat’s Last Theorem! Nowadays Cohen–Macaulay rings is, in combination with David Eisenbud’s book Commutative algebra with a view towards algebraic geometry, the backbone of any advanced course in commutative algebra.

A substantial part of Cohen–Macaulay rings is devoted to the cross-fertilization between combinatorics and commutative algebra, following the seminal work of Richard Stanley. The fascinating interplay between algebra and combinatorics has had a tremendous influence on Jürgen’s work in the last 30 years. Annetta Aramova and Dorin Popescu have been companions in this area, and in particular Takayuki Hibi, Jürgen’s coauthor of a record prone number of more than 60 joint papers. Jürgen’s third book *Monomial ideals* (2011, with Hibi) gives an excellent overview of an area in which objects of commutative algebra directly represent finite multi-sets, simplicial complexes, graphs and their higher dimensional analogues. Jürgen’s book Gröbner bases in commutative algebra (2012, with Viviana Ene) reflects Jürgen’s profound interest in computational and effective methods. The book emphasizes the structural consequences that one can transfer to an ideal or subalgebra from its initial ideal or initial subalgebra, respectively. Jürgen has contributed significantly to other aspects of combinatorial commutative algebra, for example semigroup rings and determinantal rings. The defining ideals of semigroup rings are treated profoundly in Jürgen’s book *Binomial ideals* (with Hibi and Ohsugi, 2018).

Free resolutions are the source of homological invariants, and graded free resolutions connect them to combinatorics. One can certainly say that free resolutions, openly or in disguise, have been the most important topic in Jürgen’s work. This interest stretches from monomial ideals to free resolutions over the exterior algebra. The seemingly innocent 1984 article On the Betti numbers of pure and linear free resolutions is one of the foundations of Boij-Söderberg theory whose creation was triggered by Jürgen’s and Hema Srinivasan’s multiplicity conjectures.

In the last decade Jürgen came back to Golod rings: together with Craig Huneke he proved the remarkable result that the powers of every ideal I from I2 on define Golod rings (Adv. Math. 2013). In his beloved area of algebraic combinatorics Jürgen introduced new classes of ideals, for example, determinantal face ideals, Freiman ideals and Lovász-Saks-Schrijver ideals. The latter are derived from orthogonal representations of graphs. But also the canonical module continued its important role in Jürgen’s work through its trace ideal that Jürgen, Dumitru Stamate and coauthors determined in many cases.

Jürgen, a keen traveler, has visited many universities all over the world. In the footsteps of his advisor Kunz, he spent the spring semester of 1993 at Purdue and this stay led to an important article with Luchezar Avramov containing a positive answer to a conjecture of Vasconcelos on complete intersections.

Jürgen’s stockpile of mathematical problems was inexhaustible, and he has generously helped 23 PhD students to find their way into mathematics. He was instrumental in building the PhD program at the Abdus Salam School of Mathematics in Lahore, Pakistan. Especially the Iranian mathematical community is grateful to Jürgen for his commitment to the education of young mathematicians. Siammak Yassemi writes on commalg.org: “We never forget his generosity and encouragement [that has] inspired countless individuals”.

Over the years, Jürgen has served in the editorial board of several journals: Mathematische Zeitschrift, Algebra and Representation Theory, Communications in Algebra, Homology, Homotopy and Applications, Journal of Prime Research in Mathematics, Bulletin Mathématique de la Societé des Sciences Mathématiques de la Roumanie and Bulletin of the Iranian Mathematical Society. He has been a vital member of the mathematical community also by organizing numerous conferences all over the world. In view of his mathematical achievements, Jürgen was elected corresponding member of the Academia Peloritana dei Pericolanti di Messina in 1999.

Jürgen died on April 23, 2024 after a sudden heart attack. He left us a rich legacy that will continue to bear fruit. With Jürgen we have lost a friend, a master, an excellent mathematician of admirable inventiveness and productivity until his very last days.

*Winfried Bruns* and *Aldo Conca*

May 2024