Ulrich Modules: Definitions & Applications

• Ulrich modules are Cohen–Macaulay modules defined by the equality of minimal generators and multiplicity, serving as extremal examples in commutative algebra.
• They feature equivalent local and ideal-relative formulations, linking numerical conditions, syzygies, and reduction criteria crucial for analyzing minimal multiplicity and surface singularities.
• Ulrich bundles extend these ideas to projective varieties through linear resolutions and moduli descriptions, offering insights in both geometric and homological contexts.

Searching arXiv for recent and foundational papers on Ulrich modules to support the encyclopedia entry. arXiv search query: "Ulrich modules local rings minimal multiplicity dimension two irreducible multiplicity" Ulrich modules are Cohen–Macaulay modules that attain an extremal generator–multiplicity bound. In the standard local formulation, a finitely generated $R$-module $M$ is Ulrich when $M\neq 0$, $M$ is maximal Cohen–Macaulay, and $\nu_R(M)=e_R(M)$; in the ideal-relative formulation one requires $M$ maximal Cohen–Macaulay, $e_I^0(M)=\ell_R(M/IM)$, and $M/IM$ free over $R/I$. On the projective side, the corresponding Ulrich bundles are ACM sheaves characterized by complete acyclicity of twists, linear resolutions, or triviality after finite linear projection. These local, singularity-theoretic, and projective viewpoints are now linked by blowups, syzygies, trace ideals, matrix factorizations, and moduli theory (Iyengar et al., 3 Oct 2025, Goto et al., 2013, Cho et al., 2017).

1. Local definitions and equivalent formulations

In the local theory used for existence results over higher-dimensional rings, a nonzero finitely generated $R$-module $M$0 is an Ulrich module if $M$1 is maximal Cohen–Macaulay and $M$2. When the residue field is infinite and $M$3 is a minimal reduction of the maximal ideal, this is equivalent to $M$4 being maximal Cohen–Macaulay and satisfying $M$5; equivalently, with $M$6 a parameter ideal reducing $M$7, one has

$M$8

in the Cohen–Macaulay setting (Iyengar et al., 3 Oct 2025, Celikbas et al., 20 May 2025).

A second, widely used formulation is relative to an $M$9-primary ideal $M\neq 0$0. In that setting, $M\neq 0$1 is Ulrich with respect to $M\neq 0$2 if $M\neq 0$3 is maximal Cohen–Macaulay, $M\neq 0$4, and $M\neq 0$5 is free over $M\neq 0$6. This ideal-relative language is central in the study of Ulrich ideals, surface singularities, and reduction-theoretic criteria (Goto et al., 2013).

Terminology is not completely uniform across the literature. One strand defines a nonzero Cohen–Macaulay $M\neq 0$7-module $M\neq 0$8 to be Ulrich if $M\neq 0$9, without building maximal Cohen–Macaulayness into the definition. In that convention, the inequality $M$0 for Cohen–Macaulay modules becomes the immediate numerical background, and Ulrich modules are precisely the Cohen–Macaulay modules on which equality holds (Dey et al., 2022).

2. Minimal multiplicity, syzygies, and trace-sensitive subclasses

A recurrent ring-theoretic environment for Ulrich modules is minimal multiplicity. For a Cohen–Macaulay local ring $M$1, minimal multiplicity means

$M$2

and, when the residue field is infinite, this is equivalent to $M$3 for some parameter ideal $M$4 reducing $M$5 (Celikbas et al., 20 May 2025). In this regime, Ulrich modules are tightly controlled by syzygies. Over a Cohen–Macaulay local ring with minimal multiplicity, first syzygies of Cohen–Macaulay modules are Ulrich, yielding the inclusion

$M$6

Moreover, $M$7 is a distinguished Ulrich module, every Ulrich module is a quotient of a finite direct sum of copies of $M$8, and the category of Ulrich modules acquires an exact-category structure with

$M$9

(Kobayashi et al., 2017).

A recent refinement isolates full-trace Ulrich modules, defined by the additional trace condition $\nu_R(M)=e_R(M)$0. Over a non-regular Cohen–Macaulay local ring, the existence of a full-trace Ulrich module is equivalent to minimal multiplicity. More precisely, for a $\nu_R(M)=e_R(M)$1-dimensional non-regular Cohen–Macaulay local ring, minimal multiplicity is equivalent to the assertion that $\nu_R(M)=e_R(M)$2 is a full-trace Ulrich module for all $\nu_R(M)=e_R(M)$3, and also equivalent to existence of some full-trace Ulrich module. In dimension one, this specializes to the statement that $\nu_R(M)=e_R(M)$4 itself is an Ulrich $\nu_R(M)=e_R(M)$5-module exactly when the ring has minimal multiplicity (Celikbas et al., 20 May 2025).

In dimension one, the comparison between Ulrich modules and syzygies becomes especially rigid. Under minimal multiplicity, equality

$\nu_R(M)=e_R(M)$6

is equivalent to the almost Gorenstein property, so the Ulrich category can reflect not only multiplicity-theoretic extremality but also fine canonical-duality behavior (Kobayashi et al., 2017).

3. Surface singularities and explicit classification

For two-dimensional rational double points, Ulrich theory admits a geometric classification via resolutions of singularities, anti-nef cycles, and the McKay correspondence. If $\nu_R(M)=e_R(M)$7 is a two-dimensional rational double point and $\nu_R(M)=e_R(M)$8 is a nonparameter $\nu_R(M)=e_R(M)$9-primary ideal, then the following are equivalent: $M$0 is an Ulrich $M$1-module with respect to $M$2; $M$3 is a special Cohen–Macaulay module with respect to $M$4; $M$5 is weakly special with respect to $M$6; and $M$7 is $M$8-free and $M$9 has no free summands. When these conditions hold, $e_I^0(M)=\ell_R(M/IM)$0 is an Ulrich ideal, and $e_I^0(M)=\ell_R(M/IM)$1 is again Ulrich with respect to $e_I^0(M)=\ell_R(M/IM)$2 (Goto et al., 2013).

The same paper gives a complete ADE classification of Ulrich ideals and indecomposable Ulrich modules over rational double points. The mechanism is intersection-theoretic: if $e_I^0(M)=\ell_R(M/IM)$3 is represented by an anti-nef cycle $e_I^0(M)=\ell_R(M/IM)$4 on the minimal resolution, Kato’s Riemann–Roch formula

$e_I^0(M)=\ell_R(M/IM)$5

reduces freeness modulo $e_I^0(M)=\ell_R(M/IM)$6 to an equality of cycle coefficients. This turns Ulrich classification into an explicit problem in surface geometry and representation theory (Goto et al., 2013).

For cyclic quotient surface singularities, the classification is equally explicit but combinatorial. Writing $e_I^0(M)=\ell_R(M/IM)$7 for a cyclic quotient singularity and $e_I^0(M)=\ell_R(M/IM)$8 for the indecomposable maximal Cohen–Macaulay modules, one has

$e_I^0(M)=\ell_R(M/IM)$9

where the $M/IM$0 are determined by the Hirzebruch–Jung continued fraction. Then

$M/IM$1

If $M/IM$2 is the number of exceptional curves in the minimal resolution, the number $M/IM$3 of indecomposable Ulrich modules satisfies

$M/IM$4

(Nakajima et al., 2015).

4. Existence theorems and nonexistence phenomena

A major recent development is that existence of Ulrich modules is neither automatic nor uniformly obstructed. On the positive side, for a complete local ring $M/IM$5 of dimension $M/IM$6, the blowup at the maximal ideal provides a geometric existence criterion. If $M/IM$7 is complete, equidimensional of dimension $M/IM$8, and the exceptional fiber

$M/IM$9

is geometrically reduced, then $R/I$0 admits an Ulrich module that is locally free of constant rank on the punctured spectrum; if the residue field is algebraically closed, one can choose such a module of rank $R/I$1 on the punctured spectrum. More generally, Ulrich modules exist whenever each one-dimensional connected component of the exceptional fiber is geometrically reduced (Iyengar et al., 3 Oct 2025).

The same blowup geometry also produces nonexistence. For a long time it was unknown whether every complete local domain admits an Ulrich module, but there are complete local domains of every dimension $R/I$2 with no Ulrich modules. One source is the $R/I$3-ification method: if a local domain $R/I$4 has an $R/I$5-ification $R/I$6 which is a regular local ring, then every maximal Cohen–Macaulay $R/I$7-module is of the form $R/I$8, and $R/I$9 has an Ulrich module if and only if $R$0 is an Ulrich $R$1-module, equivalently if $R$2 for a minimal reduction $R$3 of $R$4 (Yhee, 2021).

The negative answer extends even to Cohen–Macaulay local rings. There exist two-dimensional Cohen–Macaulay local rings with no Ulrich modules, including Gorenstein normal domains and complete intersection local domains. The obstruction is again blowup-theoretic: if on the blowup of $R$5 at $R$6 the exceptional fiber is a nonreduced multiple Cartier divisor, then neither the local ring nor its completion admits an Ulrich module. This shows that good singularity classes such as Gorenstein, normal, and complete intersection do not by themselves force Ulrich existence (Iyengar et al., 2024).

5. Projective-geometric realizations and moduli

On a smooth polarized projective variety, Ulrich bundles are the geometric avatars of linear maximal Cohen–Macaulay modules. For a smooth variety $R$7, an ACM sheaf $R$8 is Ulrich if it is initialized and has the maximal possible number of sections; equivalently, for $R$9,

$M$00

Another equivalent criterion is that for any finite linear projection $M$01,

$M$02

(Cho et al., 2017).

This projective theory is now highly developed in concrete classes. On a nonsingular cubic surface, stable Ulrich bundles of rank $M$03 and first Chern class $M$04 exist precisely under explicit line and twisted-cubic intersection inequalities, excluding the exceptional multiples $M$05; when they exist, the moduli space is smooth and irreducible of dimension

$M$06

and consists entirely of stable Ulrich bundles. As a consequence, stable Ulrich bundles of every rank exist on nonsingular cubic threefolds (Casanellas et al., 2011).

For the smooth complete intersection of two $M$07-dimensional quadrics in $M$08, stable Ulrich bundles exist in every rank $M$09. The moduli space of stable rank-$M$10 Ulrich bundles is identified with a nonempty open subscheme of

$M$11

where $M$12 is the associated genus-$M$13 curve, giving expected dimension $M$14 and a derived-categorical description via the Bondal–Orlov semiorthogonal decomposition (Cho et al., 2017).

Recent work on threefold scrolls over Hirzebruch surfaces pushes the moduli perspective further. For suitable scrolls $M$15, the modular component $M$16 parameterizing rank-$M$17 Ulrich vector bundles with fixed Chern classes is generically smooth and unirational, while the associated surface moduli space $M$18 is generically smooth, irreducible, and unirational. The proofs use explicit cokernel presentations on the surface and the correspondence

$M$19

between Ulrich bundles on the base and on the scroll (Fania et al., 2024).

6. Numerical, homological, and tensorial refinements

A one-dimensional numerical refinement comes from irreducible multiplicity. For a finitely generated module $M$20 of dimension $M$21 and a parameter ideal $M$22, the irreducibility coefficients $M$23 are defined from the eventual polynomial behavior of

$M$24

In dimension one,

$M$25

for some parameter ideal $M$26. This is a genuinely one-dimensional phenomenon: the paper explicitly shows that the corresponding criterion does not extend to higher dimension in that form (An et al., 2021).

From the homological viewpoint, Ulrich modules are extremal. Over a Cohen–Macaulay local ring, every Ulrich module of dimension $M$27 is $M$28-Tor-rigid-test, but not $M$29-Tor-rigid in general. The same work proves that Ulrich modules over Cohen–Macaulay local rings have maximal complexity and maximal curvature: $M$30 Consequently, complete intersection, regularity, and Gorenstein criteria can be phrased in terms of the homological dimensions of Ulrich modules (Dey et al., 2022).

A further generalization replaces exact extremality by a controlled defect. A Cohen–Macaulay module $M$31 is $M$32-Ulrich with respect to $M$33 if $M$34 is free over $M$35 and

$M$36

Unlike classical Ulrich modules, $M$37-Ulrich modules always exist: every Cohen–Macaulay module is $M$38-Ulrich. These modules retain many standard Ulrich applications, especially as obstructions to Cohen–Macaulay tensor products and as test modules for finiteness of homological dimensions (Celikbas et al., 2023).

The tensor-product problem has also been analyzed directly for classical and generalized Ulrich modules. Criteria are given for when $M$39 is Ulrich in terms of $M$40-vanishing, transpose techniques, and freeness modulo the defining ideal; the same framework yields freeness criteria, complete-intersection characterizations, an Ulrich-based approach to Berger’s conjecture, and positive solutions of the Auslander–Reiten and Huneke–Wiegand problems for the class of Ulrich modules (Miranda-Neto et al., 2023).

7. Asymptotic and relative extensions

When actual Ulrich modules are unavailable, asymptotic substitutes can still carry multiplicity-theoretic content. A sequence $M$41 is lim Ulrich if it is lim Cohen–Macaulay and

$M$42

and weakly lim Ulrich if the same asymptotic equality holds under the weaker weakly lim Cohen–Macaulay condition. Such sequences were introduced in connection with Lech’s conjecture, and weakly lim Ulrich sequences were constructed for all standard graded domains over perfect fields of positive characteristic; their existence implies Lech’s conjecture for flat local extensions from the given domain (Ma, 2020).

These asymptotic objects do not exist universally. Complete local domains of dimension $M$43 may have no Ulrich modules, and in dimension $M$44 there are complete local domains with no weakly lim Ulrich sequences. In the presence of a regular $M$45-ification, the dimension-two theory is especially rigid: $M$46 This shows that asymptotic replacement is powerful but not automatic (Yhee, 2021).

A different extension is relative rather than asymptotic. For a relative hypersurface $M$47, a relatively Ulrich bundle is a vector bundle whose restriction to every fiber is an ordinary Ulrich bundle. In this setting there is a functorial equivalence between the category of relatively Ulrich bundles on $M$48 and the category of linear representations of the associated generalized Clifford algebra $M$49. The same work proves that relative hypersurfaces are Ulrich-wild, in the sense that there exist indecomposable relatively Ulrich bundles $M$50 with

$M$51

and that relative hyperplanes have minimal Ulrich complexity one (Mondal et al., 2 Apr 2026).