Homological Minimal Programme

• The Homological Minimal Programme is a framework that uses homological algebra, mutation, and derived equivalence to algorithmically navigate and simplify minimal models.
• It detects birational contractions by relying on finite-dimensional factors, noetherian overring conditions, and Ext-group computations instead of classical negativity.
• The approach extends to noncommutative surfaces and monomial algebras, providing explicit models via tilting bundles, Tor computations, and A∞-structures.

The Homological Minimal Programme designates a family of approaches in which operations analogous to the Minimal Model Program are controlled by homological algebra, noncommutative algebra, or derived-categorical data rather than by direct manipulation of cones of curves alone. In the most explicit usage, it is a dimension-three procedure that allows one to “jump between all the minimal models” of a complete local Gorenstein $3$-fold by mutating a noncommutative algebra derived equivalent to a minimal model (Wemyss, 2014). In a broader and closely related sense, it includes noncommutative surface theories in which minimality is formulated through the absence of proper noetherian overrings inside a graded quotient ring, and explicit homological minimal models for associative algebras built from $\operatorname{Tor}$, $A_\infty$-structures, and homotopy transfer (Rogalski et al., 2018, Tamaroff, 2018). The unifying feature is that birational or resolution-type phenomena are encoded in homological or categorical structures and then recovered algorithmically or combinatorially.

1. Historical placement and conceptual scope

The classical Minimal Model Program seeks birational classification by controlled operations such as divisorial contractions, flips, and Mori fiber spaces, with finite generation of canonical or adjoint rings functioning as a structural organizing principle (Cascini et al., 2010, Corti et al., 2010). A parallel line of work describes MMPs by explicit combinatorics, for example through one-parameter families of moment polytopes on projective $\mathbf Q$-Gorenstein horospherical varieties, where the MMP is “literally the evolution of the polytope as $\epsilon$ increases” (Pasquier, 2012).

Against that background, the Homological Minimal Programme replaces some of the usual geometric detection mechanisms by homological data. In dimension three, once a minimal model $f\colon X\to \operatorname{Spec} R$ has a tilting bundle, Van den Bergh’s derived equivalence produces a noncommutative ring

$A:=\operatorname{End}_R(N),$

and the resulting module category retains enough information to determine which curves can be flopped, what the flop is, and how stability chambers change (Wemyss, 2014). In noncommutative projective surface theory, minimality is reformulated by saying that an elliptic algebra $T$ is minimal precisely when there is no proper connected graded noetherian overring between $T$ and $T(g)$, where $\operatorname{Tor}$0 is central (Rogalski et al., 2018).

This suggests that the phrase does not denote a single formalism. Rather, it denotes a programmatic viewpoint: birational simplification, contraction, and minimality are recast in terms of mutation, localization, Ext-groups, $\operatorname{Tor}$1-structures, maximal orders, or finitely generated graded rings. The strongest common theme is that geometric moves are made computable or classifiable by passing through algebraic or homological invariants.

2. Dimension three: mutation, flops, and algorithmic navigation of minimal models

In the threefold setting, the Homological MMP is developed for a complete local normal Gorenstein $\operatorname{Tor}$2-fold $\operatorname{Tor}$3 over $\operatorname{Tor}$4, together with a crepant projective birational morphism $\operatorname{Tor}$5 whose fibres are at most one-dimensional; the ideal case is that $\operatorname{Tor}$6 is a minimal model, meaning $\operatorname{Tor}$7 is $\operatorname{Tor}$8-factorial terminal and crepant over $\operatorname{Tor}$9 (Wemyss, 2014). The starting point is the derived equivalence

$A_\infty$0

with $A_\infty$1 for a tilting bundle $A_\infty$2 on $A_\infty$3 (Wemyss, 2014).

A central criterion detects when a chosen collection of curves contracts to a point without contracting a divisor. If $A_\infty$4 is a collection of curves above the closed point and

$A_\infty$5

equivalently $A_\infty$6, then

$A_\infty$7

(Wemyss, 2014). The finite-dimensional factor $A_\infty$8 is the homological detection mechanism for floppability.

Mutation then implements the birational move. Given a basic modifying module $A_\infty$9 and a summand $\mathbf Q$0, right and left mutation are defined by

$\mathbf Q$1

and the associated tilting module induces a derived equivalence

$\mathbf Q$2

(Wemyss, 2014). When $\mathbf Q$3 flops, equivalently $\mathbf Q$4, mutation and the Bridgeland–Chen flop functor coincide: $\mathbf Q$5 (Wemyss, 2014). In that sense, mutation on the algebra side is the same operation as a flop on the geometric side.

The resulting procedure is algorithmic. Starting with $\mathbf Q$6, one tests which factors $\mathbf Q$7 are finite-dimensional, mutates along the corresponding summands, replaces $\mathbf Q$8 by $\mathbf Q$9, and repeats (Wemyss, 2014). The paper explicitly describes this as “jumping between all the minimal models” without recomputing the geometry at each stage. It further proves that maximal modifying $\epsilon$0-module generators are in bijection with minimal models for complete local cDV singularities, and that the mutation graph of MM generators matches the flops graph of minimal models (Wemyss, 2014). This is presented as a lift of the Auslander–McKay correspondence to dimension three.

3. Noncommutative surfaces: minimal elliptic surfaces and homological contraction theory

For noncommutative projective surfaces, the relevant objects are connected graded noetherian domains $\epsilon$1 with

$\epsilon$2

studied through the quotient category

$\epsilon$3

(Rogalski et al., 2018). In this setting, the noncommutative analogue of a minimal model is expressed in terms of overrings. If $\epsilon$4 is an elliptic algebra with central element $\epsilon$5, then $\epsilon$6 is a minimal elliptic surface if whenever

$\epsilon$7

with $\epsilon$8 connected graded and noetherian, one has $\epsilon$9 (Rogalski et al., 2018).

This replaces the commutative criterion “no $f\colon X\to \operatorname{Spec} R$0-curves” by an overring-maximality condition. The analogue of a $f\colon X\to \operatorname{Spec} R$1-curve is a line module $f\colon X\to \operatorname{Spec} R$2 with Hilbert series

$f\colon X\to \operatorname{Spec} R$3

and self-intersection

$f\colon X\to \operatorname{Spec} R$4

(Rogalski et al., 2018). If

$f\colon X\to \operatorname{Spec} R$5

then $f\colon X\to \operatorname{Spec} R$6 can be blown down: there exists an elliptic algebra $f\colon X\to \operatorname{Spec} R$7 with

$f\colon X\to \operatorname{Spec} R$8

and $f\colon X\to \operatorname{Spec} R$9 is a direct sum of shifts of $A:=\operatorname{End}_R(N),$0 (Rogalski et al., 2018). Conversely, $A:=\operatorname{End}_R(N),$1 can be blown up at a point to recover $A:=\operatorname{End}_R(N),$2. Thus blowing up corresponds to passing to a subring, blowing down corresponds to passing to an overring, and minimality means there are no nontrivial blowdowns remaining.

The main examples are the $A:=\operatorname{End}_R(N),$3-dimensional Sklyanin algebra, Van den Bergh quadrics, and a quantum analogue of the Hirzebruch surface $A:=\operatorname{End}_R(N),$4. The paper proves that

$A:=\operatorname{End}_R(N),$5

are minimal elliptic surfaces (Rogalski et al., 2018). More conceptually, if $A:=\operatorname{End}_R(N),$6 is hereditary and $A:=\operatorname{End}_R(N),$7 has no line modules of self-intersection $A:=\operatorname{End}_R(N),$8, then $A:=\operatorname{End}_R(N),$9 is a minimal elliptic surface (Rogalski et al., 2018). A complementary contraction theorem states that when $T$0 is hereditary, every connected graded noetherian overring $T$1 of $T$2 inside $T$3 is, after a finite module-finite extension, obtained by contracting finitely many line modules $T$4 with

$T$5

(Rogalski et al., 2018).

The homological input is decisive. Elliptic algebras in the paper are Auslander–Gorenstein and Cohen–Macaulay, and the proofs exploit homological grade $T$6, maximal Cohen–Macaulay modules, torsionfree extensions, Ext-duality of line modules, hereditary localization, and singularity categories (Rogalski et al., 2018). The paper therefore shows that, in this noncommutative surface context, the minimal model story can be driven by homological algebra rather than by the geometry of points and curves.

4. Explicit homological minimal models for associative algebras

A second use of “minimal model” in the broader homological program concerns associative algebras rather than birational models of varieties. For a monomial algebra $T$7, one can write down an explicit minimal model

$T$8

where $T$9 is the cobar construction on $T$0, equipped with a canonical minimal $T$1-coalgebra structure (Tamaroff, 2018). The construction uses the combinatorics of Anick chains and algebraic discrete Morse theory applied to the reduced bar complex

$T$2

with differential

$T$3

and deconcatenation coproduct

$T$4

(Tamaroff, 2018).

The homotopy transfer theorem yields higher coproducts

$T$5

and, for monomial algebras, only the right comb planar tree contributes to the higher coproducts (Tamaroff, 2018). The resulting differential on the minimal model is completely explicit: if $T$6 is a chain, then

$T$7

where the sum is over all decompositions of $T$8 into chains $T$9 (Tamaroff, 2018). Dualizing gives a canonical $T(g)$0-structure on $T(g)$1, and the higher multiplication is nonzero exactly when the chains concatenate to a chain of the expected length (Tamaroff, 2018).

This usage is not birational in the geometric sense, but it belongs to the same homological perspective: a complicated algebra is replaced by a small, explicit, quasi-free differential graded model that encodes its higher operations. The paper explicitly situates this inside a broader “homological minimal model” program in which one starts from $T(g)$2, computes $T(g)$3, transfers the coalgebra structure using homotopy retracts, and recovers invariants and higher operations from the resulting minimal model (Tamaroff, 2018). A plausible implication is that the phrase Homological Minimal Programme can denote not only birational operations on noncommutative spaces, but also explicit homological replacement procedures that play an analogous structural role.

5. Relation to the classical Minimal Model Program

The Homological Minimal Programme is best understood in relation to the classical MMP rather than as its wholesale replacement. In the standard framework, finitely generated divisorial rings determine a geography of models, and finite generation together with the “gen” condition yields a decomposition

$T(g)$4

into rational polyhedral chambers, each carrying an optimal model (Kaloghiros et al., 2012). Likewise, finite generation of adjoint rings implies the Rationality, Cone and Contraction theorem, the existence of flips, and termination of flips with scaling in the presence of a big boundary (Corti et al., 2010). In recent algorithmic work, the threefold MMP over $T(g)$5 is turned into an actual algorithm by explicit graded-ring computations, Stein factorization, and symbolic Rees algebras (Yasuda, 14 Mar 2026).

These classical or computational formulations remain geometric even when they are heavily ring-theoretic. By contrast, the Homological MMP in the sense of flops and clusters replaces repeated geometric reconstruction by mutation of endomorphism algebras and derived equivalences (Wemyss, 2014). In noncommutative surface theory, it replaces exceptional-curve negativity by maximality inside a graded quotient ring and by line-module contraction theory controlled through Ext and localization (Rogalski et al., 2018). In the monomial-algebra setting, it replaces a presentation of $T(g)$6 by a canonical dg or $T(g)$7-model built from $T(g)$8 (Tamaroff, 2018).

The relation is therefore complementary. The classical MMP provides the geometric archetype: divisorial contractions, flips, minimal models, Mori fiber spaces, chamber decompositions, and finite generation. The homological versions preserve the same pattern of controlled simplification, but shift the mechanism to endomorphism rings, contraction algebras, line modules, hereditary localizations, or transferred $T(g)$9-structures. This suggests that the Homological Minimal Programme is not a rejection of Mori theory; it is a recoding of parts of Mori theory in homological or categorical language.

6. Main themes, limitations, and present status

Several themes recur across the literature. One is detectability: floppability can be detected by finite-dimensional factors $\operatorname{Tor}$00 of a noncommutative endomorphism algebra (Wemyss, 2014), and minimality can be detected by the absence of proper connected graded noetherian overrings inside $\operatorname{Tor}$01 (Rogalski et al., 2018). A second is algorithmicity: mutation makes it possible to navigate all minimal models in dimension three without recomputing geometry at each step (Wemyss, 2014), while explicit chain combinatorics and discrete Morse theory make the minimal model of a monomial algebra computable (Tamaroff, 2018). A third is replacement of negativity by homology: in noncommutative surface theory, homological criteria replace classical negativity arguments (Rogalski et al., 2018).

The scope is nevertheless specific. The threefold Homological MMP of flops and clusters is formulated for complete local Gorenstein $\operatorname{Tor}$02-folds with one-dimensional fibres and depends on derived equivalences coming from tilting bundles (Wemyss, 2014). The noncommutative surface theory is developed for elliptic algebras and distinguished Sklyanin/quadric examples, with hereditary or $\operatorname{Tor}$03-type conditions on $\operatorname{Tor}$04 playing a crucial role (Rogalski et al., 2018). The explicit minimal-model theory for monomial algebras is combinatorial and does not by itself produce birational models of varieties (Tamaroff, 2018).

A common misconception is that “homological” here simply means “algorithmic” or “ring-theoretic.” The proposal to program the classical MMP via finitely generated multigraded rings, convex geometry, and asymptotic valuations is explicitly not a homological theory in the sense of derived categories, Ext, or Tor (Lazić, 2023). Conversely, the homological approaches discussed above are not merely reformulations of section-ring finite generation. They use mutation, tilting, localization, contraction algebras, hereditary categories, and $\operatorname{Tor}$05-transfer as primary mechanisms.

In present usage, the most precise meaning of Homological Minimal Programme remains the dimension-three program of navigating minimal models by mutation and derived equivalence (Wemyss, 2014). The broader literature shows that the same idea extends naturally to noncommutative surfaces and to explicit minimal models for associative algebras (Rogalski et al., 2018, Tamaroff, 2018). Taken together, these works establish a coherent research direction: birational simplification, contraction, and minimality can often be formulated, detected, and computed through homological structures.