Complete-local smooth threefold flops and their slice data

Work over $\mathbb{C}$, with the Axiom of Choice. All schemes below are integral Noetherian schemes. In this complete-local setting, nonsingular (the authors' smooth-source convention) means that every local ring is regular: its Krull dimension equals the minimal number of generators of its maximal ideal. The Krull dimension of a nonzero commutative ring is the supremum of the lengths of its strict chains of prime ideals, where length counts strict inclusions. Projective means admitting a closed immersion into some finite-dimensional projective space over the base. A resolution of a normal scheme is such a projective morphism from a nonsingular scheme, isomorphic over the nonsingular locus; in particular it is birational. A partial resolution permits a normal singular source, retaining projectivity and this isomorphism condition. Its exceptional locus is the locus where the morphism is not an isomorphism.

A complete-local Du Val surface singularity means $\mathbb{C}\mathrm{⟦}a,b,c\mathrm{⟧}\mathrm{/}(q)$, up to formal change of coordinates, with $q$ one of $A_r:a^2+b^2+c^{r+1}(r\ge 1),D_r:a^2+b^{2}c+c^{r-1}(r\ge 4),E_6:a^2+b^3+c^4,E_7:a^2+b^3+b{c}^3,E_8:a^2+b^3+c^5\mathrm{.}$ Its minimal resolution $\pi :Y_{\min }\to S$ is the resolution through which every resolution of $S$ factors over $S$. Its exceptional curves are the reduced irreducible components $E_i$ of ${\pi }^{-1}(0)$, each isomorphic to ${\mathbb{P}}^1$ with $E_i^2=-2$. Their dual graph has one vertex per $E_i$ and an edge for each intersection of distinct curves; it is the corresponding finite ADE diagram. These existence and configuration facts are imported below.

A complete-local isolated compound Du Val (cDV) singularity here is a three-dimensional isolated hypersurface singularity $R=\mathbb{C}\mathrm{⟦}u,v,x,y\mathrm{⟧}\mathrm{/}(F)$ whose sufficiently general hyperplane section $R\mathrm{/}(g)$ is a Du Val surface singularity. Fix a projective crepant resolution $f:X\to \mathrm{Spec}R$ whose exceptional locus is a nonempty union of curves and which is an isomorphism away from the closed point. Here crepant means $K_X=f^{\ast }{K}_{\mathrm{Spec}R}$, with compatible canonical divisors; equivalently the canonical (dualizing) sheaves satisfy ${\omega }_X\cong f^{\ast }{\omega }_R$, compatibly over the isomorphism locus. This is the complete-local smooth flopping setting used below.

Choose a sufficiently general section $g$ for which $Y=X{\times }_{\mathrm{Spec}R}\mathrm{Spec}(R\mathrm{/}(g))$ is a partial crepant resolution of $S=\mathrm{Spec}(R\mathrm{/}(g))$. The minimal resolution factors as $Y_{\min }\stackrel{h}{\to }Y\to S$. Write $\mathrm{\Delta }$ for its finite ADE diagram and $J\subseteq \mathrm{\Delta }$ for precisely the vertices $i$ whose curve $E_i$ is mapped to a point by $h$. The images of the remaining curves are the exceptional curves of $Y\to S$, corresponding to those of $f$; write $n=\mathrm{\mid }\mathrm{\Delta }\setminus J\mathrm{\mid }$. Here exceptional curves of $f$ are the reduced one-dimensional irreducible components of its closed fibre.

For a unital ring $B$ and a left $B$-module $U$, a projective resolution is an exact augmented complex $\cdots \to P_1\to P_0\to U\to 0$ with projective terms. Its length is at most $d$ when $P_j=0$ for $j>d$. Set ${\mathrm{pd}}_{B}U=\inf \{d\ge 0:U\text{ has such a resolution of length at most }d\}$, with value $\mathrm{\infty }$ when no finite length occurs. Supply projective resolutions for the modules under consideration and define ${\mathrm{Ext}}_B^q(U,V)=H^q{\mathrm{Hom}}_B(P_{\bullet }(U),V)(q\ge 0),d^q(f)=f\circ d_{q+1}\mathrm{.}$ These resolutions exist under our Axiom of Choice, and the comparison with derived-category morphisms identifies the resulting Ext groups canonically for different supplied resolutions. Right $B$-modules use these same constructions as left $B^{\mathrm{o}\mathrm{p}}$-modules. Thus global dimension below takes the supremum of $\mathrm{pd}$ over all right modules, without restricting to finite modules.

An NCCR in this three-dimensional setting is an algebra $\mathrm{\Lambda }={\mathrm{End}}_{R}M$, where $M$ is a finitely generated reflexive $R$-module, $\mathrm{\Lambda }$ is maximal Cohen--Macaulay over $R$, and $\mathrm{gl.dim}\mathrm{\Lambda }=3$. Reflexive means that the evaluation map $M\to {\mathrm{Hom}}_R({\mathrm{Hom}}_R(M,R),R)$ is an isomorphism; maximal Cohen--Macaulay means ${\mathrm{depth}}_R\mathrm{\Lambda }=\dim R=3$, where ${\mathrm{depth}}_{R}L=\inf \{i\ge 0:{\mathrm{Ext}}_R^i(R\mathrm{/}\mathfrak{m},L)\mathrm{\ne }0\}$ and $\mathfrak{m}$ is the maximal ideal; global dimension is the supremum of projective dimensions of right modules.

For an ${\mathcal{O}}_X$-module sheaf $\mathcal{F}$, let ${\mathcal{F}}_{\mathrm{a}\mathrm{b}}$ be its underlying abelian sheaf. Under our Axiom of Choice, fix the functorial injective-resolution datum on abelian sheaves supplied by the sheaf-cohomology definition cited below. Write $0\to {\mathcal{F}}_{\mathrm{a}\mathrm{b}}\to {\mathcal{I}}^0({\mathcal{F}}_{\mathrm{a}\mathrm{b}})\to {\mathcal{I}}^1({\mathcal{F}}_{\mathrm{a}\mathrm{b}})\to \cdots$ for its resolution and ${\mathcal{I}}^{\bullet }({\mathcal{F}}_{\mathrm{a}\mathrm{b}})$ for the deleted complex. Define $H^q(X,\mathcal{F})=H^q(\mathrm{\Gamma }(X,{\mathcal{I}}^{\bullet }({\mathcal{F}}_{\mathrm{a}\mathrm{b}})))(q\ge 0)\mathrm{.}$ Here $\mathrm{\Gamma }(X,-)$ is global sections and the right-hand $H^q$ is cochain cohomology. For $a\in R$, multiplication through the structure morphism $f$ gives an abelian-sheaf endomorphism ${\mu }_a$ of ${\mathcal{F}}_{\mathrm{a}\mathrm{b}}$; set $a\cdot \xi =H^q(X,{\mu }_a)(\xi )$. This defines the $R$-module structure used below, independently of the resolution datum up to canonical $R$-linear isomorphism. In degree zero this identifies $H^0(X,\mathcal{F})$ with $\mathrm{\Gamma }(X,\mathcal{F})$. In particular $H^1(X,{\mathcal{L}}_i^{-1})$ below is sheaf cohomology with this $R$-action; its finite generation and the associated bundle extension remain part of the cited geometric construction.

For the arrangement statements fix the following basic Van den Bergh choice, rather than an arbitrary NCCR. Put $I=\mathrm{\Delta }\setminus J$ and label the exceptional curves of $f$ by $C_i$ for $i\in I$. Choose line bundles ${\mathcal{L}}_i$ with $\mathrm{deg}({\mathcal{L}}_i{\mathrm{\mid }}_{C_k})={\delta }_{ik}$. Let $m_i=r_i-1$ be the minimal number of $R$-generators of $H^1(X,{\mathcal{L}}_i^{-1})$. Choose an ordered minimal generating set $({\xi }_{i1},\dots ,{\xi }_{i{m}_i})$ of this $R$-module and take the bundle extension associated to this chosen generating set by the Van den Bergh construction recorded in External input: $0⟶{\mathcal{O}}_X^{\oplus (r_i-1)}⟶{\mathcal{E}}_i⟶{\mathcal{L}}_i⟶0.$ Set $\mathcal{V}={\mathcal{O}}_X\oplus {⨁}_{i\in I}{\mathcal{E}}_i$ and $\mathcal{T}={\mathcal{V}}^{\vee }$, where $(-{)}^{\vee }=\mathcal{H}o{m}_{{\mathcal{O}}_X}(-,{\mathcal{O}}_X)$. These are basic tilting bundles: their indecomposable summands are pairwise nonisomorphic and occur once; each bundle has vanishing positive self-Ext, and its derived Hom detects zero objects of $D^b(\mathrm{coh}X)$. The bundle $\mathcal{V}$ is generated by global sections. The existence and stated properties are imported below. We use usual function composition for endomorphism multiplication and fix $M=H^0(X,\mathcal{V})=R\oplus {⨁}_{i\in I}{M}_i,\mathrm{\Lambda }={\mathrm{End}}_{R}M\cong {\mathrm{End}}_X\mathcal{V},A={\mathrm{End}}_X\mathcal{T}\cong {\mathrm{End}}_R{H}^0(X,\mathcal{T})\mathrm{.}$ Here $M_i=H^0(X,{\mathcal{E}}_i)$, ${\mathrm{rank}}_R{M}_i=r_i>0$, and $r_0=1$ labels the free summand. The pushdown $M$ is basic and $\mathrm{\Lambda }$ is an NCCR. This is the globally generated bundle choice in Hirano--Wemyss Setup 4.1; Wemyss Notation 2.10 uses $\mathcal{T}$ as ${\mathcal{V}}_X$.

For either $\mathrm{\Gamma }=A$ or $\mathrm{\Gamma }=\mathrm{\Lambda }$, write $e_0,(e_i{)}_{i\in I}$ for the summand projections and $r=(1,(r_i{)}_{i\in I})$. The rank-vector real stability space is ${\mathrm{\Theta }}_r(\mathrm{\Gamma })=\{\theta \in {\mathbb{R}}^{\{0\}\cup I}:{\theta }_0+{\sum }_{i\in I}{r}_i{\theta }_i=0\}\mathrm{.}$ For a finite-dimensional right $\mathrm{\Gamma }$-module $Q$, its dimension vector has entries ${\dim }_{\mathbb{C}}Q{e}_i$, and $\theta (Q)={\sum }_i{\theta }_i{\dim }_{\mathbb{C}}Q{e}_i$. A module $Q$ of dimension vector $r$ is $\theta$-semistable when $\theta (N)\ge 0$ for every submodule $N\subseteq Q$, and stable when this is strict for every nonzero proper submodule. A weight is generic when all its semistable modules of dimension vector $r$ are stable. The real chamber convention is specified in the next paragraph. The independent curve coordinates are $({\theta }_i{)}_{i\in I}$; ${\theta }_0=-{\sum }_{i\in I}{r}_i{\theta }_i$. No assertion for all dimension vectors or for an arbitrary NCCR is intended.

Put $W_r(\mathrm{\Gamma })=\{\theta \in {\mathrm{\Theta }}_r(\mathrm{\Gamma }):\text{some dimension-}r\text{ module is semistable but not stable}\}$. This is the wall locus. A GIT chamber is a connected component, in the real Euclidean topology, of ${\mathrm{\Theta }}_r(\mathrm{\Gamma })\setminus W_r(\mathrm{\Gamma })$. Precisely, a wall hyperplane is a codimension-one linear subspace $H$ of ${\mathrm{\Theta }}_r(\mathrm{\Gamma })$ for which there exist $\theta \in H$ and an open ball $B$ about $\theta$ with $W_r(\mathrm{\Gamma })\cap B=H\cap B$. A wall face in $H$ is a connected component of the set of such points $\theta$; two distinct chambers are adjacent when their closures contain the same wall face. Thus the hyperplane and its open wall faces are distinguished. These conventions use the actual nongeneric locus, rather than every possible submodule-dimension hyperplane. On rational weights they use Wemyss's rank-vector GIT convention; their continuation to real weights by the same inequalities is justified below.

Let $\mathrm{coh}X$ be the category of coherent ${\mathcal{O}}_X$-modules and $\mathrm{mod}\mathrm{\Lambda }$ the category of finitely generated right $\mathrm{\Lambda }$-modules, with their usual module morphisms. For these essentially small abelian categories we use small equivalent models. Write $D^b(\mathrm{coh}X)$ and $D^b(\mathrm{mod}\mathrm{\Lambda })$ for the localizations of their bounded cochain homotopy categories at quasi-isomorphisms, with the cochain shift and distinguished cone triangles. Exact functors are additive functors equipped with a shift isomorphism and preserving distinguished triangles. A natural isomorphism is an invertible natural transformation; $\mathrm{Auteq}$ denotes exact autoequivalences modulo natural isomorphism. The size condition and abelian-category hypotheses are justified below.

For the tilting assertions, view a bounded coherent complex $K$ as a complex in the abelian category $\mathrm{Mod}({\mathcal{O}}_X)$ of all ${\mathcal{O}}_X$-module sheaves, using the exact inclusion of coherent sheaves. For a finite locally free sheaf $E$ in degree zero, choose a bounded-below injective replacement $K\to I_K^{\bullet }$ in this ambient category and put $\mathbf{R}{\mathrm{Hom}}_X(E,K)={\mathrm{Hom}}_{{\mathcal{O}}_X}(E,I_K^{\bullet })\in D^{+}(\mathbf{A}\mathbf{b}),{\mathrm{Ext}}_X^p(E,F)=H^p\mathbf{R}{\mathrm{Hom}}_X(E,F)(p\ge 0)$ for a coherent sheaf $F$ in degree zero. Here ${\mathrm{Hom}}_{{\mathcal{O}}_X}$ means global sheaf morphisms, and the complex differential is postcomposition with $d_I$. The resolution construction is independent up to natural isomorphism. These are global Ext groups and global derived Hom. In particular, positive self-Ext vanishing for $E$ means ${\mathrm{Ext}}_X^p(E,E)=0$ for every $p>0$; detection of zero means that $\mathbf{R}{\mathrm{Hom}}_X(E,K)\cong 0$ implies $K\cong 0$ in $D^b(\mathrm{coh}X)$. Apply these conventions to each of $\mathcal{V}$ and $\mathcal{T}$. Injectives are taken in $\mathrm{Mod}({\mathcal{O}}_X)$; their existence and the replacement hypotheses are checked below.

Projective morphisms before Proj

Derived category of an abelian category

Coherent module sheaves

Coherent sheaves on a locally Noetherian scheme

Affine quasi-coherent sheaves are modules

Compatible local sheaves glue uniquely up to unique isomorphism

Finite modules over Noetherian rings are Noetherian

Every algebra of finite type over a Noetherian ring is a Noetherian ring

Every quotient and every localisation of a Noetherian ring is Noetherian

The derived category inherits a triangulated structure

Exact functor between triangulated categories

Natural isomorphism

The Axiom of Choice

Projective dimension of an object

Ext via a projective resolution of the first variable

Under the Axiom of Choice, every module admits a projective resolution

Ext is hom in the derived category

Enough injective sheaves of modules

Bounded below complexes admit injective replacements

Derived hom in the bounded setting

Cohomology of derived hom is ext

Sheaf cohomology as right derived global sections

Krull dimension of a nonzero ring

Wemyss, Definition 1.1 and §2.1, supply the cDV and crepancy conventions; §3.3 supplies the generic slice, its domination by the minimal surface resolution, and the surviving-curve correspondence; §4.1 and Definition 4.2 supply the NCCR convention. Hirano--Wemyss Setup 4.1 supplies the smooth complete-local flop scope. Reid, §1.5, Table 1 and Theorem 2.1(2), supplies the ADE surface equations and crepant resolution data; Exercise 4 states the minimal-resolution curve configuration. These geometric facts are used as established inputs; their proofs are not reconstructed here. Stacks Lemma 10.31.2 supplies Noetherianity of the four-variable formal power series ring over $\mathbb{C}$. No flop-equivalence or arrangement theorem is used to define the categories.

Wemyss's cDV survey, opening of §2 and §2.2, supplies that a complete-local cDV ring is normal and that an isolated cDV singularity is Gorenstein terminal, hence Gorenstein. Wemyss's Homological MMP, Setup 2.9, states that a crepant projective birational morphism in dimension at most three to a complete-local normal Gorenstein base, with fibres of dimension at most one and, in dimension three, only Gorenstein terminal singularities on its source, satisfies $\mathbf{R}{f}_{\ast }{\mathcal{O}}_X\cong {\mathcal{O}}_{\mathrm{Spec}R}$. Here $\mathbf{R}{f}_{\ast }$ denotes total right-derived pushforward; this equality means $f_{\ast }{\mathcal{O}}_X\cong {\mathcal{O}}_{\mathrm{Spec}R}$ and $R^q{f}_{\ast }{\mathcal{O}}_X=0$ for $q>0$. The source attributes the equality to Kawamata vanishing; that theorem is imported, not reconstructed here. This is the hypothesis imposed in Van den Bergh §3.1 for the following bundle construction. Step 1.1 checks its application to the fixed resolution.

The additional bundle construction is imported from Wemyss §2.2, Notation 2.10 and Lemma 2.12, and Van den Bergh Lemma 3.5.2, Proposition 3.5.4 and Theorem 3.5.5. Specifically, Van den Bergh Lemma 3.5.2, equation (3.8) and its proof, and equation (3.10) immediately before Proposition 3.5.4 construct the bundle extension from a chosen minimal $R$-generating set of $H^1(X,{\mathcal{L}}_i^{-1})$. The proof of Lemma 3.5.2 states that different minimal generating sets give isomorphic middle terms. This precise extension construction is imported along with its existence; no local reconstruction of an Ext--cohomology correspondence is claimed. Van den Bergh §3.2, Corollary 3.2.8, Proposition 3.2.10 and Corollary 3.2.11 supply the tilting and crepant endomorphism comparison for the globally generated choice; Hirano--Wemyss Setup 4.1 and §4.1 use its basic pushdown and NCCR. Wemyss §1.3 and §5.1, Definition 5.1, supply rank-vector weights and stability inequalities. His §1.6 uses left modules and endomorphisms multiplied in order of application; converting both conventions gives right $A$-modules with our usual composition. These construction and stability inputs are imported without reconstructing the geometric proofs. No arrangement conclusion is used to construct either algebra or to define stability.

Wemyss Remark 5.16 and Notation 5.17 supply the rational rank-vector wall-locus convention in [F8]. The connected-component and local wall-face definitions above specify its real interpretation; the finite-dimensional argument below checks that interpretation without importing a geometric arrangement identification.

1.1 The field $\mathbb{C}$ is Noetherian, so [F7] and [F4] make $R$ Noetherian. A projective embedding from [F1] covers $X$ by finitely many closed subschemes of affine spaces over $R$, whose coordinate rings are finite-type $R$-algebras. They are Noetherian by [F4]. Thus $X$ is Noetherian, and [F2] applies to make $\mathrm{coh}X$ abelian. The fixed isolated cDV base is normal Gorenstein by [F11]. The resolution is projective crepant birational of dimension three with nonsingular source, so meets Setup 2.9's source condition. Its closed fibre is at most one-dimensional by the exceptional-curve hypothesis; all other fibres are points. Thus [F11] gives $\mathbf{R}{f}_{\ast }{\mathcal{O}}_X\cong {\mathcal{O}}_{\mathrm{Spec}R}$, supplying Van den Bergh's structure-sheaf hypothesis for the basic-bundle construction over this complete-local complex base. [given, F1, F2, F4, F7, F11]

1.2 Choose finitely many generators of $M$. Evaluation on them embeds ${\mathrm{End}}_{R}M$ into a finite direct sum of copies of $M$ as an $R$-module. By [F4], $\mathrm{\Lambda }$ is finite over the central ring $R$. Every finite right $\mathrm{\Lambda }$-module is then finite over $R$. The kernel of a map between two such modules is an $R$-submodule of a finite $R$-module, hence is finite over $R$, and those generators also generate it over $\mathrm{\Lambda }$. Cokernels and finite direct sums are finite as well. Kernels and cokernels are the usual module ones, so $\mathrm{mod}\mathrm{\Lambda }$ is abelian. [given, F4]

1.3 Fix either algebra $\mathrm{\Gamma }$ and its vector $r$. There are finitely many integer vectors $\beta$ with $0\le {\beta }_i\le r_i$; let $\mathcal{B}$ exclude $0$ and $r$. For a dimension-$r$ module $Q$, let $D(Q)\subseteq \mathcal{B}$ be the vectors of its nonzero proper submodules. Then $Q$ is semistable exactly on the closed cone $C_{D(Q)}=\{\theta \in {\mathrm{\Theta }}_r(\mathrm{\Gamma }):\theta \cdot \beta \ge 0\text{ for all }\beta \in D(Q)\}$, and is stable exactly when all those inequalities are strict. Only finitely many subsets $D$ can occur. Consequently $W_r(\mathrm{\Gamma })$ is the finite union, over occurring $D$ and $\beta \in D$, of $C_D\cap \{\theta :\theta \cdot \beta =0\}$. These are closed cones defined by rational linear inequalities. Each displayed hyperplane is proper: a vector proportional to $r$ with integer zeroth entry between zero and $r_0=1$ is either $0$ or $r$. Hence the generic locus is open. For each fixed $Q$, its semistable locus is closed; within the generic locus it equals its stable locus, which is open. It is therefore both open and closed there, and is constant on every connected component. Thus every defined chamber has constant semistable and stable modules, as required by the GIT convention [F8]. [given, F8, construct, algebra]

1.4 The module categories for $R$ and for a unital algebra $B$ are abelian with their usual kernels and cokernels. By [F9], our Choice assumption gives a projective resolution for each module considered, using $B^{\mathrm{o}\mathrm{p}}$ for right modules. Supply these resolutions objectwise when applying the projective-dimension and Ext definitions; no simultaneous choice over a proper class is required. Their projective-resolution and supplied-data hypotheses are therefore met. The comparison in [F9] applies because Choice supplies Dependent Choice; it identifies each resolution Ext with the same derived-category morphism group, so the suppressed resolution choice changes the group only by its canonical comparison isomorphism. In particular the depth formula uses module Ext over $R$, and the NCCR global dimension uses projective dimensions of all right $\mathrm{\Lambda }$-modules. [given, F9]

1.5 The scheme $X$ has an underlying topological space and every ${\mathcal{O}}_X$-module sheaf has an underlying abelian sheaf, so [F12] applies under the stated Choice assumption and supplies the resolution datum used in the Definition. Left exactness of global sections identifies the kernel of $\mathrm{\Gamma }(X,{\mathcal{I}}^0)\to \mathrm{\Gamma }(X,{\mathcal{I}}^1)$ with $\mathrm{\Gamma }(X,\mathcal{F})$, proving the asserted degree-zero identification. For the multiplication maps, ${\mu }_{a+b}={\mu }_a+{\mu }_b$, ${\mu }_{ab}={\mu }_a\circ {\mu }_b$ and ${\mu }_1=\mathrm{i}\mathrm{d}$. Additivity and functoriality of $H^q$ therefore give a unital ring homomorphism $R\to {\mathrm{End}}_{\mathbf{A}\mathbf{b}}{H}^q(X,\mathcal{F})$, which is exactly an $R$-module structure. The natural comparison between resolution data commutes with each ${\mu }_a$, hence is $R$-linear; the degree-zero comparison is also $R$-linear. This checks the cohomology notation and scalar structure used in the bundle construction, while its finite-generation and extension assertions remain the independent geometric imports. [given, F12, algebra]

1.6 The ring $R$ and the local rings at points of the integral schemes in the Definition are nonzero commutative rings, so the dimension interface [F13] applies to each. In particular it supplies the dimensions in the regular-local-ring condition and in ${\mathrm{depth}}_R\mathrm{\Lambda }=\dim R=3$; the value three is already part of the fixed three-dimensional hypersurface setting. No assertion about the zero ring is required. [given, F13]

2.1 Fix the finite affine cover $U_i=\mathrm{Spec}{A}_i$ of $X$ from step 1.1. Every coherent sheaf restricts, by [F2] and [F3], to the sheaf of a finite $A_i$-module. Each such module is isomorphic to $A_i^{r_i}\mathrm{/}{N}_i$ for an integer $r_i\ge 0$ and a submodule $N_i\subseteq A_i^{r_i}$. These quotients form a set of models. Between two fixed sheaves the morphisms form a set: they are families of maps on the set of open subsets, satisfying the restriction and module conditions. Thus overlap isomorphisms between the model sheaves also form sets, and the cocycle-compatible gluing data on this fixed finite cover form a set. By [F3] their glued sheaves give a set of representatives up to isomorphism for all coherent sheaves on $X$; coherence is local. The full category on these models is therefore small and equivalent to $\mathrm{coh}X$. [F2, F3, F6, step 1.1, construct]

2.2 Partition ${\mathrm{\Theta }}_r(\mathrm{\Gamma })$ by the signs and zeros of the finitely many forms $\theta \cdot \beta$ for $\beta \in \mathcal{B}$. Each nonempty stratum has dense rational points: its zero equations have rational coefficients, so Gaussian elimination gives a rational basis for their solution subspace; approximate basis coefficients by rationals closely enough to retain the finitely many strict signs. Step 1.3 shows that every module's stability and semistability, and membership in $W_r(\mathrm{\Gamma })$, are constant on each stratum. It also expresses the wall locus as a finite union of closed rational cones, each recovered by closure from its rational points using the same argument on its sign strata. Therefore the real wall locus is exactly the closure of the rational nongeneric locus. Taking connected components of its complement and the local hyperplane pieces specified in the Definition gives the stated real continuation of [F8], without treating the whole collection of possible numerical hyperplanes as actual walls. This argument uses only the fixed-vector stability inequalities, independently of any later slice or flop-arrangement theorem. [F8, step 1.3, construct, algebra]

3.1 Similarly, every finite right $\mathrm{\Lambda }$-module is isomorphic to ${\mathrm{\Lambda }}^r\mathrm{/}N$ for some nonnegative integer $r$ and right submodule $N\subseteq {\mathrm{\Lambda }}^r$. There is a set of these models, and all their module homomorphisms form sets, so their full category is a small equivalent model for $\mathrm{mod}\mathrm{\Lambda }$. The equivalences here and in step 2.1 preserve kernels, cokernels and finite sums by their universal properties, hence the small models are abelian. [step 1.2, step 2.1, F6, construct]

4.1 In either small abelian model there is a set of bounded cochain complexes and of maps between them, hence a small bounded homotopy category. The small-complexes alternative of [F5] applies to its localization at quasi-isomorphisms, giving actual Hom sets. Closure under shifts and cones gives the bounded version of the triangulated structure in [F5]. These constructions define both displayed derived categories without using a tilting or flop equivalence to establish their existence. They also make the exact-autoequivalence notation meaningful by [F6]. Choice is used through [F2] and [F3] and for representative choices in the set-sized model constructions; no global choice over a proper class of sheaves or modules is used. [F2, F3, F5, F6, step 2.1, step 3.1]

5.1 The scheme $X$ is a ringed space, so [F10] applies to all ${\mathcal{O}}_X$-module sheaves. By [F2], the inclusion of coherent sheaves is exact and therefore takes bounded coherent complexes and quasi-isomorphisms to those in this ambient abelian category. Such complexes are bounded below; [F10] and our Choice assumption supply their bounded-below injective replacements. The sheaf $E$ in degree zero is bounded above, so these replacements meet the injective alternative of the derived Hom definition in [F10]. Its cohomology comparison supplies precisely the global Ext groups used in the tilting convention, independently of the replacement. This establishes applicability of the homological constructions; the vanishing and detection properties of $\mathcal{V}$ and $\mathcal{T}$ remain the independent geometric imports already specified in External input. [given, F2, F10] ∎