On quiver Grassmannians and orbit closures for gen-finite modules

We show that endomorphism rings of cogenerators in the module category of a finite-dimensional algebra A admit a canonical tilting module. The tilted algebra B is related to A by a recollement. We call an A-module M gen-finite if there are only finitely many indecomposable modules generated by M. Using the canonical tilts of endomorphism algebras of suitable cogenerators associated to M, and the resulting recollements with A, we construct desingularisations of the orbit closure and quiver Grassmannians of M, thus generalising all results from previous work of Crawley-Boevey and the second author in 2017. We provide dual versions of the key results, in order to also treat cogen-finite modules.


Introduction
Let A be a finite-dimensional basic algebra over an algebraically closed field K. There are various algebraic varieties whose points parametrise A-modules; those we focus on here are orbit closures (in the representation space of A), and quiver Grassmannians. These varieties often have singularities, and our aim here is to construct desingularisations, in a representation-theoretic way. Constructing a desingularisation for a variety can be useful in studying the variety itself, particularly with respect to questions concerning its singularity type. We recommend Zwara's survey [46] for an overview of some results in this area.
In the late 70s, Kraft and Procesi [25] constructed desingularisations of orbit closures for K[t]/(t n ), under the assumption that K has characteristic zero. Similar methods were used much later by Cerulli Irelli, Feigin and Reineke [12] to construct desingularisations of quiver Grassmannians for path algebras of Dynkin quivers. These results were unified and extended to arbitrary representation-finite algebras, by Crawley-Boevey and the second author [14]. Here, we give a more general construction, for arbitrary finite-dimensional algebras, using instead suitable finiteness condition on the module M defining the orbit closure or quiver Grassmannian; precisely, we ask that M generates finitely many indecomposable modules. Desingularisations of orbit closures of such modules were constructed by Zwara [45] using different methods.
Intriguingly, our desingularisations for both orbit closures and quiver Grassmannians of M are given in terms of varieties of modules for a second algebra B, associated to M via a homological construction. Moreover, our results may also be applied to give new desingularisations in the representation-finite case; the constructions of [12,14] involve the basic additive generator of A-mod, whereas we show that this may often be replaced with a smaller module, depending on the particular orbit closure or quiver Grassmannian under consideration.
A crucial ingredient in the constructions of [14], proved therein, is the fact that Auslander algebras have a unique tilting and cotilting module generated and cogenerated by projective-injectives. In our wider context, we prove the following more general statement by way of a replacement. For a module M , let gen(M ) be the category of A-modules generated by M (see Definition 2.10). Lemma 1.1 (cf. Lemma 2.7(2)). Let A be a finite-dimensional algebra, E a basic finite-dimensional cogenerating A-module, and Γ = End A (E) op . Then P = Hom A (E, DA) is a faithful projective Γmodule and there is a unique (up to isomorphism) basic tilting Γ-module T P with gen(T P ) = gen(P ).
In the situation of this lemma we study the tilted algebra B = End Γ (T P ) op , which we call the cogenerator-tilted algebra of the cogenerator E.
Since P is a summand of T P , there is an idempotent e ∈ B defined by projection onto P , which we call special. This has the property that eBe ∼ = A, and we prove (Theorem 2.12) that the resulting intermediate extension functor c : A-mod → B-mod (see Section 2) satisfies c(DA) = D(eB), c(E) = DT P .
The proof of this result uses a description of B as the endomorphism ring of the morphism viewed as a 2-term complex in the homotopy category of A (see Section 3) where E → Q(E) denotes a minimal injective envelope of E.
This description in terms of the homotopy category of A also provides us with a connection between the representation space of B and rank varieties in the representation space R A (d) of A-modules with dimension vector d. These are defined for E = t j=1 E j with E j indecomposable and m = ( When Q is injective, dim Hom A (N, Q) is constant as N runs over R A (d), and we denote its value by [d, Q]. Since eBe ∼ = A, the dimension vector of a B-module X may be written as a pair (d, s), where d is the dimension vector of the A-module eX. Moreover, the representation space R B (d, s) of (d, s)-dimensional B-modules is acted on by the product Gl d × Gl s of general linear groups, and we may restrict to the action by Gl s . Proposition 1.2 (cf. Proposition 4.6). Assume K has characteristic zero. Let E be a cogenerating A-module and B its cogenerator-tilted algebra. Let d ∈ Z n ≥0 be a dimension vector for A, and let m ∈ Z t ≥0 . If C E m = ∅, then we may extend d to a dimension vector (d, s) ∈ Z t ≥0 for B such that the special idempotent e of B induces an isomorphism For any M ∈ R A (d) there exists a cogenerator E = t j=1 E j such that the orbit closure O M is an irreducible component of the variety C E m for m j = dim Hom A (M, E j ), but it is usually difficult to compute E explicitly. We call M gen-finite if there is a finite-dimensional module E such that gen(M ) = add E. Zwara proved in [45,Thm. 1.2(4)] that if M is gen-finite, then O M = C E m for E = t j=1 E j the unique basic module with add E = gen(M ) and m j = dim Hom A (M, E j ). A little further analysis (see Remark 4.2) shows that it is enough to have gen(M ) ⊆ add E, so we may assume E is a cogenerator. Thus the orbit closure of a gen-finite module may be realised as an affine quotient variety as in the previous proposition.
From now on, assume M is a gen-finite module and let E be the basic cogenerator formed by taking the direct sum of all indecomposables in gen(M ), together with any remaining indecomposable injectives. In this case, we may use the cogenerator-tilted algebra B, together with the special idempotent e and its corresponding intermediate extension functor c, to construct desingularisations of various varieties of modules associated to M .
Firstly, we construct a desingularisation of the orbit closure O M . Call the B-modules in cogen(D(eB)) stable, and let (d, s) := dim c(M ). We write O induced by e. Each such map is projective since it is an algebraic map between projective varieties, and we may restrict it to a projective map for each i. Combining these maps, we obtain  [46]. (Q2) There are several different ways of realizing arbitrary projective varieties as quiver Grassmannians; see independent work of Hille [18,19], Huisgen-Zimmermann (including collaborations with Bongartz, Derksen and Weyman) [8,15,22], Reineke [32] and Ringel [38,39]. From the point of view of our work, it is naturally interesting to discover which projective varieties are isomorphic to a quiver Grassmannian for a gen-finite module, and thus can be desingularised by our construction. Since we suspect that this will not always be possible, it would also be of interest to extend our results to other quiver Grassmannians-optimistically, such considerations could lead to a representation-theoretic proof of Hironaka's theorem [20], stating that every projective variety admits a desingularisation. Throughout the paper, all algebras are finite-dimensional K-algebras over some field K, and without additional qualification 'module' is taken to mean 'finitely-generated left module'. Morphisms are composed from right-to-left.

Special (co)tilting
The goal of this section is to characterise certain tilting (and cotilting) modules which are generated by a projective summand (or respectively cogenerated by an injective summand). These modules will form the basis of our subsequent constructions. This section and the next, being purely homological, require no additional assumptions on the field K.
Definition 2.1. Let Γ be a finite-dimensional K-algebra. Recall that T ∈ Γ-mod is a tilting module (or sometimes classical tilting module) if (T1) pd T ≤ 1, (T2) Ext 1 Γ (T, T ) = 0, and (T3) there is an exact sequence 0 We say that T is P -special, for some projective Γ-module P , if P ∈ add T and the module T 0 in (T3) can be chosen to lie in add P . Dually, C ∈ Γ-mod is a cotilting module if (C1) id C ≤ 1, (C2) Ext 1 Γ (C, C) = 0, and (C3) there an exact sequence 0 and we say C is Q-special, for some injective Γ-module Q, if Q ∈ add C and C 0 in (C3) can be chosen to lie in add Q. We say that a tilting module is special if it is P -special for some P , and define special cotilting modules analogously.
Proposition 2.2. If T and T ′ are P -special tilting modules, then add T = add T ′ . In particular, any two basic P -special tilting modules are isomorphic. The analogous results hold for Q-special cotilting modules.
Proof. Since the middle term in the exact sequence from (T3) may be chosen to lie in add P in both cases, we have gen(T ) = gen(P ) = gen(T ′ ), so write T for this subcategory. These sequences each provide a monomorphism from Γ to an object of add P , so P ∈ T is faithful and ann(T ) = 0. Hence by [41], the direct sum T 0 of indecomposable Ext-projectives in T is a tilting Γ-module. But the tilting modules T and T ′ are also Ext-projective in T . Since any two tilting modules have the same number of pairwise non-isomorphic direct summands, it follows that add T = add T 0 = add T ′ . The statement for cotilting modules is proved dually.
Definition 2.3. Let Γ be a finite-dimensional algebra, P a projective Γ-module and Q an injective Γ-module. Bearing in mind Proposition 2.2, we denote (when they exist) the unique basic P -special tilting module by T P and the unique basic Q-special cotilting module by C Q . Their endomorphism algebras are denoted by Remark 2.4. If add P = add P ′ , then there is a P -special tilting module if and only if there is a P ′ -special tilting module, and T P = T P ′ . The analogous statement also holds for Q-special cotilting modules, so without loss of generality we may always assume that P and Q in Definitions 2.1 and 2.3 are basic.
Example 2.5. (1) The first examples of tilting modules were APR-tilting modules [3]. Let Γ be the path algebra of an acyclic quiver and a a sink in the quiver, with at least one incoming arrow. Let P = i =a P (i). Then the unique basic P -special tilting module T P = P ⊕ τ − S(a) is precisely the APR-tilting module.
(2) The 1-shifted and 1-coshifted modules for an algebra Γ of positive dominant dimension, studied by the authors in [29] (see also [21,27]) are Π-special, where Π additively generates the category of projective-injective Γ-modules. (3) The characteristic tilting module T of a right ultra-strongly quasihereditary algebra is special cotilting, by a theorem of Conde stated in the introduction to [13]. In the notation of loc. cit., in which the indecomposable injective modules are indexed by pairs (i, j) with j ≤ ℓ i for some ℓ i , the module T is special cotilting for the injective Q = i Q i,ℓ i , the theorem showing that each indecomposable injective Q i,j with j < ℓ i fits into an exact sequence for T (i, j) a summand of T . Dually, the characteristic tilting module of a left ultra-strongly quasihereditary algebra is special tilting.
Remark 2.6. Let Γ be a finite-dimensional algebra, Q ∈ Γ-mod an injective module and P ∈ Γ-mod a projective module. Assume that there exists a Q-special cotilting module and a P -special tilting module. As usual, we denote the unique basic such modules by C Q and T P , and their endomorphism algebras by B Q and B P .
(1) Since Q ∈ add C Q , the B Q -module P = Hom Γ (C Q , Q) is projective. It then follows by applying Hom Γ (C Q , −) to the exact sequence in (C3) that DC Q is the unique basic P -special tilting B Q -module. (2) Dually, the B P -module Q = D Hom Γ (P, T P ) is injective, and DT P is the unique basic Q-special cotilting B P -module.
The following lemma provides our most important source of special tilting and cotilting modules. (1) If E is a cogenerator, then P = DE is a projective Γ-module and there is a unique basic P -special tilting module T P for Γ. Moreover, End Γ (P ) op ∼ = A. (2) If E is a generator, then Q = DE is an injective Γ-module and there is a unique basic Q-special cotilting module C Q for Γ. Moreover, End Γ (Q) op ∼ = A.
Proof. We prove only (1), statement (2) being dual. First, observe that is projective, since E is a cogenerator so DA ∈ add E. By Proposition 2.2, it is enough to show the existence of a P -special cotilting module. Let f : E → Q(E) be an injective envelope. Applying Hom A (E, −) to this map and taking the cokernel yields a short exact sequence Moreover, P 0 = Hom A (E, Q(E)) ∈ add Hom A (E, DA) = add P. Let T = P 0 ⊕ T 1 ; we claim that T is a P -special cotilting module. Property (T3) is immediate from sequence (2.1). Since E is a cogenerator, DA is a summand of Q(E), and so P ∈ add P 0 ⊆ add T , establishing (T1). To show that (T2) holds, it is enough to prove that (i) Ext 1 Γ (T 1 , P 0 ) = 0, and (ii) Ext 1 Γ (T 1 , T 1 ) = 0, since P 0 is projective. For (i), apply Hom Γ (−, P 0 ) to (2.1) to obtain an exact sequence 0 Hom We wish to show that g is surjective. Consider the commutative diagram in which the vertical maps are isomorphisms from Yoneda's lemma. Since f is an injective envelope of E, the map h = Hom A (f, Q(E)) is surjective, so g is also surjective as required. We now show that (ii) follows from (i). Applying various Hom-functors to sequence (2.1) yields the commutative diagram 0 Hom Γ (T 1 , P 0 ) Hom Γ (P 0 , P 0 ) By (i) we know that g is surjective, and q is also surjective since Γ is projective. Thus p is surjective, and (ii) follows. The final statement follows by Yoneda's lemma, since using that E is a cogenerator so DA ∈ add E.
(1) If E is a cogenerator, let P be as in Lemma 2.7(2). We call B P = End Γ (T P ) the cogeneratortilted algebra of E, and the idempotent e ∈ B P given by projection onto P is called special. (2) If E is a generator, let Q be as in Lemma 2.7(1). We call B Q = End Γ (C Q ) the generator-cotilted algebra of E, and the idempotent e ∈ B Q given by projection onto Q is called special.
It follows from Lemma 2.7 that the idempotent subalgebra defined by the special idempotent is isomorphic to A in both cases, and from Remark 2.6 that the canonical tilting B Q -module DC Q is the unique B Q e-special tilting module and the canonical cotilting B P -module DT P is the unique D(eB P )-special cotilting module. We note that if E is both a generator and a cogenerator then, in the terminology of [29], the generator-cotilted algebra of E is the 1-coshifted algebra of Γ, and the cogenerator-tilted algebra of E is the 1-shifted algebra of Γ. Example 2.9. Let A be a finite-dimensional algebra, and fix a natural number L such that rad L (A) = 0. Let E be a basic A-module such that and write R A = End A (E) op . This construction is due to Auslander [2], and Dlab and Ringel showed that R A is quasihereditary [16]; hence R A is often called the ADR-algebra of A. Since A/ rad L (A) = A, the module E is a generator, and so R A admits a unique basic DE-special cotilting module C by Lemma 2.7.
The algebra R A is even right ultra-strongly quasihereditary [13], so by Example 2.5(3), its characteristic tilting module is special cotilting for an injective module Q = i Q i,ℓ i . By [13,Lem. 4.4], Q = DE, and so in fact the characteristic tilting module is the special cotilting module C from Lemma 2.7. Thus in this case the generator-cotilted algebra of E is, by definition, the Ringel dual of the quasihereditary algebra R A [36, §6].
Let B be a finite-dimensional algebra, let e ∈ B be an idempotent element and write A = eBe. We obtain from e a diagram Such data is known as a recollement of abelian categories, and can be defined in abstract, but we will only consider recollements of module categories determined by idempotents as above (cf. [30]). For a B-module M , one obtains the same A-module eM either by applying the functor e in this diagram, or by multiplying on the left by the idempotent e, hence the abuse of notation. Since ℓ and r are left and right adjoints of e respectively, and eℓ = er = id, there is a natural isomorphism functorial in M , and so determining a canonical map of functors ℓ → r. This map is equivalently described as the composition of the counit of the adjunction (ℓ, e) with the unit of the adjunction (e, r). Taking its image yields a seventh functor c : A-mod → B-mod, called the intermediate extension [26]. This functor will be particularly important in our geometric constructions, and so much of the algebraic part of the paper is devoted to studying it. We recall from [29] a description of the images and kernels of some of the functors appearing in the above recollement. This description uses the following notation. Definition 2.10. Let X ∈ A-mod be a module. We write gen(X) for the full subcategory of A-mod consisting of modules admitting an epimorphism from an object of add X, and gen 1 (X) for the full subcategory of A-mod consisting of modules Z fitting into an exact sequence such that X i ∈ add X and is exact. We define cogen(X) and cogen 1 (X) dually.

Moreover, the image of the intermediate extension c = im(ℓ → r) is given by
im c = ker p ∩ ker q = gen(P ) ∩ cogen(Q).
The main conclusion of the algebraic part of the paper is the following theorem, which we will prove at the end of Section 3. To prove this theorem, we will give a different description of the algebra B in each part, as the endomorphism algebra of a bounded complex of A-modules in the homotopy category (cf. [29, §4.3]).

Endomorphism rings in the homotopy category
3.1. Homotopy categories and derived equivalence. We begin by repeating some general principals from [29, §4.3]. Let A be a finite-dimensional algebra, E ∈ A-mod, and Γ = End A (E) op . The bounded homotopy categories H b (Γ-proj) and H b (Γ-inj) of complexes of projective and injective Γ modules respectively admit tautological functors to D b (Γ), equivalences onto their images, which we treat as identifications. These subcategories may be characterised intrinsically as the full subcategories of D b (Γ) on the compact and cocompact objects (in the context of additive categories) respectively. Extending the Yoneda equivalences to complexes, one sees that both of these subcategories of D b (Γ) are equivalent to the full subcategory thick(E) of H b (A), i.e. the smallest triangulated subcategory of the homotopy category H b (A) closed under direct summands and containing (the stalk complex) E.
be any equivalence of triangulated categories. It follows from the intrinsic description of H b (Γ-proj) and H b (Γ-inj) above that F induces respective equivalences from the subcategories of compact and cocompact objects of T to these subcategories of D b (Γ), and thus allows us to realise thick E as a full subcategory of T (in two ways). This holds in particular when T = D b (B) for some algebra B, such as the endomorphism algebra of a tilting or cotilting Γ-module.
Whenever B is derived equivalent to Γ, it follows from Rickard's Morita theory for derived categories [33] that the image in H b (Γ-proj) of the stalk complex B ∈ H b (B-proj) is a tilting complex with endomorphism algebra B, inducing the derived equivalence. The preimage of this tilting complex under the Yoneda equivalence is an object of thick E ⊆ H b (A), again with endomorphism algebra B. Similarly, the image of DB ∈ H b (B-inj) in H b (Γ-inj) is a cotilting complex, and its preimage under the dual Yoneda equivalence is another object of thick E with endomorphism algebra B. Our conclusion is that when Γ is the endomorphism algebra of an A-module E (or more generally an object E ∈ H b (A)), any algebra B derived equivalent to Γ must also appear as an endomorphism algebra in thick E ⊆ H b (A). In general, B need not be an endomorphism algebra in A-mod.
In the context of Theorem 2.12, we may compute the relevant objects of thick E explicitly. This calculation generalises [14,Prop. 5.5] for the case that A is representation-finite and add E = A-mod, a connection that we will expand on in the next subsection.
Proposition 3.1. Let A be a finite-dimensional basic algebra and let E ∈ A-mod be a basic module.
(1) Assume E is a cogenerator, and write P = DE (see Lemma 2.7(2)). Then The special idempotent e ∈ B P given by projection onto P corresponds under this isomorphism to projection onto the summand 0 → DA. (2) Assume E is a generator, and write Q = DE (see Lemma 2.7(a)). Then where g : P (E) → E is a minimal projective cover. The special idempotent e ∈ B Q given by projection onto Q corresponds under this isomorphism to projection onto the summand A → 0.
Proof. As usual, we prove only (1), since (2) is dual. By definition B P is the endomorphism algebra of the unique basic P -special tilting Γ-module T P , so that the image of B P in K b (proj Γ) is given by a projective resolution of T P . By the proof of Lemma 2.7, we have T P = T 1 ⊕ DE and an exact sequence Thus a projective resolution of T P is given by the direct sum of the map Γ → P 0 above with the zero map 0 → DE, treated as a 2-term complex. Taking the preimage of this complex under the Yoneda equivalence Hom A (E, −) yields and the required isomorphism follows. Since 0 → DE, corresponding under Yoneda to 0 → DA, is the part of the projective resolution of T P contributed by the summand P , we have the claimed relationship between idempotents.
Remark 3.2. We did not specify degrees in the complexes on the right-hand side of the isomorphisms of Proposition 3.1, since such a choice plays no role in the statement. When such concreteness is required, we take the term E to be in degree 0 in each case. The assumptions on basicness of A and E and minimality of the relevant projective cover and injective envelope are necessary since B I and B P are basic algebras by construction. However, one can remove all these assumptions from the statement at the cost of replacing the isomorphisms by Morita equivalences.
3.2. The category of injective envelopes. In [14, §3], the category H of 'projective quotients' of an algebra A is defined, in order to give a useful description of the recollement arising from the special idempotent of the generator-cotilted algebra of an additive generator E of A-mod, when A is representation-finite, so that such a generator exists. We will briefly recall this construction, and generalise some of the results to our setting, in Section 3.3 below. However, since in the geometric applications to follow we have opted to use cogenerator-tilted algebras instead, we give more details for this case, in which we require instead the dual notion of a categoryȞ of injective envelopes. We defineȞ via the following construction, dual to that in [14, §3]. Let Q be the category with objects the monomorphisms X → Q of A-modules for which Q is injective, and morphisms given by commuting squares. Then H is obtained from Q as the quotient by those morphisms factoring through an object of the form id Q : Q → Q for Q injective.
We may view this category in a different way; first we identify Q with a full subcategory of the category C b (A) of bounded chain complexes of A-modules, by interpreting the objects X → Q of Q as complexes with X in degree 0. It is then straightforward to check that a map between such complexes factors through a complex of the form id Q : Q → Q if and only if it is null-homotopic, so thatȞ is identified with the full subcategory of K b (A) on the same objects as Q.
Now consider case (1) from Proposition 3.1, so that E ∈ A-mod is a cogenerator, and write so that the cogenerator-tilted algebra B of E satisfies for P = DE. Since f is an injective envelope, Q E ∈Ȟ (under our convention that the E term is in degree 0). We writeȞ E = add Q E for the additive closure of Q E inȞ, or equivalently in K b (A). By We are now able to give an explicit description and several properties of the functor c, by exploiting similar calculations for c in [14]. for (X → Q) ∈Ȟ E .
Proof. Since e = e E e, it follows by uniqueness of adjoints that ℓ = ℓ E ℓ and r = r E r. Post-composing with e E , we see that ℓ = e E ℓ, r = r E r. By definition of c, there is an epimorphism π : ℓ → c and a monomorphism ι : c → r, with ιπ equal to the canonical map ℓ → r. We can obtain the canonical map ℓ → r by precomposing all functors with the exact functor e E ; the induced map ℓ → e E c is again an epimorphism, and e E c → r is again a monomorphism. It follows that c = e E c is given by restriction, as claimed. Dualising By Proposition 3.1, there is an equivalence of categoriesȞ E -mod → B-mod, given by evaluation on the additive generator Q E ofȞ E , which we will often treat as an identification. Using this identification, we may now prove Theorem 2.12(1).
Proof of Theorem 2.12 (1). We first show that c(DA) ∼ = D(eB) for e the special idempotent. The identity map DA → DA is an injective envelope, and is isomorphic to the zero object inȞ E . Thus, after evaluating on Q E to identifyȞ E -mod with B-mod, the injective resolution of c(DA) from Theorem 3.4 provides an isomorphism Recall that B = End Γ (T P ) op , where Γ = End A (E) op and P is the projective Γ-module DE. By Remark 2.6, the canonical cotilting B-module DT P is the unique basic D(eB)-special cotilting module. Thus to show that c(E) = DT P , it is enough to show that it is such a module.
Since 3.3. The category of projective quotients. All of the results of the previous section have dual analogues, leading to a proof of Theorem 2.12 (2). We merely state the dual results, which correspond more closely to those of [14]. Let H be the category with objects given by surjective maps P → X of A-modules for which P is projective, and morphisms by commuting squares modulo maps factoring through an object of the form id P : P → P . Just as forȞ, we may view H as a full subcategory of K b (A) on the maps P → X, thought of as 2-term complexes with X in degree 0. Assume E ∈ A-mod is a generator, and write Γ = End A (E) op . Writing B for the generator-cotilted algebra of E, and where g : P (E) → E is a minimal projective cover, Proposition 3.1 shows that H E -mod and B-mod are equivalent categories via evaluating functors in H E -mod on the above additive generator. This identifies the restriction functor e : B-mod → A-mod, induced from the special idempotent, with the restriction of functors in H E -mod to add(A → 0). In particular, pd c(M ) ≤ 1. Furthermore, Ext 1 H E -mod (c(M ), c(N )) = 0 for any N ∈ A-mod. Theorem 2.12(2) then follows from Proposition 3.1 and Theorem 3.6 via a dual argument to that given above for part (1).

Rank varieties and orbit closures
4.1. Rank varieties as affine quotient varieties. We now turn to the geometric part of the paper. Let A be as before a finite-dimensional algebra over a field K, now assumed to be algebraically closed. In this section we also assume that K has characteristic zero for compatibility with [14], and that A = KQ/I for a finite quiver Q and an admissible ideal I. For d ∈ Z n ≥0 , we denote by the representation space of Q, each point of which defines a d-dimensional KQ-representation in the usual way. The representation space of A is then the closed subvariety of collections of maps satisfying the relations in I. This space carries a natural action of the algebraic group Gl d := n i=1 Gl d i , with orbits corresponding to isomorphism classes of d-dimensional A-modules. are determined by dim M when P is projective and Q is injective, and τ gives a bijection between non-projective indecomposables and non-injective indecomposables, we have C τ E M = D E M . It follows in the same way that C M = D M ; we call this space the rank variety of M , and typically opt for the notation C M . This variety has been studied by Bongartz [10] and (as a scheme) by Riedtmann-Zwara [34], among others. in these cases. Thus, when discussing C E M or C E m we may always assume that E is a cogenerator, and when discussing D E M or D E m we may always assume that E is a generator, without any loss of generality.
Moreover, if C M = C E M , we even have C M = C E⊕X M for any X ∈ A-mod. Thus in this case we are able to assume without loss of generality that the module E in question has any property that may be acquired by taking the direct sum with another module, such as being generating, cogenerating, or satisfying gldim End A (E) op < ∞ By Hilbert's basis theorem, for any M there exist modules E and E ′ such that C M = C E M = D E ′ M , although E and E ′ are neither explicitly nor uniquely determined. As a result, it is rarely clear how to find such modules, an obvious exception being when A is representation-finite, in which case both can be taken to be additive generators of A-mod.   [14], the first step in constructing a desingularisation of C M for an A-module M is to realise it as an affine quotient of some variety of representations for another algebra B. In fact, we may do this for any of the varieties C E m or D E m . We begin with C E m . Assuming without loss of generality that E is a basic cogenerator (see Remark 4.2), we may decompose E = t j=1 E j , with E j = D(e j A) indecomposable injective for 1 ≤ j ≤ n, and indecomposable non-injective otherwise.
Let B be the cogenerator-tilted algebra of E, and let e be its special idempotent. Since A ∼ = eBe, we may choose a complete set of primitive orthogonal idempotents of B extending that of A, and thus write the dimension vector of a B-module X as (d, s) ∈ Z n × Z t−n , where d = dim eX and s = dim(1 − e)X. We number the components of d from 1 to n, and those of s from n + 1 to t; this is compatible with our numbering of the indecomposable summands of E (cf. Proposition 3.1(1)). We have Gl (d,s) = Gl d × Gl s , so it makes sense to consider only the Gl s action on R B (d, s). Now the restriction functor e provides a map and by [14,Lem. 6.3] [14,Lem. 7.2]. For any injective A-module Q and any dimension vector d, let

an induced isomorphism of varieties
where N ∈ R A (d) is arbitrary, noting that [N, Q] depends only on dim N = d by injectivity of Q. Since this quantity also only depends on Q up to isomorphism, for any X ∈ A-mod we get a well-defined integer [d, Q(X)], where X → Q(X) is a minimal injective envelope. We may now state the main result of this subsection. Proposition 4.6. Let E be a cogenerating A-module, with indecomposable summands labelled as in the preceding paragraph, and B its cogenerator-tilted algebra. Let d ∈ Z n ≥0 be a dimension vector for A, and let m ∈ Z t ≥0 .
In this case, we may extend d to a dimension vector (d, s) ∈ Z t ≥0 for B by defining s j := [d, Q(E j )] − m j for n + 1 ≤ j ≤ t, and the special idempotent e of B induces an isomorphism On the other hand, if n + 1 ≤ j ≤ t then, by Lemma 3.3, we have dim c(N ) = (d, s ′ ), where so it is enough to prove that C E m coincides with the codomain. We have [N, E j ] = d j ≥ m j for any N ∈ R d (A) if 1 ≤ j ≤ n, and so by using the calculation of dim c(N ) from above, we see that dim c(N ) ≤ (d, s) if and only if [N, E j ] ≥ m j for n + 1 ≤ j ≤ t.
We now state the dual result for D E m , which may be proved similarly. In this case we may assume E is a basic generator, and decompose E = t j=1 E j so that E j = e j A is indecomposable projective for 1 ≤ j ≤ n, and E j is indecomposable non-injective otherwise. Let B be the generator-cotilted algebra of E, and e its special idempotent, so that again we have A ∼ = eBe. Dual to the earlier statement for injective envelopes, we get a well defined integer [P (X), d] for any X ∈ A-mod and any dimension vector d, by taking P (X) → X to be a minimal projective cover. The dual of Proposition 4.6 is then the following. As remarked earlier, Hilbert's basis theorem allows us to apply Propositions 4.6 and 4.7 to the rank variety C M of a module M , by expressing it either as C E M for some cogenerator E, or as D E ′ M for some generator E ′ .

4.2.
Desingularisation of orbit closures for gen-finite modules. As above, let A be a finitedimensional algebra, and let B be any finite-dimensional algebra possessing an idempotent e with A ∼ = eBe. Recall that this data induces a recollement B/BeB-mod B-mod A-mod i e q p ℓ r analogous to (2.2). We call the B-modules in cogen(D(eB)) stable, and those in gen(Be) costable; note that the category of stable modules is closed under taking submodules, and the category of costable modules is closed under taking quotients. By Lemma 2.11, the image of the intermediate extension functor c corresponding to e is the category of modules which are both stable and costable. For any subset Z ⊆ R B (d, s) we write Z st for the set of stable modules in Z, and Z cst for the set of costable modules in Z. In be the short exact sequence obtained by splitting off a maximal direct summand of the form 0 0 Z ′′ Z ′′ 0 id from (4.1); we claim that this is our desired sequence. To see this, observe that the morphism γ ∈ End Λ (Z ′ ) is nilpotent by Fitting's lemma, and by induction on i one may prove that The main step in our argument is the following theorem, characterising the stable (d, s)-dimensional B-modules in O c(M ) and giving a sufficient condition for them to be smooth points of this variety. We are now ready to describe our desingularisation for the orbit closure of a gen-finite module.

Desingularisation of quiver Grassmannians
Let A = KQ/I for K an algebraically closed field. As before, it is convenient for us to write Q 0 = {1, . . . , n}. Let M be a finite-dimensional A-module and d ∈ Z n ≥0 a dimension vector. We denote by Gr A M d the quiver Grassmannian of d-dimensional submodules of M . The majority of this section is to proving the following theorem, from which we obtain smooth varieties to use in constructing our desingularisations. Before proving this theorem, we require some preparation. . Then a Λ-module is given by a B-linear morphism between two B-modules, so X defines a Λmodule X = (X 1 X − − → X), and each U ∈ Gr B X d defines a Λ-module U = (U → X), which has a natural inclusion U → X such that X/ U = (X/U → 0). We first prove that Ext 2 Λ ( U , U ) = 0.
Applying Hom Λ ( U , −) to the short exact sequence where the last space is zero because id Λ X = id B X ≤ 1. We now claim that Ext 1 Λ ( U , X/ U ) ∼ = Ext 1 B (U, X/U ) = 0. To see this, chose an exact sequence with Q injective. This induces an exact sequence of Λ-modules, to which we apply Hom Λ ( U , −) to obtain using that (Q → 0) is injective. We can also apply Hom B (U, −) to the exact sequence (5.2) to obtain The first three terms of the preceding four-term exact sequences are isomorphic, therefore also the fourth, i.e. Ext 1 Λ ( U , X/ U ) ∼ = Ext 1 B (U, X/U ), the latter being zero by assumption. Looking back at (5.1), we see that Ext Since H is smooth, H is smooth.
By definition we have a projective Gl s -equivariant map π : H → O X mapping (Z ⊂ Y ) to Y . The fibre over X is π −1 (X) = Gr B X d . Consider the multiplication map Gl s × π −1 (X) → H given by (g, U ⊂ X) → (gU ⊂ gX). The composition of this map with π induces surjective maps on tangent spaces, since the map Gl s → O X by g → gX has this property, and hence so does π. Thus we have a short exact sequence and from equality of the outer terms we conclude that π −1 (X) is smooth.
Lemma 5.3. Let B be a finite-dimensional basic algebra and X ∈ B-mod. Assume C ∈ B-mod has the properties that Ext i B (C, C) = 0 for all i > 0 and X ∈ add C. Then if U ∈ Gr B X d fits into a short exact sequence 0 (i) Applying Hom B (−, C) to the short exact sequence we see that Ext i B (U, C) for i ≥ 1. (ii) It then follows that Ext i B (U, U ) = 0 for i ≥ 2 by applying Hom B (U, −) to the same exact sequence and using (i). Now apply Hom B (U, −) to the short exact sequence for each i > 0. Since X ∈ add C, we have Ext i B (U, X) = 0 by (i), and Ext i+1 B (U, U ) = 0 by (ii). Thus we conclude Ext i B (U, X/U ) = 0. We are now ready to prove Theorem 5.1. d,s . We claim that there is an exact sequence 0 U C 0 C 1 0 with C 0 , C 1 ∈ add C, so that the result follows by Lemma 5.3. We write E = N ⊕ Q with add N = gen(M ) and Q injective, and let f : U → C 0 be a left add(c(N ))approximation of U . Since U ⊆ c(M ) with c(M ) ∈ add c(N ), and f must factor over this inclusion, f is a monomorphism. We complete it to the short exact sequence in which C 0 ∈ add C by construction, and C 1 ∈ add C as we now show. By applying Hom B (−, c(N )) to the short exact sequence and using that f is an add(c(N ))-approximation and c(N ) is rigid, we see that Ext 1 B (C 1 , c(N )) = 0. Since c(Q) is injective by Theorem 2.12 we even have Ext 1 B (C 1 , C) = 0. As C is a cotilting module, this means C 1 ∈ cogen(C), but we also have C 1 ∈ gen(C) by the short exact sequence.
We claim that gen(C) ∩ cogen(C) ⊆ im c. To see this, note that whenever we have c(X) p − → Y i − → c(Z) with p an epimorphism and i a monomorphism, it follows from the fully-faithfulness of c that ip = c(g) for some morphism g : X → Z. But c preserves epimorphisms and monomorphisms, so Y ∼ = im c(g) ∼ = c(im g) ∈ im c.
Recalling that C = c(E), we conclude that C 1 = ceC 1 ∈ add C, completing the proof.

Gen-finite modules
Our constructions above, both for orbit closures and quiver Grassmannians, involve the assumption of gen-finiteness of a module. In this section we recall some basic facts about gen-finite modules, including methods for easily constructing examples. Recall from Definition 4.4 that an A-module M is gen-finite if there exists X ∈ A-mod with gen(M ) = add X, and cogen-finite if there exists X ∈ A-mod with cogen(M ) = add X. Lemma 6.1. If A = KQ for an acyclic quiver Q and M is a gen-finite A-module, then τ M and M ⊕ DA are also gen-finite A-modules.
We observe that gen(τ M ) ⊆ τ (τ − gen(τ M )) ⊕ add DA because every module N can be decomposed as N = τ τ − N ⊕ Q for some injective module Q. Putting these two observations together we get Thus if M is gen-finite, then τ M is also gen-finite. Now we prove the second claim. Decompose Z ∈ gen(M ⊕ DA) into Q ⊕ Z ′ with Q ∈ add DA maximal, so that Z ′ has no injective summands. Since A = KQ is hereditary, it follows that Hom A (DA, Z ′ ) = 0, and so Z ′ must be in gen(M ). Thus if M is gen-finite, M ⊕ DA is also genfinite.
As a corollary, we obtain the following well-known result.
Corollary 6.2. If A = KQ for an acyclic quiver Q, then every preinjective module is gen-finite. Also, τ j S is gen-finite for every semi-simple module S and every j ≥ 0.
More generally, we recall the following definition [35]. Thus we may assume in (2) that M is faithful, so ann(M/IM ) = I and we are in a special case of (1). Finally, (3) follows from Lemma 2.11 since eAe-mod is equivalent to gen 1 (P ) ⊆ gen(P ) via the left adjoint ℓ to e. If rad n (A) = 0 and A/ rad n−1 (A) is representation-finite, then A is torsionless-finite [37, Special cases (1)]. In particular, if rad 2 (A) = 0 then A is torsionless-finite. 7. Example 7.1. A module for the n-subspace quiver. Let A be the path algebra of the n-subspace quiver: 1 2 · · · n − 1 n 0 When treating an A-module X as a representation of this quiver, we denote by X i the linear map carried by the arrow i → 0. 7.2. The cogenerator-tilted algebra. For our constructions, we choose E = M , noting that M is a cogenerator, and add M = gen(M ). Then Γ = End A (E) op ∼ = KQ Γ / rad 2 (Q Γ ) for Q Γ the quiver Here each vertex i ′ corresponds to the summand Q(i) of M , noting that Q(i) = S(i) for i ≥ 1, and corresponds to S(0). The projective Γ-module P = Hom A (E, DA) = n i=0 P (i ′ ) is faithful. Since [S( ), P (i ′ )] = 0 for 1 ≤ i ≤ n, a minimal left add P -approximation of S( ) is given by a monomorphism S( ) → P (0 ′ ), with cokernel S(0 ′ ). This implies T P = P ⊕ S(0 ′ ) is the P -special tilting Γ-module. Then we calculate that the cogenerator-tilted algebra B = End Γ (T P ) op of E is isomorphic to the path algebra KQ B for Q B the quiver [1] [2] · · · [n − 1] where, under this identification, U is the image of the monomorphism on the arrow ♦ → [0]. We can check that X is smooth and irreducible by considering the projection pr 2 : X → P 1 . The fibre over [1 : 0] consists of all tuples (N 1 , . . . , N n ) ∈ Mat 2×2 (K) n such that each N i has lower row zero, and hence this fibre is an affine space (of dimension 2n). In fact, it is a B-representation where B ⊆ Gl 2 (K) denotes the upper triangular matrices operating by conjugation. Since pr 2 is a Gl 2equivariant map into the homogeneous space P 1 , it follows that X is is a vector bundle over P 1 , and so is smooth and irreducible. In particular, since c(M ) is rigid, X = O