On quiver Grassmannians and orbit closures for representation-finite algebras

We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective-injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke.


Introduction
Starting from a finite-dimensional module M over a finite-dimensional associative algebra A one can obtain an algebraic variety either by forming the so-called quiver Grassmannian Gr A M d parameterizing submodules of M of dimension vector d, or by taking the orbit closure O M associated to M in the representation variety parameterizing all A-modules of the same dimension vector as M . In the situation where A is the path algebra of a Dynkin quiver, Cerulli Irelli, Feigin and Reineke constructed desingularizations of quiver Grassmannians [CIFR13a] and realized orbit closures as affine quotient varieties [CIFR13b]. One would like to generalize these constructions to other algebras, and Keller and Scherotzke [KS14] and Scherotzke [Sch15] obtained some results in the cases when A is an iterated tilted algebra of Dynkin type or a self-injective algebra of finite representation type. In this paper we generalize to the case when A is an arbitrary finite-dimensional algebra of finite representation type. We again construct desingularizations of quiver Grassmannians and realize orbit closures as affine quotient varieties. For the latter, our construction unifies the work of Cerulli Irelli, Feigin and Reineke with a construction of closures of conjugacy classes used by Kraft and Procesi [KP79] to prove their normality. In addition, we construct desingularizations of orbit closures.
In order to study a Dynkin quiver Q, a certain algebra (denotedΛ Q or B Q ) has been introduced independently by Hernandez and Leclerc [HL15] and by Cerulli Irelli, Feigin and Reineke. We generalize this to an algebra B = B A , which we call the projective quotient algebra, associated to any finite-dimensional algebra A of finite representation type. We begin by discussing this algebra. We write A-Mod for the category of (left) A-modules and A-mod for the category of finitedimensional modules. Let Q be the category whose objects are surjective morphisms f : P → X in A-mod with P projective, and whose morphisms from f : P → X to f ′ : P ′ → X ′ are given by pairs (t, s) ∈ Hom A (P, P ′ ) × Hom A (X, X ′ ) such that f ′ t = sf . We can consider Q as a category of complexes with two terms, and define H to be the corresponding homotopy category. Assuming that A has finite representation type, the category H has only finitely many indecomposable objects, and we set G to be the direct sum of them. Thus G is an additive generator for H. The projective quotient algebra for A is defined to be B = End H (G) op .
There is another characterization of the projective quotient algebra which shows how natural it is. Recall that if A has finite representation type, its Auslander algebra is End A (E) op , where E is the direct sum of one copy of each indecomposable A-module. Auslander [Aus74] characterized the algebras which arise this way as those of global dimension ≤ 2 and dominant dimension ≥ 2. In the same spirit we have the following. (All tilting and cotilting modules are assumed to be classical, so with projective dimension ≤ 1 and injective dimension ≤ 1 respectively.)  We use this result together with standard properties of intermediate extension functors to obtain our geometric results. Henceforth let K be an algebraically closed field. Still A is a basic algebra of finite representation type, and let e 1 , . . . , e n be a complete set of primitive orthogonal idempotents in A. The dimension vector of an A-module N is d = (dim e i N ) ∈ N n 0 . Let e 1 , . . . , e m be a complete set of orthogonal idempotents in B, with the first n ≤ m being the corresponding idempotents for A. Dimension vectors for B are given by pairs (d, r) with d ∈ N n 0 and r ∈ N m−n 0 . Let M be an A-module. Recall that Gr A M d is a projective variety, possibly with singularities. If N is an A-module of dimension d, then there is a map of varieties from the set of injective maps in Hom A (N, M ) to the Grassmannian, whose image is the set of submodules isomorphic to N . We denote it S [N ] ; it is locally closed and if non-empty it is irreducible. Note that Gr A M d may have several irreducible components, but using that A has finite representation type, they are of the form S [N i ] for some A-modules N 1 , . . . , N ℓ . We consider S [c(N i )] ⊆ Gr B c(M ) d,r i , where (d, r i ) is the dimension vector of c(N i ). A desingularization of a variety X is a birational projective morphism from a smooth variety to X. Our first geometric application is the following. Modules of any given dimension vector d ∈ N n 0 are parameterized by a variety R A (d) and the group Gl d acts with the orbits being the isomorphism classes. We write O M for the orbit corresponding to a module M . Our second geometric application is as follows.
Theorem 1.5. If K has characteristic zero and M is an A-module, then O M is isomorphic to the affine quotient variety R B (d, r) / / Gl r where (d, r) is the dimension vector of c(M ), and also to O c(M ) / / Gl r .
Associated to the recollement there is a stability notion for B-modules. We denote by R B (d, r) s the open subset of stable modules in R B (d, r) and by R B (d, r) s /Gl r the corresponding geometric quotient (the GIT moduli space of stable B-module). Using this we obtain another desingularization.
Theorem 1.6. If K has characteristic zero and M is an A-module, then the natural map Part of this work was completed while the first author was visiting CRC 701 at Universität Bielefeld. The author thanks his hosts for their hospitality.

Recollements
Recall (see [BBD82]) that a recollement of abelian categories is a diagram consisting of of three abelian categories and six functors satisfying the following conditions.
(3) The natural transformations P : id A → pi and Q : qi → id A are isomorphisms.
(4) The functor i is an embedding onto the full subcategory of B with objects b such that eb = 0. The condition (1) implies that i and e are exact,and the conditions (2) and (3) are equivalent to the fully faithfulness of ℓ, r and i.
(2) Let N be in B with e(N ) ∼ = M . Then c(M ) is a subquotient of N . More precisely, from the above diagram we get short exact sequences with e(ker β N ) = 0 = e(coker(j)). (5) If B is a category with sums and products in which the natural map α N α → α N α is monic for all indexed sets of objects {N α }, then c commutes with direct sums.
(6) The functor c maps simples in C to simples in B. There is a bijection between sets of isomorphisms classes of simples giving by sending a simple N in A to i(N ) and a simple M in C to c(M ). (7) There are short exact sequences of natural transformations The next lemma is essentially in [FP04].
where the e in the lower row is the adjunction isomorphism. Since p M is an epimorphism, the lower row is an injective linear map. The diagram commutes since e(p M ) is the identity. Analoguously, in (2) the morphism identifies with Hom B (F, c(M )) where e stands for the adjunction isomorphism. Since i M is a monomorphism, the lower line is an injective linear map. For the fully faithfulness, the following map is the identity ) is injective and the second map Hom B (c(M ), c(N )) → Hom C (M, N ) is surjective. Now, the second map is injective by the previous part of the proof, therefore the second map is an isomorphism. It follows that also Hom C (M, N ) → Hom B (c(M ), c(N )) is an isomorphism. Since c is fully faithful, it preserves indecomposables.
Stable and costable objects. Associated to a recollement there are the notions of stable and costable objects (see [KS13, §2.6, §4.8] for the case of Kan extensions). We consider a recollement as above with intermediate extension functor c. We characterize stable and costable objects in the following two lemmas. For an additive functor f : S → T we denote by ker f the full subcategory in S whose objects are send to zero under f . Lemma/Definition 2.3. We say that an object F in the middle category B is stable (or e-stable) if one of the following equivalent conditions is fulfilled (1) Hom B (G, F ) = 0 for every object G with e(G) = 0.
(2) The natural map F → re(F ) is a monomorphism.
(4) F ∈ ker p. From (3) it follows that every subobject of a stable object is stable. Furthermore, if F is stable, then there is a (natural) monomorphism ce(F ) → F (compare Lemma 2.1 part (2)).
Lemma/Definition 2.4. We say that an object H in the middle category B is costable (or e-costable) if one of the following equivalent conditions is fulfilled (1) Hom B (H, G) = 0 for every G with e(G) = 0.
Proof Definition 2.5. We say F is bistable if it is stable and costable.
It follows directly from Lemma 2.1 that F is bistable if and only if F ∼ = ce(F ). Now, every recollement of abelian categories with middle term B is determined by its associated TTF-triple, which is a triple (X , Y, Z) of subcategories of B such that (X , Y) and (Y, Z) are torsion pairs, compare e.g. [PV14]. For our given recollement, the TTF-triple is (ker q, ker e, ker p) = (costables, ker e, stables).
Functor categories. We are interested in Krull-Schmidt categories, by which we mean a small additive K-category R with finite-dimensional Hom sets and split idempotents. We denote by R the category of K-linear contravariant functors R → K-Mod. If X is an object in R, then the representable functor Hom R (−, X) is a projective object of R, and projective objects isomorphic to one of this form are said to be finitely generated. The functor D Hom R (X, −) is an injective object of R, and injective objects of this form are said to be finitely cogenerated.
Any K-linear functor f : S → R of Krull-Schmidt categories induces a restriction functor R → S. Using tensor products over categories, and the usual hom-tensor adjointness, one obtains left and right adjoints ℓ, r : S → R given by Recall that if S is a full Krull-Schmidt subcategory of R, then R/S denotes the quotient category whose objects are the same as in R and with morphisms given by the quotient vector space where I S (X, X ′ ) is the vector subspace with elements the morphisms X → X ′ which factor through an object in S. In this situation we have a recollement of the following form.
Lemma 2.6. For R a Krull-Schmidt category and S a full additive subcategory of R, there is a recollement where e is the restriction functor given by the inclusion S ⊂ R and i is given by composition with the natural functor R → R/S.

The categories Q and H and two auxiliary recollements
Let A be a basic finite-dimensional K-algebra and let e 1 , . . . , e n be a complete set of primitive orthogonal idempotents in A. Since A is basic, the modules P i = Ae i are a complete set of non-isomorphic indecomposable projective A-modules and their tops S i are a complete set of non-isomorphic simple A-modules.
The category Q of quotients of projectives and a first auxiliary recollement. Let Q be the category defined as in the introduction. Let E be the full subcategory of Q consisting of the objects 1 : P → P , and let P be the full subcategory of Q consisting of the objects 0 : P → 0. The following is clear.
Lemma 3.1. The category Q is Krull-Schmidt and its indecomposable objects are of the following three types up to isomorphism (1) a projective cover f U : (2) indecomposables in E, so of the form 1 : Remark 3.2. Clearly Q is a full subcategory of the category T whose objects are the morphisms f : Y → X in A-mod and whose morphisms are given by commutative diagrams. Now T is abelian, indeed it is equivalent to the category of finite-dimensional modules for the algebra of 2 × 2 upper triangular matrices with entries in A. It is not difficult to see that Q is functorially finite and extensionclosed in T . Thus it is an exact category and, by [AS81, Theorem 2.4], the category Q has Auslander-Reiten sequences. It is easy to see that the indecomposable Ext-projectives are the objects of the form 1 : P i → P i and 0 : P i → 0, and the indecomposable Ext-injectives are the objects of the form Considering P as a full subcategory of Q, the first auxiliary recollement we consider is The functor e ′ is given by e ′ (F ) = F (A → 0) with its induced A-module structure.
Proof. For the moment let ℓ ′ , r ′ and c ′ be defined by the formulas in the statement of the theorem. Since P is finitely generated and projective, ℓ ′ is right exact and commutes with direct sums. Thus, to show it agrees with the left adjoint, it suffices to check this on the free module A, that is, Since e ′ is right exact and commutes with direct sums, by considering a projective presentation of F ∈ Q by direct sums of representable functors, in order to prove that r ′ is right adjoint to e ′ it suffices to prove that Hom Q (R, r ′ (M )) ∼ = Hom A (e ′ (R), M ) for any representable functor R = Hom Q (−, (f : P → X)). This is clear since on the left hand side Yoneda's Lemma gives and on the right hand side we have e ′ (R) = Hom Q ((A → 0), (f : P → X)) ∼ = ker f . Now the isomorphism gives the result for c ′ .
The homotopy category H and a second auxiliary recollement. We define H = Q/E. It is easy to see that a morphism (θ, φ) from the object f : P → X to the object f ′ : P ′ → X ′ in Q factors through E if and only if there is a map h : X → P ′ with θ = hf and φ = f ′ h. Thus, if one considers Q as a category of complexes with two terms, then H is the corresponding homotopy category. The category H is Krull-Schmidt and up to isomorphism its indecomposable objects are those of types (1) and (3) in Lemma 3.1. Associated to the quotient H = Q/E there is a second auxiliary recollement Lemma 3.4. The functor q 0 : Q → H sends a representable functor Hom Q (−, (P → X)) to the representable functor Hom H (−, (P → X)).
giving the claim.
Proof. Say (θ, φ) is a morphism from f : P → X to f ′ : P ′ → X ′ , and let r : P ′ → P be a retraction for θ, so rθ = 1 P . Take a morphism in H from an object f ′′ : P ′′ → X ′′ to f : P → X, and let it be represented by a morphism ( defines a map from f : P ′′ → X ′′ to 1 : P → P , and its composition with the map (1, f ) from 1 : is also a morphism from f ′′ : P ′′ → X ′′ to f : P → X, and since these two maps have the same first component, and f ′′ is surjective, they must be equal, This shows that (θ ′ , φ ′ ) is the zero map in H, as required.
Lemma 3.6. If G is a functor in H and i 0 (G) is finitely presented in Q, then proj. dim Q i 0 (G) ≤ 2. Moreover G is finitely presented in H and proj. dim H G ≤ 2.
Proof. By assumption there is a projective presentation for some morphism (θ ′ , φ ′ ) from the object f ′ : P ′ → X ′ to the object f ′′ : P ′′ → X ′′ in Q. Now the functor i 0 (G) vanishes on every object of E, so in particular on 1 : P ′′ → P ′′ , so the map is onto. Thus (1, f ′′ ) lifts to a map from 1 : P ′′ → P ′′ to f ′ : P ′ → X ′ . In particular θ ′ is a split epimorphism. Thus φ ′ is also an epimorphism, and we obtain the diagram Now the map f : P → X need not be onto, but factorizing it as f 0 : P → Im(f ) followed by the inclusion, we get an exact sequence Applying q 0 one obtains a sequence of functors Since q 0 is a left adjoint, this sequence is right exact. To show it is exact at the term Hom H (−, (f ′ : P ′ → X ′ )), we consider a morphism (α, β) from g : Q → M to f ′ : P ′ → X ′ . which is sent to zero in Hom H ((g : Q → M ), (f ′′ : P ′′ → X ′′ )). Thus there is a map h : M → P ′′ with hg = θ ′ α and f ′′ h = φ ′ β. Let s be a section for θ ′ and r a retraction for θ with 1 P ′ = θr + sθ ′ . Then α − shg = θα ′ for some α ′ : Q → P and β − f ′ sh = φβ ′ for some β ′ : M → X, which actually has image contained in Im(f ) since and since φ is a monomorphism and g is an epimorphism, we have β ′ = f r. Then, up to a morphism factoring through a projective, (α, β) comes from the map (α ′ , β ′ ) from g : Q → M to f 0 : P → Im(f ). Now, because the map from P to P ′ is a split monomorphism, the map from f 0 : P → Im(f ) to f ′ : P ′ → X ′ induces a monomorphism in H, and it follows that the sequence is exact on the left, too.

The main recollement
Observe that there are no non-zero maps from an object 0 : P → 0 in P to an object 1 : P ′ → P ′ in E, so the natural functor from P to H = Q/E is fully faithful, and hence P can be considered as a full subcategory of H. Again, identifying P ∼ = A-Mod, we have a recollement The identity map on the right hand side for F = ℓ(M ) gives a natural morphism ℓ ′ (M ) → i 0 ℓ(M ). Now for any G ∈ Q, by [FP04, Proposition 4.2] the natural map i 0 p 0 (G) → G is a monomorphism. Since e ′ is exact and e = e ′ i 0 , we deduce that for any A-module M , the map is a monomorphism. By adjointness we can rewrite this as giving the result.
Lemma 4.2. We have c ′ = i 0 c, so that c is given by the same formula as c ′ , that is, Proof. For an A-module M , recall that c ′ (M ) is the image of the map from ℓ ′ (M ) to r ′ (M ). Now, using that i 0 is exact, this factors as Proof. Let f : P → X be an object in Q. By diagram chasing one obtains an exact sequence This gives an exact sequence Applying q 0 and using that q 0 preserves representables by Lemma 3.4 It is right exact since q 0 is a left adjoint, and it is exact on the left by Lemma 3.5 about monomorphisms in H.
Rigidity. In general, we do not know when intermediate extension functors map all modules to rigid modules but in our situation, we can prove this statement, generalizing [CIFR13a, Theorem 5.6].
Using the fully faithfulness of c we can write it as , c(N )) → 0 and by the definition of c(N ) the middle map is which is tautologically surjective, giving the result.
We observe the following special property that c maps injectives to injectives and projectives to projectives. More precisely, one has  (2) It suffices to prove this for M a finitely generated projective module (or even for M = A). In this case it follows from the projective resolution of c(M ) since one can take Q = M and the first term is zero.
We remark that c ′ sends injectives to injectives, but it need not send projectives to projectives.
Injective dimension. The Nakayama functors ν and ν − for A-modules are defined by ν(M ) = D Hom A (M, A) and ν − (M ) = Hom A (DM, A). They define inverse equivalences between the category of finite-dimensional projective A-modules and the category of finite-dimensional injective A-modules. For any modules X and Y there is a functorially defined map which is an isomorphism if X is a finite-dimensional projective module.
Lemma 4.6. The following triangle is commutative for any finite-dimensional projective A-modules P and P ′ .
Proof. One can reduce to the case when P = P ′ = A. Proof. Choose a minimal injective copresentation 0 → M → I → J. Applying ν − and using the definition of the Auslander-Reiten translate τ − we get an exact sequence This gives an object ν − J → τ − M in Q and a morphism from For an object f : P → X in Q, consider the sequence Stable and costable functors. We want to characterize stable and costable objects for our main recollement.
Lemma 4.8. Let F in H. The following are equivalent.
(2) There is an injective A-module I such that F is a subfunctor of c(I).
(3) There is an A-module M such that F is a subfunctor of c(M ). Proof.
(3) implies (1) because submodules of stable modules are stable. Assume F fulfills (1). Let I be an injective hull of e(F ). We apply the left exact functor r and get a monomorphism re(F ) → r(I).
By Lemma 4.5 we have r(I) = c(I) and by Lemma 2.3, we have a monomorphism F → re(F ). Composition gives a monomorphism F → c(I) and (2) is fulfilled. Now, (2) implies clearly (3).
Lemma 4.9. Let H in H. The following are equivalent.
(2) There is a projective A-module P such that H is a quotient of c(P ).
(3) There is a A-module M such that H is a quotient of c(M ).
We remark the following. H). Proof. The first part is straightforward. Apply Hom H (−, F ) to the sequence to get a long exact sequence

The projective quotient algebra
Let A be a basic finite-dimensional algebra as in §3 and §4. We now suppose that A has finite representation type. In this case H has only finitely many indecomposable objects up to isomorphism, those of the form P U → U , where U is a non-projective indecomposable A-module and P U is a projective cover, and those of the form P i → 0. We denote by G the direct sum of all these indecomposable objects and define B = B A = End H (G) op . It is a finite-dimensional basic algebra, and H is equivalent to the category of finitely generated projective B-modules. Now the object A → 0 in H is the direct sum of the objects P i → 0, so it is is a summand of G, and e ∈ B is the projection of G onto this summand. Then eBe ∼ = End H (A → 0) op ∼ = End A (A) op ∼ = A. The recollement given at the start of this section can be identified with the one stated in the introduction to the paper, given by the idempotent e ∈ B.
Recall that E is the direct sum of all indecomposable A-modules and the Auslander algebra is Γ = End A (E) op . We define C = c(E). Since c is fully faithful, we have End B (C) op ∼ = Γ.
Lemma 5.1. The projective quotient algebra B has global dimension at most 2 and the module C is a tilting and cotilting module.
Proof. Any finite-dimensional B-module has projective dimension ≤ 2 by Lemma 3.6, giving the global dimension claim. The number of indecomposable summands of C is equal to the number of indecomposable A-modules, which is also the number of indecompsable objects in H, so the number of simple B-modules. Thus C is a tilting and cotilting module by Lemmas 4.3, 4.7 and Theorem 4.4. Proof. The assignment Hom A (E, X) → Hom(c(E), c(X)) induces a map h X : c(E) ⊗ Γ Hom A (E, X) → c(X) which is functorial in X. Now h E is an isomorphism, both sides commute with direct sums, and any A-module is a direct sum of summands of E, so h X is an isomorphism for all X. The assertion about extensions follows from Theorem 4.4 since every A-module is a direct sum of finite-dimensional A-modules.
The next lemma follows from Lemmas 4.8 and 4.9.
Lemma 5.3. The stable B-modules are those cogenerated by C and the costable modules are those generated by C.
We now recall some results from tilting theory (which apply to the projective quotient algebra). For the moment let B be an arbitrary finite dimensional algebra. A tilting and cotilting B-module C induces two torsion pairs on B-mod respectively (swapping torsion pairs and torsion and torsion-free classes). The first two equivalences hold since C is tilting, the second two since C is cotilting.
This leads to another characterization of projective quotient algebras. Let B again be the projective quotient algebra. Recall from the introduction that there is a unique basic tilting cotilting Γ-module T generated and cogenerated by projective-injectives.
Proof. By Lemma 1.1 it suffices to show that D C is generated and cogenerated by projective-injective left Γ-modules, or equivalently that C is generated and cogenerated by projective-injective right Γmodules. By the theory of Auslander algebras (or the Nakayama isomorphism Θ X,Y ), E = e(C) is a projective-injective right Γ-module. We look at the epimorphism p and monomorphism i These are maps of left B-modules, but they are functorial, so also maps of right Γ-modules. By definition of the right adjoint, re(C) = Hom A (e(B), e(C)) is a Γ-submodule of Hom K (e(B), e(C)) ∈ Add(e(C)) = Add(E). By definition of the left adjoint, ℓe(C) = Be ⊗ A e(C), which is a quotient of Be⊗ K e(C) ∈ Add(E) as a Γ-module. This proves that C is generated and cogenerated by Add(E).
The same argument gives another characterization of projective quotient algebras.
Theorem 5.6. If B is an arbitrary finite-dimensional algebra and C is a basic tilting and cotilting B-module satisfying the properties in Theorem 5.4, then B is a projective quotient algebra.
Proof. The condition on the kernels gives a strong TTF triple from which it follows that A is representation-finite and E = e(C) is an Auslander generator. This implies that Γ = End B (C) op = End A (E) op = End A (e(C)) op is the Auslander algebra of A. Now C is generated and cogenerated by projective-injective right Γ-modules as in the proof of Theorem 5.5. The result follows. Now, let Γ be the Auslander algebra and ǫ ∈ Γ = End A (E) op be the idempotent element corresponding to the projection on the direct sum of all projective A-modules (which is isomorphic to A, so ǫΓǫ ∼ = End A (A) op ∼ = A). We look at the map given by left multiplication with this idempotent, We remark that, ker ǫ is equivalent to (A-mod) ∧ , which is also equivalent to ker e ∼ = H/P in our main recollement. It means we can find two recollements with the same outer terms and modules over the two tilted algebras B and Γ in the middle. The next result shows that T appears naturally in the recollement given by the idempotent ǫ ∈ Γ.
Proposition 5.7. One has ker ǫ = ker Hom Γ (−, T ). In particular, we can find Add(T ) as the Extinjectives in the corresponding torsionfree class to ker ǫ.
Proof. Using the previous remark and the Brenner Butler theorem, we know that ker ǫ ∼ = ker e = ker Hom B (C, −) ∼ = ker Hom Γ (−, T ). Therefore, it is enough to prove one inclusion. So, let X be a Γ-module with Hom Γ (X, T ) = 0. Since T is a tilting module, it has as a direct summand every finite dimensional projective-injective module. So in particular, every I(a) with a being a simple Γ-module corresponding to a projective A-module is a summand of Γ. Therefore, Hom Γ (X, I(a)) = 0 for those a. But this implies dim a X = 0 for those a and therefore ǫX = 0.

Representation schemes and varieties
In this section we prove some general results about representation varieties and schemes for algebras. We prove a version of the First Fundamental Theorem for quivers, and then we apply it to finitedimensional algebras. Then we prove a criterion for smoothness in representation schemes. These results are used in later sections.
We begin by recalling the basic definitions. Let K be an algebraically closed field, let A be a finitely generated K-algebra and let e 1 , . . . , e n be a complete set of orthogonal idempotents in A. For d ∈ N n 0 , there is an affine scheme R A (d) of d-dimensional representations of A. For a commutative K-algebra R, its R-points are the K-algebra homomorphisms θ : corresponds to projection onto the summand K d i .
The K-points of R A (d) are K-algebra maps θ : A → M d (K), and the corresponding A-module is X = K d with the action of a ∈ A given by left multiplication by θ(a). The condition that θ(e i ) = ǫ i ensures that e i X = K d i under the identification K d = n i=1 K d i . We write R A (d) for the affine variety of d-dimensional representations of A. Its points are the K-points of R A (d), ignoring any non-reduced structure on the scheme. The algebraic group Gl d acts on by conjugation, and its orbits are the isomorphism classes of representations of A of dimension d, that is, A-modules X with dim e i X = α i for all i.
Suppose that K has characteristic zero. For an algebraic group G acting on an affine variety X, we denote by X / / G the categorical quotient, the variety with coordinate ring K[X] G , where K[X] is the coordinate ring of X.
First Fundamental Theorem for quivers with partial group action. Let Q B be a finite quiver with vertices 1, . . . , n, n + 1, . . . , m (n ≤ m) and let (d, r) ∈ N n 0 × N m−n 0 be a dimension vector for Q. Let Q >n be the full subquiver of Q B supported on the vertices n + 1, . . . , m. We call a non-trivial path in Q B primitive if starts and ends at a vertex in the range 1, . . . , n but does not otherwise pass through any such vertex. Let e := n i=1 e i , and observe that eKQ B e is isomorphic to the path algebra of a quiver Q ′ A with vertices 1, . . . , n whose arrows are the primitive paths. Note that Q ′ A may be infinite, but if we only consider primitive paths of length at most N 2 , where N = 1 + m i=n+1 r i , we obtain a finite subquiver Q A of Q ′ A .
We Proof of Lemma 6.1. By breaking matrices into their rows and columns it is easy to see that there is a Gl r -equivariant isomorphism of varieties Moreover, the diagonal copy of the multiplicative group K * in Gl (1,r) ∼ = K * × Gl r acts trivially on We have to prove that the map on the coordinate rings is surjective. By the First Fundamental Theorem [LBP90] the ring K[R KQ ∞ (1, r)] Gl (1,r) is generated by traces along oriented cycles in Q ∞ of length at most N 2 . Moreover, since the dimension vector at the vertex ∞ is equal to 1, we can reduce to oriented cycles of the following two types (i) An oriented cycle in passing only through the vertices n + 1, . . . , m. This invariant is the image under ϕ of the corresponding trace invariant for Q >n .
(ii) An oriented cycle of length at most N 2 starting and ending at vertex ∞ and not otherwise passing through ∞. Each primitive path of length at most N 2 from i to j in Q B gives rise to d i d j oriented cycles of this type and all arise this way. Such a primitive path correspond to an arrow from i to j in Q A and therefore to a matrix of size d j × d i in R KQ A (d). The maps R KQ A (d) → K picking out the entries of this matrix are sent by ϕ to the required trace invariant.
Thus each generator of K[R KQ ∞ (1, r)] Glr is in the image of ϕ, so ϕ is onto.
First Fundamental Theorem for finite dimensional algebras with an idempotent. Let K be an algebraically closed field of characteristic zero. Let B be a basic finite-dimensional algebra and let e 1 , . . . , e n , e n+1 , . . . , e m (n ≤ m) be a complete set of primitive orthogonal idempotents in B. Let e = n i=1 e i and let A = eBe. For any dimension vector (d, r) ∈ N m 0 we have a natural restriction map e : R B (d, r) → R A (d) given by left multiplication with e, and it induces a map R B (d, r) / / Gl r → R A (d).
Note that here we must use the reduced scheme structure on R A (d) and R B (d, r).
Proof. We write B = KQ B /I with Q B a quiver with vertices 1, . . . , m corresponding to the idempotents e 1 , . . . , e m in B and I an admissible ideal. Then there is a closed immersion R B (d, r) → R KQ B (d, r). Now we find a commutative diagram where f = m i=n+1 e i and the right hand map is induced by the algebra homomorphisms KQ A → A and KQ >n → f Bf . By Lemma 6.1 the top map is a closed immersion, and the Reynolds operator ensures that the left hand vertical map is a closed immersion. It follows that the bottom map is a closed immersion. Now, the variety R f Bf (r) / / Gl r classifies semi-simple f Bf -modules of dimension vector r. Since B is basic, so is f Bf and therefore this quotient variety is just a point. The claim follows.
Smooth points of module schemes. Let A be a finitely generated algebra over an algebraically closed field K, let e 1 , . . . , e n be a complete set of orthogonal idempotents in A and let d ∈ N n be a dimension vector. (Note that we allow the possibility that n = 1, so we just work with the total dimension of a representation). Recall that the closed points of the scheme R A (d) are identified with its K-points, so correspond to A-module structures X on n i=1 K d i . Lemma 6.4. If Ext 2 A (X, X) = 0, then X is a smooth point of R A (d). We use the following version of the lifting criterion for smoothness, see [Sta15, Tag 02HW].
Lemma 6.5. Suppose Λ is a finitely generated commutative K-algebra and m a maximal ideal in Λ.
The following are equivalent.
(1) The scheme Spec Λ is smooth at m.
(2) Any K-algebra map Λ → R/I with R is a finite dimensional commutative local K-algebra with maximal ideal n and I ⊂ R is an ideal with I 2 = 0 and I ∼ = R/n as an R-module, and the preimage of n/I is m, lifts to a K-algebra map Λ → R.
Proof of Lemma 6.4. The module structure X corresponds to a homomorphism θ : A → M d (K) with θ(e i ) = ǫ i for all i, and we can identify End K (X) with M d (K), with the left and right actions of a ∈ A corresponding to multiplication on the left or right by θ(a). One knows that R A (d) = Spec Λ, for a suitable finitely generated commutative K-algebra Λ with the property that K-algebra maps Λ → R correspond to K-algebra maps θ : A → M d (R) with θ(e i ) = ǫ i for all i.
In the setup of the lifting criterion we are given a finite dimensional commutative local K-algebra R with maximal ideal n and an ideal I with K-dimension 1 and I 2 = 0.
Since p and φ are algebra homomorphisms, P becomes a K-algebra, and it is an extension of A by M d (I), so it must be the trivial extension. Thus the map P → A has a section A → P . Composing it with the homomorphism P → M d (R) we get a lifting ψ 0 : A → M d (R) with pψ 0 = φ.
We now need to adjust ψ 0 to obtain a lifting ψ with ψ(e i ) = ǫ i for all i. Let S = K n with the coordinatewise multiplication, so it is a separable algebra over K, and let σ : S → A be the map sending the coordinate vectors in S to the idempotents e i . Let η : S → M d (K) be the map sending the coordinate vectors to ǫ i . Then the condition that θ(e i ) = ǫ i for all i can be rewritten as θσ = η. Similarly φσ = η, and we need to find ψ with ψσ = η.
For s ∈ S define d(s) = ψ 0 (σ(s)) − η(s). This defines a map from S to M d (R), and it has image contained in M d (I) since pd = φσ − η = 0. Now the bimodule action of A on M d (I) gives a bimodule action of S, in which, by the discussion above, the left or right action of s ∈ S can be given by left or right multiplication by any of η(s) or φ(σ(s)) or ψ 0 (σ(s)). Now it is easy to see that d : S → M d (I) is a derivation, so, since S is separable, it is an inner derivation, so there is γ ∈ M d (I) with d(s) = η(s)γ − γη(s) for all s ∈ S. Letting g = 1 + γ ∈ Gl d (R), it follows that gψ 0 (σ(s))g −1 = η(s). Then defining ψ(a) = gψ 0 (a)g −1 we obtain the required lift of φ.

Geometric applications of the projective quotient algebra
Desingularizations of quiver Grassmannians for representation-finite algebras. This is a straightforward generalization of [CIFR13a,§7]. Let A be a finite-dimensional, representation-finite, basic associative K-algebra and M a finite-dimensional A-module and let d be a dimension vector. is smooth (not just as a variety, but also with its natural scheme structure) and S [c(N )] is a connected component. (2) The map π is projective since Gr B c(M ) d,r is projective. We claim that we have It is enough to show that for F ⊂ c(M ) with dim F = (d, r) one has the following equivalence: So, given F ⊂ c(M ) with dim F = (d, r) and e(F ) = U . Then, we know that F is in H and stable because it is the submodule of the stable module c(M ) and by Lemma 2.3, we have On the other hand, given F ⊂ c(M ). with dim F = (d, r) and c(U ) ⊂ F . Since e is exact, we get U ⊂ e(F ). But we also have for every simple A-module S i and therefore U = e(F ).
(3) Since π is projective, its image is closed. For every U ∈ S [N ] we have c(U ) ∈ S [c(N )] and π(c(U )) = U , therefore S [N ] is contained in the image. By (2), we get that the morphism restricts to a bijection π −1 (S [N ] ) = S [c(N )] → S [N ] . To see that it is an isomorphism over an open set we just need to see that it is an isomorphism on tangent spaces over an open set. Now, the map π has a scheme-theoretic version π ′ -straightforward to define on commutative K-algebra valued points-such that the map on underlying reduced schemes coincides with π. Let U ∈ Gr Since the variety Gr A M d has to have some smooth points in S [N ] , we conclude that for these the map on tangent spaces is an isomorphism.
Orbit closures as quotients. Let R B (d, r) be the representation space of all (d, r)-dimensional Bmodules. The functor e induces a map R B (d, r) → R A (d), also denoted e. We observe the following.
Lemma 7.2. Assume R B (d, r) = ∅ and N ∈ R A (d). Then the following are equivalent.
(1) N ∈ Im e (2) dim c(N ) ≤ (d, r) (pointwise at every idempotent for B) In particular, Im e is a closed subset of R A (d).
Proof. Assume N = e(F ) for some F ∈ R B (d, r). By Lemma 2.1 we get dim c(N ) ≤ dim F = (d, r). Now assume dim c(N ) ≤ (d, r). Because the algebra B is basic and finite-dimensional, there is a unique semi-simple B-module S 1 of dimension dim c(N ) and a unique semi-simple B-module S 2 of dimension (d, r). Since dim c(N ) ≤ (d, r) one has S 1 is a submodule of S 2 , set S = S 2 /S 1 . We have e(S) = 0 and the dimension of S is (d, r) − dim c(N ). Set F = c(N ) ⊕ S, this gives a point in R B (d, r) with e(F ) = N , so N ∈ Im e. Since we are in the representation-finite case, we just need to see the following: Let r). We look at the (not exact) sequence since c is an additive functor we have Im c(i) ⊂ ker c(p) and since intermediate extensions preserve mono-and epimorphisms we have that c(i) is a monomorphism and c(p) is an epimorphism. This implies dim c(N 1 ⊕ N 2 ) ≤ dim c(M ).
Let M be a (K-valued) point in R A (d). We consider the closure of its Gl d -orbit O M ⊂ R A (d) (with the reduced scheme structure). Desingularization of orbit closures. Let K be an algebraically closed field of characteristic zero. Let Q B be a quiver as in §6 and let Q ∞ be the deframed quiver constructed from a given dimension vector (d, r). We consider the stability notion on R KQ ∞ (1, r) given by the vector θ ∈ R m−n+1 defined as r) is θ-stable if and only if for any non-zero proper subrepresentation N of M one has θ · dim N > 0. One can also consider θ-semi-stable representations M defined by the condition θ · dim N ≥ 0. But for this particular θ these two notions are equivalent. By geometric invariant theory (compare [Kin94]) there is a projective morphism of varieties where () s denotes the subset of θ-stable points and the quotient on the left hand side is a geometric quotient. Now recall, that we have an Gl r -equivariant isomorphism of varieties R KQ B (d, r) → R KQ ∞ (1, r). It is easy to see that under this isomorphism the θ-stable points correspond to the points of R KQ B (d, r) which are stable KQ B -modules with respect to the recollement given by the idempotent e = n i=1 e i . Thus the projective morphism becomes R KQ B (d, r) s /Gl r → R KQ B (d, r) / / Gl r where () s denotes the stable points. Now, let B be a finitely generated algebra and let e 1 , . . . , e n , e n+1 , . . . , e m (n ≤ m) be a complete set of orthogonal idempotents in B. By writing B as a quotient of a path algebra KQ B one obtains a projective morphism R B (d, r) s /Gl r → R B (d, r) / / Gl r . The left hand side is a geometric quotient-to see that it is a good quotient, use e.g. [BB02, Theorem 7.1.4] and the other properties check directly. r) s , the geometric quotient by Gl r exists and by composing with the closed immersion and the isomorphism from the previous theorem one gets a projective map Theorem 7.5. ϕ is a desingularization.
Proof. We claim that R B (d, r) s /Gl r is smooth. Since the global dimension of B is at most 2, we can use Lemma 4.10 to see that the second self-extension group of every stable point vanishes. By Lemma 6.4, we get that this is a smooth point in R B (d, r), therefore the open subvariety R B (d, r) s is smooth. It follows that R B (d, r) s /Gl r is smooth because R B (d, r) s → R B (d, r) s /Gl r is a principal For orbit closures of Dynkin quivers there already exists a resolution of singularities constructed by Reineke using the directedness of the Auslander-Reiten quiver of Dynkin quiver, see [Rei03].

Examples of projective quotient algebras
In this section, we choose representation-finite algebras A and describe the associated projective quotient algebras B.
8.1. Hereditary algebras of finite representation type. In this case A = KQ with Q Dynkin quiver (i.e. the underlying graph is Dynkin of type A,D or E). The homotopy category H is equivalent to the full subcategory of Q given by the objects f : P → X such that X has no non-zero projective summand and so it is equivalent to the category H Q of [CIFR13a] and [CIFR13b]. Our algebra B coincides with the algebra B Q from loc. cit. where they calculate its Ext-quiver and the relations from extensions between simple modules. Our results for this case are as loc. cit. The desingularization of orbit closures is not explicitly considered there, but other desingularizations are already known, see [Rei03], and the Hernandez-Leclerc construction [HL15] relates orbit closures to Nakajima quiver varieties, and in this context moduli spaces have previously been used to obtain desingularizations, cf. [Sch15, Theorem 3.2].
8.2. Self-injective algebras of finite representation type. In this case the functor ker from H to A-mod sending an object f : P → X to ker(f ) is an equivalence of categories. So, B is isomorphic to the Auslander algebra Γ of A. Moreover, we have a commuting diagram of functors 8.3. The truncated polynomial ring. Let A = K[X]/(X n ). This algebra is representation-finite and self-injective. The indecomposables in Q are the objects U r := (A → K[X]/(X r )) with 0 ≤ r ≤ n and the indecomposables in H are those with r < n. (But note that H is not equivalent to the full subcategory of Q containing these indecomposables.) The Auslander algebra of A (and therefore also B) has a unique structure as quasi-hereditary algebra. Explicitly, we can describe B with the quiver and relations p r j r = j r−1 p r−1 , 0 < r − 1 < n − 1, and p n−1 j n−1 = 0. The brackets [−] indicate the idempotent e such that eBe = A. The stable modules are given by the modules F with Hom(U r , F ) = 0 for 1 ≤ r ≤ n, this means in F all maps p * have to be monomorphisms. The costable modules are given by modules H with Hom(H, U r ) = 0 for 1 ≤ r ≤ n, this means in H all maps j * have to be epimorphisms. The stable modules coincide with the ∆-filtered modules and the costable modules with the ∇-filtered modules for the unique quasi-hereditary structure (see next example or e.g. [BHRR99]). In particular, the tilting module C coincides with the characteristic tilting module.
Our geometric construction of the orbit closures as affine quotient varieties coincides with the one from Kraft and Procesi in [KP79,§3.3]. Their variety Z equals our R B (d, r) and the union of the stable and the costable locus is contained in their smooth variety Z 0 . 8.4. The nilpotent oriented cycle. Let Q be the quiver with vertices {1, . . . , N } identified with their residue classes in the additive group Z/N Z. For each vertex i, we have one arrow x i : i → i + 1. Let I be the ideal given by all path of length n in Q. Then, the algebra A = KQ/I is a representationfinite, self-injective Nakayama algebra, for N = 1 we reobtain the previous example. We denote by E i [r] the indecomposable A-module with top S i of dimension r, i ∈ Z/N Z, r ∈ {1, . . . , n}. If we set E i [0] := 0, the indecomposable objects in H are U i,r := (E i [n] → E i [r]), i ∈ Z/N Z, 0 ≤ r ≤ n − 1. The algebra B can be described by the quiver with vertices (i, r), i ∈ Z/N Z, 0 ≤ r ≤ n − 1 and arrows and relations see below (for N = 3, n = 4 with identification of the left and right boundary) (2, 4)