Desingularizations of Quiver Grassmannians for the Equioriented Cycle Quiver

We construct torus equivariant desingularizations of quiver Grassmannians for arbitrary nilpotent representations of an equioriented cycle quiver. We apply this to the computation of their torus equivariant cohomology.


Introduction
Quiver Grassmannians are projective varieties parametrizing subrepresentations of quiver representations.Originating in the geometric study of quiver representations [Schofield 1992] and in cluster algebra theory [Caldero and Chapoton 2006], they have been applied extensively in recent years in a Lie-theoretic context, namely as a fruitful source for degenerations of (affine) flag varieties [Cerulli Irelli et al. 2013;2017;Feigin et al. 2017;Pütz 2022].This approach allows for an application of homological methods from the representation theory of quivers to the study of such degenerate structures.
The resulting varieties being typically singular, a construction of natural desingularizations is very desirable.For quiver Grassmannians of representations of Dynkin quivers this was accomplished in [Cerulli Irelli et al. 2013] building on [Feigin and Finkelberg 2013], and for Grassmannians of subrepresentations of loop quivers in [Feigin et al. 2017] (in other directions, this construction was generalized to representations of large classes of finite dimensional algebras in [Crawley-Boevey and Sauter 2017; Keller and Scherotzke 2014; Leclerc and Plamondon 2013]).
In the present paper, we synthesize the approaches of [Cerulli Irelli et al. 2013;Feigin et al. 2017] and construct desingularizations of quiver Grassmannians for nilpotent representations of equioriented cycle quivers, thereby, in particular, desingularizing degenerate affine flag varieties [Pütz 2022].
As an important application, this allows us to describe the equivariant cohomology of degenerate affine flag varieties and more general quiver Grassmannians, in continuation of the program started in [Lanini and Pütz 2023a;2023b].
In the first section, we recall some background material on quiver Grassmannians for nilpotent representations of the equioriented cycle quiver.In Section 3 we give an explicit construction for desingularizations of quiver Grassmannians for nilpotent representations of the equioriented cycle, along the lines of [Cerulli Irelli et al. 2013].We prove that the desingularization has a particularly favorable geometric structure, namely it is isomorphic to a tower of Grassmann bundle.Consequently, it, admits a cellular decomposition which is compatible with the cellular decomposition of the singular quiver Grassmannian.In Section 4, we recall the definition of certain torus actions on cyclic quiver Grassmannians, together with the necessary framework to compute torus equivariant cohomology.Finally, in Section 5 we prove that the desingularization is equivariant with respect to the torus action as introduced in [Lanini and Pütz 2023a].This allows to use the construction from that paper for the computation of torus equivariant cohomology to all quiver Grassmannians for nilpotent representations of the equioriented cycle.

Quiver Grassmannians for the equioriented cycle
In this section we recall some definitions concerning quiver Grassmannians and representations of the equioriented cycle.We refer to [Kirillov Jr. 2016;Schiffler 2014] for general representation theoretic properties, and to [Cerulli Irelli et al. 2012] for basic properties of quiver Grassmannians.
Generalities on quiver representations.Let Q be a quiver, consisting of a set of vertices Q 0 and a set of arrows Q 1 between the vertices.A Q-representation M consists of a tuple of ‫-ރ‬vector spaces M (i) for i ∈ Q 0 and tuple of linear maps M α : M (i)  → M ( j) for (α : i → j) ∈ Q 1 .We denote the category of finite dimensional Q-representations by rep ‫ރ‬ (Q).The morphisms between two objects M and N are tuples of linear maps ϕ i : M (i)  → N (i) for i ∈ Q 0 such that ϕ j • M α = N α • ϕ i holds for all (α : i → j) ∈ Q 1 .
Definition 2.1.For M ∈ rep ‫ރ‬ (Q) and e ∈ ‫ގ‬ Q 0 , the quiver Grassmannian Gr e (M) is the closed subvariety of i∈Q 0 Gr e i (M (i) ) of all subrepresentations U of M such that dim ‫ރ‬ U (i)  = e i for i ∈ Q 0 .
For an isomorphism class [N ] of Q-representations, the stratum S [N ] is defined as the set of all points (that is, subrepresentations) U ∈ Gr e (M) such that U is isomorphic to N .By [Cerulli Irelli et al. 2012, Lemma 2.4], S [N ] is locally closed and irreducible.If there are only finitely many isomorphism classes of subrepresentations of M, as will be the case in the following, the S [N ] thus define a finite stratification of the quiver Grassmannians.
Every basis B of M ∈ rep ‫ރ‬ (Q) consists of bases for each vector space M (i) of the Q-representation M, where m i := dim ‫ރ‬ M (i)  for all i ∈ Q 0 , and [m] := {1, . . ., m}.
Definition 2.2.Let M ∈ rep ‫ރ‬ (Q) and B a basis of M. The coefficient quiver Q(M, B) consists of: Representations of the equioriented cycle.For n ∈ ‫,ގ‬ by n we denote the equioriented cycle quiver on n vertices.Hence the set of arrows and the sets of vertices are in bijection with ‫ޚ‬ n := ‫;ޚ‪/n‬ޚ‬ more precisely, we have ( n ) 0 = ‫ޚ‪/n‬ޚ‬ and arrows α i : i → i + 1 for all i ∈ ‫.ޚ‪/n‬ޚ‬Here and in the following, we consider all indices modulo n unless specified differently.
for all i ∈ ‫ޚ‬ n , i.e., all concatenations of the maps of M along the arrows of n vanish after at most N steps.M is called nilpotent if it is N -nilpotent for some N .
From now on we use the short hand notation M i for the map along the arrow α i .
Example 2.3.Let i ∈ ‫ޚ‬ n and let ℓ ∈ ‫ޚ‬ ≥1 .Consider the ‫-ރ‬vector space V with basis B = {b 1 , . . ., b ℓ } equipped with the ‫ޚ‬ n -grading given by deg Take the operator A ∈ End(V ) uniquely determined by setting Ab k = b k+1 for any k < ℓ and Aw ℓ = 0.The vector space of the corresponding n -representation over the j-th vertex is spanned by the elements of B of degree j.Let m j be the number of these basis elements.In this basis, the map from vertex j to j + 1 is given by a m j × m j+1 matrix with ones on the diagonal below the main diagonal and all other entries equal to zero.It is immediate to check that this n -representation is nilpotent.We denote this representation by U i (ℓ).
Example 2.5.Observe that the basis B from Example 2.3 can be obviously rearranged into the union of ordered bases where k i is the number of elements b ∈ B with deg(b) = i.With respect to B, the coefficient quiver of U i (ℓ) has the form: By [Kirillov Jr. 2016, Theorem 1.11], every nilpotent n -representation is isomorphic to a n -representation of the form Here N is the nilpotence parameter of the representation and the tensor product with the ‫-ރ‬vector spaces counts the multiplicities of the indecomposable summands.
Let ‫ރ‬ n be the path algebra of n and define the path for all i ∈ ‫ޚ‬ n and some fixed N ∈ ‫.ގ‬We define the path algebra ideal generated by all paths of length N , and we denote the truncated path algebra The following is a special case of [Schiffler 2014, Theorem 5.4].Proposition 2.6.The category rep ‫ރ‬ ( n , I N ) of bounded quiver representations is equivalent to the category mod ‫ރ‬ (A (N )  n ) of modules over the truncated path algebra.Remark 2.7.The U i (N ) for i ∈ ‫ޚ‬ n are the longest indecomposable nilpotent representations in rep ‫ރ‬ ( n , I N ).
Let P i ∈ rep ‫ރ‬ ( n , I N ) be the n -representation starting at vertex i ∈ ‫ޚ‬ n corresponding to the projective indecomposable A (N )  n -module, and let I j ∈ rep ‫ރ‬ ( n , I N ) be the n -representation ending at vertex j ∈ ‫ޚ‬ n corresponding to the injective indecomposable A (N )  n -representation.We can identify bounded projective and bounded injective representations of the cycle, via indecomposable nilpotent representations (see [Pütz 2022, Proposition 4.2]): Proposition 2.8.For n, N ∈ ‫ގ‬ and all i, j ∈ ‫ޚ‬ n the projective and injective representations P i and I j of the bound quiver ( n , I N ) satisfy Remark 2.9.In particular, U i (N ) is projective and injective in rep ‫ރ‬ ( n , I N ).If we want to emphasize the injective nature of an indecomposable n -representation we sometimes use the notation U ( j; ℓ) := U j−ℓ+1 (ℓ).
Parametrization of irreducible components.In Section 3 we will construct desingularizations of all quiver Grassmannians associated to nilpotent representations of the equioriented cycle, which requires knowledge of their irreducible components.Let us first recall the approach: since there are only finitely many isomorphism classes of nilpotent n -representations in any fixed dimension, the stratification of every quiver Grassmannian into strata S [N ] is finite.Since the strata are irreducible, the irreducible components of quiver Grassmannians are therefore of the form S [N ] for certain isomorphism classes [N ], which provide a natural labeling (and a canonical representative) of the components.
For arbitrary nilpotent representations of the equioriented cycle the structure of the irreducible components of the associated quiver Grassmannians is not known.In the special case that all indecomposable direct summands of the n -representation M have length N = ωn (for ω ∈ ‫)ގ‬ and e = (ωk, . . ., ωk) ∈ ‫ޚ‬ n , we have an explicit description of the irreducible components of the quiver Grassmannian Gr e (M) [Pütz 2022, Lemma 4.10 k .The dimension of the irreducible components is ωk(n − k).

Construction of the desingularization
The approach to the construction of desingularizations of quiver Grassmannians for the equioriented cycle quiver carried out in this section is a synthesis of the approach of Cerulli Irelli et al. [2013] for Dynkin quivers and the approach of Feigin et al. [2017] for the loop quiver.We will construct another quiver for which certain quiver Grassmannians yield desingularizations, which relies on certain favorable homological properties similar to those in [Cerulli Irelli et al. 2013, Section 4].Note that the present case is not immediately covered by the generalizations [Crawley-Boevey and Sauter 2017; Keller and Scherotzke 2014; Leclerc and Plamondon 2013], and our construction has the advantage of being of a very explicit linear algebra nature.
Bounded representations of the equioriented cylinder.In this subsection we introduce a map : We start with the definition of Q and the ideal I. Let ˆ n,N be the quiver with vertices which we call an equioriented cylinder quiver.We define În,N as the ideal in the path algebra ‫ރ‬ ˆ n,N generated by the relations Example 3.1.ˆ 4,4 is the following quiver: (1, 1) α 1,1 (2, 1) (1, 2) (2, 2) (1, 3) We define a functor : where the inclusions in the last row arise naturally from the definition of the vector spaces of the representation M.Here M i denotes the map along the arrow α i .
and hence is a fully faithful functor.
Proof.By construction of , the vector spaces constituting M ∈ rep ‫ރ‬ ( ˆ n,N , În,N ) are subspaces of the corresponding vector spaces constituting M. Hence each morphism in Hom n (N , M) induces a morphism in Hom ˆ n,N ( N , M) whose components at the additional vertices are obtained by restriction.It is immediate to check that this induces the desired bijection N ,M , N ,M (id M ) = id M and that Now we want to describe the image of the indecomposable U i (ℓ) under .Let A ∞×N be the infinite band quiver of height N , that is, the quiver with vertices (i, k) for i ∈ ‫ޚ‬ and k ∈ [N ] and arrows α i,k : (i, k) → (i, k+1) and β i,k : (i, k) → (i+1, k−1) whenever both vertices exist.Define a map of quivers φ : A ∞×N → ˆ n,N , induced by sending each index i ∈ ‫ޚ‬ to its equivalence class i ∈ ‫ޚ‬ n .This extends to a push-down functor : and zero otherwise.The maps along the arrows of A ∞×N are identities if both the source and target space are one-dimensional and zero otherwise.Using the explicit definitions of the functors and , and the explicit descriptions of the representations U i (ℓ) and V (i; ℓ), we can now directly verify that Analogously, we define A ∞×N -representations V (i, k; ℓ) consisting of vector spaces V (i, k; ℓ) ( j,r )  = ‫ރ‬ for ( j, r ) ∈ (A ∞×N ) 0 with j ≥ i and i +k ≤ j +r ≤ i +k+ℓ and W (i, k; ℓ) with vector spaces W (i, k; ℓ) ( j,r )  = ‫ރ‬ for ( j, r ) From now on we erase all zeros and arrows connected to zeros from the picture.Hence we obtain

‫ރ‬
Homological properties of the category of cylinder representations.In this section, we follow closely the approach of Cerulli Irelli et al. [2013] to establish certain favorable homological properties of the image of the functor .
Proposition 3.5.The simple, projective and injective objects in rep ‫ރ‬ ( ˆ n,N , În,N ) are given by Proof.For the simple objects this is immediate.
It remains to prove vanishing of all We apply the functor Hom ˆ n,N , În,N (_, Ûj (ℓ ′ )) to the above projective resolution of Ûi (ℓ), simplify the terms involving projectives, and obtain the exact sequence By definition of , the map α is the canonical surjection proving the desired Ext 1 -vanishing.□ Example 3.9.For N = 4 we obtain the following projective and injective resolutions of U i (3) = U (i − 2; 3): This induces maps res V,W : Remark 3.13.In general, there is no explicit description of the gsub e (M).But if the indecomposable summands of M are all of length ωn for n, ω ∈ ‫,ގ‬ we can apply Lemma 2.10.
Example 3.14.Let n = 3, N = 2 and consider the quiver Grassmannian for ) and e = (1, 2, 3).It has eight isomorphism classes of subrepresentations but only two irreducible components.Namely S In particular Û is contained in π −1 N (U ) ⊂ Gr dim N ( M).It remains to show that π −1 N (U ) = { Û }: By construction of res and it follows that ,2) holds for all V ∈ π −1 N (U ) since U and N are isomorphic.This implies that Û (i,2) = V (i,2) holds for all i ∈ ‫ޚ‬ n .Inductively, it follows that ) for all i ∈ ‫ޚ‬ n and the first nontrivial choice of a subspace U is clear by definition of F U and the construction of the restriction functor.The other inclusion follows since every point V of the fiber π −1 N (U ) has to contain the vector spaces of Û in its vector spaces V (i,k) over each vertex (i, k) of ˆ n,N , in order to map to U .The isomorphism between F U and the quiver Grassmannian is a direct consequence of the explicit description of the fiber.□ We are now ready to state the main result of the paper, which is proved after the next proposition.
Remark 3.18.Using Proposition 3.16, we can compute the fiber dimensions for the desingularization to examine whether it is small, in the spirit of [Feigin and Finkelberg 2013, Section 2].This is the case for the quiver Grassmannian Gr 2 (M) from [Lanini and Pütz 2023b, Example 3.13] where Q = 1 and M = U 1 (2) ⊕ S 2 1 .In general, desingularizations of quiver Grassmannians for the cycle are not small.It already fails for the loop quiver (i.e., 1 ) and the quiver Grassmannian Gr 2 (N ) where N = U 1 (2) 2 .
For the proof of Theorem 3.17 we recollect the main properties of the maps π N : Proposition 3.19.Let M ∈ rep ‫ރ‬ ( n , I N ) and [N ] ∈ gsub e (M).Then: (i) The variety Gr dim N( M) is smooth with irreducible equidimensional connected components.
(ii) The map π N is one-to-one over S [N ] .
(iii) The image of π N is closed in Gr e (M) and contains S [N ] .
(iv) The map π N is projective.
Theorem 3.21.For each [N ] ∈ gsub e (M) the quiver Grassmannian Gr dim N ( M) is isomorphic to a tower of fibrations where n := dim N and m := dim M and each map is a fibration with fiber isomorphic to a product of ordinary Grassmannians of subspaces.
Proof.Every point U of the quiver Grassmannian Gr dim N ( M) is parametrized by a collection of subspaces Define X k as the image of Gr dim N ( M) in the variety Gr n( m) (k) which is defined analogous to Gr n(‫ރ‬ m), with the only difference that the second product runs over We proceed by decreasing induction starting from k = N .Every point in the product of Grassmannians of subspaces Gr n( m) (N ) can be extended to an element of Gr dim N ( M) since the upper vector spaces of an element in the quiver Grassmannian are not related.This implies X N = Gr n( m) (N ) as desired.Now assume that the vector spaces U (i,k ′ ) are fixed for all i ∈ ‫ޚ‬ n and k ′ > k.Since U has to be contained in the quiver Grassmannian it has to satisfy the relations ,k+1) for all i ∈ ‫ޚ‬ n .
Hence the next layer of vector spaces U (i,k) requires This is equivalent to the choice of a point in the Grassmannian because every map Mα i,k is a projection where the last m(i,k) − m(i,k+1) coordinates are sent to zero and each Mβ i,k is an inclusion.□ Remark 3.22.The explicit description of the desingularization in Theorem 3.21 allows to construct a cellular decomposition of Gr dim N ( M) (see Theorem 5.5).In particular, it implies that Gr dim N ( M) is smooth.

Torus equivariant cohomology and equivariant Euler classes
In this section we briefly recall definitions and constructions concerning torus actions on quiver Grassmannians, torus equivariant cohomology and torus equivariant Euler classes.More details on the general theory is found in [Arabia 1998;Brion 1998;Goresky et al. 1998;Gonzales 2014].The application to quiver Grassmannians is introduced in [Lanini and Pütz 2023a; 2023b].In Section 5 we provide examples and apply our desingularizations to the computation of equivariant cohomology of quiver Grassmannians for the equioriented cycle.
Moment graph and torus equivariant cohomology.Let X be a projective algebraic variety over ‫.ރ‬The action of an algebraic torus T ∼ = ‫ރ(‬ * ) r on X is skeletal if the number of T -fixed points and the number of one-dimensional T -orbits in X is finite.We call a cocharacter χ ∈ X * (T ) generic for the T -action on X if X T = X χ ‫ރ(‬ * ) .By X * (T ) we denote the character lattice of T .The T -equivariant cohomology of X with rational coefficients is denoted by H • T (X ).Definition 4.1.The pair (X, T ) is a GKM-variety if the T -action on X is skeletal and the rational cohomology of X vanishes in odd degrees.
The closure E of every one-dimensional T -orbit E in a projective GKM-variety admits an T -equivariant isomorphism to ‫ސރ‬ 1 .Thus each one-dimensional T -orbit connects two distinct T -fixed points of X .Definition 4.3.Let (X, T ) be a GKM-variety, and let χ ∈ X * (T ) be a generic cocharacter.The corresponding moment graph G = G(X, T, χ ) of a GKM-variety is given by the following data: Remark 4.5.The characters from (MG2) are only unique up to a sign.This sign does not play a role in Theorem 4.4.Hence we can fix our favorite convention.
BB-filterable varieties.In this subsection we describe a class of varieties which admit an explicit formula for the computation of their equivariant cohomology.Let X be a ‫ރ‬ * -variety.By X ‫ރ‬ * we denote its fixed point set and X 1 , . . ., X m denote the connected components of X ‫ރ‬ * .This induces a decomposition (4.6) where W i is called attracting set of X i .Since decompositions of this type were first studied by Bialynicki-Birula [1973], we call it a BB-decomposition.
Definition 4.7.We say that W i from (4.6) is a rational cell if it is rationally smooth at all w ∈ W i .This in turn holds if [Gonzales 2014, p. 292, Definition 3.4]).
Definition 4.8.A projective T -variety X is BB-filterable if: (BB1) The fixed point set X T is finite.
Theorem 4.9 (see [Lanini and Pütz 2023a, Theorem 1.15]).Let X be a BB-filterable projective T -variety.Then: (1) X admits a filtration into T -stable closed subvarieties Z i such that (3) The singular rational cohomology of Z i vanishes in odd degrees, for i ∈ [m].
(4) If , additionally, the T -action on X is skeletal, each Z i is a GKM-variety.
Euler classes and cohomology module bases.For the precise definition of Euler classes we refer the reader to [Arabia 1998, Section 2.2.1].Instead we give three properties which are enough to determine the equivariant Euler classes in our setting.
(1) If Y is smooth at y then Eu T (y, Y ) = (−1) dim(Y ) det T y Y , where det T y Y is the product of the characters by which T acts on the tangent space T y Y . ( Remark 4.11.Lemma 4.10 differs from [Brion 1998] by using Euler classes instead of equivariant multiplicities which are inverse to each other up to a sign. Definition 4.12 (see [Gonzales 2014, Lemma 6.7]).Let The next theorem gives an explicit formula to compute a basis for H • T (X ) as free module over H • T ( pt).Observe that everything depends on the order of the fixed points which is in general not unique.
Theorem 4.13 (see [Lanini and Pütz 2023a, Theorem 2.12]).Let (X, T ) be a BB-filterable GKM-variety with filtration There exists a unique basis {θ (i)  } i∈[m] of H • T (X ) as free module over H • T ( pt), such that for any i ∈ [m] the following properties hold: (1) θ (i)  x j = 0 for all j < i.
Observe that this action extends to the quiver Grassmannian Gr e (M) only under some additional assumptions about the grading (see [Lanini and Pütz 2023a, Lemma 5.12]).
Remark 4.18.If the desingularizations constructed in Theorem 3.17 are T -equivariant, this theorem implies that we can compute the T -equivariant cohomology of all quiver Grassmannians for nilpotent representations of n , using Theorem 4.13.
From now on we assume that T := ‫ރ(‬ * ) d+1 acts on Gr e (M) as in [Lanini and Pütz 2023a, Lemma 5.12].Here d is the number of connected components in Q(M, B) and the additional parameter comes from cyclic symmetry.The weight functions of the action are defined implicitly by the formula used in [Lanini and Pütz 2023a, Section 5.2].

Torus equivariant desingularization and application
In this section we apply the methods from the previous section to compute Euler classes at singular points and torus equivariant cohomology of quiver Grassmannians for the equioriented cycle using their desingularizations as constructed in Theorem 3.17.It remains to show that these desingularizations are torus equivariant.
Remark 5.1.A choice of basis B of M ∈ rep ‫ރ‬ ( n , I N ) induces a basis B of M ∈ rep ‫ރ‬ ( ˆ n,N , În,N ) such that the connected components of Q( M, B) are in bijection with the images of the indecomposable summands of M. In particular the basis B(i,k) over the vertex (i, k) of ˆ n,N is a subset in the basis B (i+k−1) of cardinality m i+k−1 − c where c is the corank of the map i) for k = 1.This allows us to extend the T -action to the vector spaces of M by extending the weight functions according to the inclusions of the basis described above.In other words, all basis vectors of B which have the same image in B get the same weight.
Proposition 5.2.The T -action on the vector spaces of M as defined in Remark 5.1 extends to every quiver Grassmannian Gr k ( M). Proof.The T -equivariance of π N from Lemma 5.3 gives the desired description of the fixed points.Now we prove that the ‫ރ‬ * -attracting sets of these fixed points from the BB-decomposition are cells.By [Carrell 2002, Lemma 4.12], they provide an α-partition, i.e., there exists a total order of the fixed points It remains to show that they are isomorphic to affine spaces.This is induced by the cellular decomposition of Gr e (M) and the T -equivariance of the desingularization: Assume p ∈ Gr dim N ( M) is a T -fixed point.The vector space p (i,k) over the vertex (i, k) of ˆ n,N is a point in the Grassmannian of subspaces Gr n(i,k) ‫ރ (‬  m(i,k) ).By construction of the ‫ރ‬ * -action (as in Remark 5.1), the attracting set of p (i,k)  in Gr n(i,k) ‫ރ (‬  m(i,k) ) is a cell.The attracting set of p in the whole quiver Grassmannian is the intersection of these cells along the maps of M. We proceed by induction on k.For k = 1 there is nothing to show because there are no maps between the vector spaces.If k = 2, we have the original vector spaces of the representation M and one additional layer of subspaces therein.The relations between the coordinates in the attracting sets are the same as for Gr e (M).Hence they are cells by [Lanini and Pütz 2023a, Theorem 5.7].The maps of M along the arrows β i,k of ˆ n,N are inclusions and the maps along α i,k are projections where the last m (i,k)  − m (i,k+1) coordinates are sent to zero (see [Lanini and Pütz 2023a, Proposition 4.8]).Thus we obtain that the intersecting relations for each k ∈ [N ] are of the form as described in [Lanini and Pütz 2023a, Theorem 5.7].This implies the desired isomorphisms to affine spaces.□ Remark 5.6.In the setting that M = i∈‫ޚ‬ n U i (ωn) and e = (ωk, . . ., ωk) ∈ ‫ޚ‬ n it is possible to strengthen the results concerning the desingularization (see [Feigin et al. 2023a, Sections 2.5 and 2.6]).Namely, Gr e (M) has n k explicitly described irreducible components (see Example 2.12) and the cells of Gr dim N ( M) are the strata of the corresponding T -fixed points.
Example.Now, we provide an explicit example for the constructions from the previous sections.Let M := U 1 (4) ⊕ U 2 (2) ⊕ U 2 (2) be a 2 -representation and fix the dimension vector e = (2, 2).The quiver Grassmannian Gr e (M) has five strata (i.e., isomorphism classes of subrepresentations) with the representatives: The stratum of V 2 is three-dimensional, the strata of V 1 , V 3 and V 4 are twodimensional and the stratum of V 5 is one-dimensional.This is computed using [Pütz 2022, Proposition 4.4] and [Cerulli Irelli et al. 2012, Lemma 2.4].
Here the dashed lines were used to highlight the symmetries of the labeling and avoid to write the labels in the picture.There are four points which are not rationally smooth.Namely the tangent spaces at p 1 , p 2 , p 6 and p 7 are four-dimensional, whereas Gr e (M) itself is threedimensional.We can read this from the picture as follows: the number of edges adjacent to a point is the dimension of its tangent space and the number of outgoing edges is the cell dimension.The irreducible components are obtained as closure of the strata of the points p 8 , p 7 and p 5 , because their strata are not contained in the closure of any other stratum.These are generic subrepresentation types of M for dimension vector e = (2, 2).Hence the desingularization of Gr e (M) consists of three components.
The extended quiver ˆ 2,4 is (1, 2) (2, 1) (2, 2) (1, 1) (2, 4) For the basis induced by the basis B of M, the coefficient quiver of M is Here the separating lines between the vertices indicate if they live over the inner or outer vertex of ˆ 2,4 in that position.Representatives for the extended representations of the generic subrepresentation types are With the explicit description of the cellular decomposition of the quiver Grassmannians Gr dim V1 ( M), Gr dim V2 ( M) and Gr dim V3 ( M) from Theorem 5.5, it is a straightforward computation that their moment graphs are with labels Here pi, j is the preimage of p i in Gr dim Vj ( M).Moreover, from the cellular decompositions we obtain the isomorphisms Gr dim V1 ( M) ∼ = Gr 1 ‫ރ(‬ 3 ), Gr dim V2 ( M) ∼ = Fl(SL 3 ), Gr dim V3 ( M) ∼ = Gr 2 ‫ރ(‬ 3 ).
Observe that the special role of p 8 in this example allows to generate more zeroentries as in the general setting of Theorem 4.13.
. For each W ∈ rep ‫ރ‬ ( ˆ n,N , În,N ) we define the representation res W ∈ rep ‫ރ‬ ( n , I N ) by res by forgetting the components of the morphisms at the vertices (i, k) with k ≥ 2. Hence we obtain a functor res : rep ‫ރ‬ ( ˆ n,N , În,N ) → rep ‫ރ‬ ( n , I N ).The proof of the following proposition is immediate by the construction of and res.Proposition 3.10.res• (M) = M holds for all M ∈ rep ‫ރ‬ ( n , I N ).The desingularization map.In this subsection we provide the construction of the desingularization map, again closely following[Cerulli Irelli et al. 2013].An example is given below.Definition 3.11.An isomorphism class [N ] of n -representations is called a generic subrepresentation type of M ∈ rep ‫ރ‬ ( n , I N ) to dimension vector e, if the stratum S [N ] is open in Gr e (M).The set of generic subrepresentation types is denoted by gsub e (M).Remark 3.12.By construction, for some [N ] ∈ gsub e (M) the closure of the stratum S [N ] is an irreducible component of Gr e (M), and all irreducible components are obtained in this way.

For
[N ]  ∈ gsub e (M) we define a map from a quiver Grassmannian of the cylinder quiver to a quiver Grassmannian of the cycle quiver π N : Gr dim N ( M) → Gr e (M) by π N (U ) := res U for all U ∈ Gr dim N ( M). Proposition 3.15.For each [N ] ∈ gsub e (M) the map

(
MG0) The T -fixed points as vertices, i.e., G 0 = X T .(MG1) The closures of one-dimensional T -orbits E = E ∪ {x, y} as edges in G 1 , oriented from x to y if lim λ→0 χ (λ).p = x for p ∈ E. (MG2) Every E is labeled by α E ∈ X * (T ) describing the T -action on E. Theorem 4.4 [Goresky et al. 1998, Theorem 1.2.2].Let (X, T ) be a GKM-variety with moment graph G = G(X, T, χ ) and set R Remark 4.15.From now on we assume the choice of a basis B of M such that the connected components of Q(M, B) are in bijection with the indecomposable direct summands of M. Such a choice is always possible by [Kirillov Jr. 2016, Theorem 1.11].A grading of M ∈ rep ‫ރ‬ ( n ) with respect to a fixed basis is a map wt : B → ‫ޚ‬ B .This induces an action of λ ∈ ‫ރ‬ * by λ.b := λ wt(b) • b.Remark 4.16.Combining several weight functions wt 1 , . . ., wt D : B → ‫ޚ‬ B , we can define the action of λ = (λ j ) j∈[D] ∈ ‫ރ(‬ * ) D by λ.b := j∈[D]

Proof.
We have to show that the T -action is compatible with the maps of the quiver representation M. By construction of the action and the representation M, this follows immediately from the compatibility of the T -action (on the vector spaces of M) with the maps of M as shown in [Lanini and Pütz 2023a, Lemma 5.12].□ Lemma 5.3.The desingularization of Theorem 3.17 is T -equivariant.Proof.With Proposition 5.2, the statement follows immediately from the construction of the grading as in Remark 5.1 together with the description of the desingularization in Theorem 3.17.□ Remark 5.4.The T -equivariance of the desingularization allows us to use [Lanini and Pütz 2023a, Lemma 2.1(3)] for the computation of equivariant Euler classes at the singular points of Gr e (M).This allows us to apply [Lanini and Pütz 2023a, Theorem 2.12] about the construction of a basis for the T -equivariant cohomology to all quiver Grassmannians for nilpotent representations of the cycle.Cellular decomposition of the desingularization.Theorem 5.5.For [N ] ∈ gsub e (M) the T -fixed points of Gr dim N ( M) are exactly the preimages of the T -fixed points of S [N ] ⊂ Gr e (M) under π N .The ‫ރ‬ * -attracting sets of these points provide a cellular decomposition of Gr dim N ( M).

] :
Lemma 2.10.Let M denote the n -representation ⊕ i∈‫ޚ‬ n U (i; ωn)⊗‫ރ‬ d i with ω ∈ ‫ގ‬ and d i ∈ ‫ޚ‬ ≥0 for all i ∈ ‫ޚ‬ n , define m := i∈‫ޚ‬ n d i and e := (ωk, . . ., ωk) ∈ ‫ޚ‬ n .The irreducible components of Gr e (M) are in bijection with the set C k (d) := p ∈ ‫ޚ‬ n ≥0 : p i ≤ d i for all i ∈ ‫ޚ‬ n , j∈I U ( j; ωn) The parametrization of the projective and injective representations is a direct computation using the formula based on paths in the quiver ˆ n,N (see[Schiffler 2014, Definition 5.3]) and their relations from În,N as described in the beginning of this section.Lemma 3.8.For M ∈ rep ‫ރ‬ ( n , I N ) the injective and projective dimension of M is at most one and Ext 1 □Theorem 3.6.The category rep ‫ރ‬ ( ˆ n,N , În,N ) has global dimension at most two.Proof.It suffices to construct projective resolutions of length at most two for all simple representations in rep ‫ރ‬ ( ˆ n,N , În,N ).These representations are denoted by S i,k and consist of a single copy of ‫ރ‬ at vertex (i, k) and all other vector spaces and the maps are zero.The projective resolutions of S i,1 are of the form 0 → P i,2 → P i,1 → S i,1 → 0 and for S i,k with k ≥ 2 this generalizes to 0 → P i+1,k → P i+1,k−1 ⊕ P i,k+1 → P i,k → S i,k → 0. □ Example 3.7.For N = 4 and S i,3 we obtain the following projective resolution:ˆ n,N , În,N ( M, M) = 0.Proof.It suffices to compute the projective and injective dimension of the image of all indecomposable representations U i (ℓ) ∈ rep ‫ރ‬ ( n , I N ), by exhibiting projective resp.injective resolutions, namely 0 [Cerulli Irelli et al. 2013,.8 we can apply[Cerulli Irelli et al. 2013, [Cerulli Irelli et al. 2013.8.With the properties of ˆ n,N -representations from Theorem 3.6 and Lemma 3.8, the maps π N as in Proposition 3.19 and as in Proposition 3.3, the rest of the proof is the same as for[Cerulli Irelli et al. 2013, Corollary 7.7].□ Remark 3.20.In particular, Cerulli Irelli et al. [2021, Proposition 37] proves the conjecture from [Cerulli Irelli et al. 2013, Remark 7.8], about the irreducibility of Gr e ( M) in [Cerulli Irelli et al. 2013, Corollary 7.7] for arbitrary representations M of a Dynkin quiver.