Minimal Resolutions Research Articles

Hochschild Homology Groups of System Quiver Algebras of Minimal Wild Type

Abstract In the paper, based on the the minimal projective bimodule resolutions of the system quiver algebras of minimal wild representation type, we calculate explicitly the dimensions of all Hochschild homology groups of the system quiver algebras of minimal wild representation type by means of combinatorics.

Generalized Borcea–Voisin Construction

C. Voisin and C. Borcea have constructed mirror pairs of families of Calabi-Yau threefolds by taking the quotient of the product of an elliptic curve with a K3 surface endowed with a non-symplectic involution. In this paper, we generalize the construction of Borcea and Voisin to any prime order and build three and four dimensional Calabi-Yau orbifolds. We classify the topological types that are obtained and show that, in dimension 4, orbifolds built with an involution admit a crepant resolution and come in topological mirror pairs. We show that for odd primes, there are generically no minimal resolutions and the mirror pairing is lost.

Resolutions of mesh algebras: periodicity and Calabi–Yau dimensions

A triangulated category $${(\mathcal{T},\Sigma)}$$ is said to be Calabi–Yau of dimension d if Σ d is a Serre functor. We determine which stable module categories of self-injective algebras Λ of finite type are Calabi–Yau and compute their Calabi–Yau dimensions, correcting errors in previous work. We first show that the Calabi–Yau property of mod-Λ can be detected in the minimal projective resolution of the stable Auslander algebra Γ of Λ, over its enveloping algebra. We then describe the beginning of such a minimal resolution for any mesh algebra of a stable translation quiver and apply covering theory to relate these minimal resolutions to those of the (generalized) preprojective algebras of Dynkin graphs. For representation-finite self-injective algebras of torsion order t = 1, we obtain a complete description of their stable Calabi–Yau properties, but only partial results for those algebras of torsion order t = 2. We also obtain some new information about the periods of the representation-finite self-injective algebras of torsion order t > 1. Finally, we describe how these questions can also be approached by realizing the stable categories of representation-finite self-injective algebras as orbit categories of the bounded derived categories of hereditary algebras, and illustrate this technique with several explicit computations that our previous methods left unsettled.

Predicted Virtual Soft Shadow Maps with High Quality Filtering

AbstractIn this paper we present a novel image based algorithm to render visually plausible anti‐aliased soft shadows in a robust and efficient manner. To achieve both high visual quality and high performance, it employs an accurate shadow map filtering method which guarantees smooth penumbrae and high quality anisotropic anti‐aliasing of the sharp transitions. Unlike approaches based on pre‐filtering approximations, our approach does not suffer from light bleeding or losing contact shadows. Discretization artefacts are avoided by creating virtual shadow maps on the fly according to a novel shadow map resolution prediction model. This model takes into account the screen space frequency of the penumbrae via a perceptual metric which has been directly established from an appropriate user study. Consequently, our algorithm always generates shadow maps with minimal resolutions enabling high performance while guarantying high quality. Thanks to this perceptual model, our algorithm can sometimes be faster at rendering soft shadows than hard shadows. It can render game‐like scenes at very high frame rates, and extremely large and complex scenes such as CAD models at interactive rates. In addition, our algorithm is highly scalable, and the quality versus performance trade‐off can be easily tweaked.

Chebyshev polynomials on symmetric matrices

In this paper we evaluate Chebyshev polynomials of the second kind on a class of symmetric integer matrices, namely on adjacency matrices of simply laced Dynkin and extended Dynkin diagrams. As an application of these results we explicitly calculate minimal projective resolutions of simple modules of symmetric algebras with radical cube zero that are of finite and tame representation type.

Note on a Geometric Isogeny of K3 Surfaces

The paper establishes a correspondence relating two specific classes of complex algebraic K3 surfaces. The first class consists of K3 surfaces polarized by the rank-sixteen lattice H ⊕ E7 ⊕ E7. The second class consists of K3 surfaces obtained as minimal resolutions of double covers of the projective plane branched over a configuration of six lines. The correspondence underlies a geometric two-isogeny of K3 surfaces. 1 Geometric Two-Isogenies on K3 Surfaces Let X be an algebraic K3 surface defined over the field of complex numbers. A Nikulin (or symplectic) involution on X is an analytic automorphism of order two Φ: X→ X such that Φ∗(ω) = ω for any holomorphic two-form ω on X. This type of involution has many interesting properties (see [17, 18]), amongst which the most important are: (a) the fixed locus of Φ consists of precisely eight distinct points, and (b) the surface Y obtained as the minimal resolution of the quotient X/Φ is a K3 surface. Equivalently, one can construct Y as follows. Blow up the eight fixed points on X obtaining a new surface X. The Nikulin involution Φ extends to an involution Φ on X which has as fixed locus the disjoint union of the eight resulting exceptional curves. The quotient X/Φ is smooth and recovers the surface Y from above. In the context of the above construction, one has a degree-two rational map pΦ : X 99K Y with a branch locus given by eight disjoint rational curves (the even eight configuration in the sense of Mehran [16]). In addition, there is a push-forward morphism (see [12, 17]) (pΦ)∗ : H(X,Z)→ HY (1) mapping into the orthogonal complement in H(Y,Z) of the even eight curves. The metamorphosis of the surface X into Y is referred to in the literature as the Nikulin construction. The most well-known class of Nikulin involutions is given by the Shioda-Inose structures [12, 17, 18]. These consist of Nikulin involutions that satisfy two additional requirements. The first condition asks for the surface Y to be Kummer. The second requirement asserts that the morphism (1) induces a Hodge isometry between the lattices of transcendental cocycles TX(2) and TY. An effective criterion for a particular K3 surface X to admit a Shioda-Inose structure was given by Morrison [17]. In this paper, we shall work with another class of Nikulin involutions: fiber-wise translations by a section of order two in a jacobian elliptic fibration. This class of involutions was discussed by Van Geemen and Sarti [9]. Let us be precise: Definition 1.1. A Van Geemen-Sarti involution is an automorphism ΦX : X → X for which there exists a triple (φX, S1, S2) such that: (a) φX : X→ P is an elliptic fibration on X, (b) S1 and S2 are disjoint sections of φX, (c) S2 is an element of order two in the Mordell-Weil group MW(φX,S1), (d) ΦX is the involution obtained by extending the fiber-wise translations by S2 in the smooth fibers of φX using the group structure with neutral element given by S1. Under the above conditions, one says that the triple (φX, S1, S2) is compatible with the involution ΦX. Any given Van Geemen-Sarti involution is, in particular, a Nikulin involution. One can naturally regard a Van Geemen-Sarti involution ΦX as a fiber-wise two-isogeny between the original K3 surface X and the newly constructed K3 surface Y. Since ΦX acts as a translation by an element of order two in each of the smooth fibers ∗Department of Mathematics and Computer Science, University of Missouri-St. Louis, St. Louis MO 63108. e-mail: clinghera@umsl.edu †Department of Mathematical and Statistical Sciences, University of Alberta. Edmonton AB T6G 2G1. e-mail: doran@math.ualberta.ca

Minimal free resolutions of general points lying on cubic surfaces in [formula omitted

Casanellas has shown that a generalized version of Lorenzini’s Minimal Resolution Conjecture (as modified by Mustaţaˇ) holds for a set of t general points on a smooth cubic surface in P 3 , for certain specific values of t . We extend her work by verifying the conjecture for all t , and by allowing the cubic surface to have isolated double points.

Cellular resolutions of cointerval ideals

Minimal cellular resolutions of the edge ideals of cointerval hypergraphs are constructed. This class of d-uniform hypergraphs coincides with the complements of interval graphs (for the case d = 2), and strictly contains the class of ‘strongly stable’ hypergraphs corresponding to pure shifted simplicial complexes. The polyhedral complexes supporting the resolutions are described as certain spaces of directed graph homomorphisms, and are realized as subcomplexes of mixed subdivisions of the Minkowski sums of simplices. Resolutions of more general hypergraphs are obtained by considering decompositions into cointerval hypergraphs.

(p, λ)-Koszul algebras and modules

In this paper, the notions of (p, λ)-Koszul algebra and (p, λ)-Koszul module are introduced. Some criteria theorems for a positively graded algebra A to be (p, λ)-Koszul are given. The notion of weakly (p, λ)-Koszul module is defined as well and let WKλp(A) denote the category of weakly (p, λ)-Koszul modules. We show that M ∈ WKλp(A) if and only if it can be approximated by (p, λ)-Koszul submodules, which is equivalent to that G(M) is a (p, λ)-Koszul module, where G(M) denotes the associated graded module of M. As applications, the relationships of the minimal graded projective resolutions of M, G(M) and (p, λ)-Koszul submodules are established. In particular, for a module M ∈ WKλp(A) we prove that ⊕i≥0 ExtAi(M,A0) ∈ gr0(E(A)), we also get as a consequence that the finitistic dimension conjecture is valid in WKλp(A) under certain conditions.

Minimal resolutions of geometric [formula omitted]-modules

In this paper, we study minimal free resolutions for modules over rings of linear differential operators. The resolutions we are interested in are adapted to a given filtration, in particular to the so-called V -filtrations (see Oaku and Takayama (2001) [18] and Granger and Oaku (2004) [9]). We are interested in the module D x , t f s associated with germs of functions f 1 , … , f p , which we call a geometric module, and it is endowed with V -filtration along t 1 = ⋯ = t p = 0 . The Betti numbers of the minimal resolution associated with this data lead to analytical invariants for the germ of space defined by f 1 , … , f p . For p = 1 , we show that, under some natural conditions on f , the computation of the Betti numbers is reduced to a commutative algebra problem. This includes the case of an isolated quasi-homogeneous singularity, for which we give the Betti numbers explicitly. Moreover, for an isolated singularity, we characterize the quasi-homogeneity in terms of the minimal resolution.

Stability Conditions on An-Singularities

We study the spaces of locally finite stability conditions on the derived categories of coherent sheaves on the minimal resolutions of $A_n$-singularities supported at the exceptional sets. Our main theorem is that they are connected and simply-connected. The proof is based on the study of spherical objects in A. Ishii and H. Uehara, Autoequivalences of derived categories on the minimal resolutions of $A_n$-singularities on surfaces, J. Differential Geom., 71(3):385–435, 2005, and the homological mirror symmetry for $A_n$-singularities.

Blow-analytic equivalence of two variable real analytic function germs

Blow-analytic equivalence is a notion for real analytic function germs, introduced by Tzee-Char Kuo in order to develop real analytic equisingularity theory. In this paper we give complete characterisations of blow-analytic equivalence in the two dimensional case, in terms of the real tree model for the arrangement of real parts of Newton-Puiseux roots and their Puiseux pairs, and in terms of minimal resolutions. These characterisations show that in the two dimensional case the blow-analytic equivalence is a natural analogue of topological equivalence of complex analytic function germs. Moreover, we show that in the two-dimensional case the blow-analytic equivalence can be made cascade, and hence satisfies several geometric properties. It preserves, for instance, the contact order of real analytic arcs. In the general n n -dimensional case, we show that a singular real modification satisfies the arc-lifting property.

McKay correspondence for canonical orders

Canonical orders, introduced in the minimal model program for orders, are simultaneous generalisations of Kleinian singularities k [ [ s , t ] ] G k[[s,t]]^G , G > S L 2 G > SL_2 and their associated skew group rings k [ [ s , t ] ] ∗ G k[[s,t]]*G . In this paper, we construct minimal resolutions of canonical orders via non-commutative cyclic covers and skew group rings. This allows us to exhibit a derived equivalence between minimal resolutions of canonical orders and the skew group ring form of the canonical order in all but one case. The Fourier-Mukai transform used to construct this equivalence allows us to make explicit the numerical version of the McKay correspondence for canonical orders noted in Chan, Hacking, and Ingalls, which relates the exceptional curves of the minimal resolution to the indecomposable reflexive modules of the canonical order.

Drug impurity profiling: Method optimization on dissimilar chromatographic systems: Part I: pH optimization of the aqueous phase

The use of dissimilar chromatographic systems in drug impurity profiling can be very advantageous. Screening a new-drug impurity mixture on those systems not only enhances the chance that all impurities are revealed, but also allows choosing a suited system for further method development. In this paper several strategies were evaluated to predict the optimal pH (of the buffer used in the mobile phase) from the screening results. Four or five dissimilar stationary phases were screened at four pH values (between 2.5 and 9.4), in order to obtain maximal information about the composition of the sample and to select one column for the subsequent optimization. Different linear models (straight lines, 2nd and 3rd degree polynomials) based on these experiments were tested for their ability to predict the retention times ( t R ) of the impurities at intermediate pH values. The predicted t R values were then used to calculate minimal resolutions and eventually to select an optimal pH at which the highest minimal resolution is predicted. None of the applied models is accurate enough to predict correctly which peaks are worst separated at the indicated optimal pH. However, the best strategy (applying a second degree polynomial describing the t R measured at 3 consecutive screening pH values) did succeed in indicating an optimal pH at which a good separation of the impurities is obtained. Unfortunately, the resulting separation quality is not or only slightly better than the best separation obtained during screening. Therefore, it can be concluded that the most (time-) efficient approach to develop an impurity profile of a new drug is to screen it on four or five dissimilar columns at four different pH values and to retain the best screening conditions (without making predictions for intermediate conditions) for further optimization of the organic modifier composition of the mobile phase, and occasionally the temperature and the gradient. This is at least the case when the profiles have a complexity similar to those studied.

Generic representation theory of quivers with relations

The irreducible components of varieties parametrizing the finite dimensional representations of a finite dimensional algebra Λ are explored, with regard to both their geometry and the structure of the modules they encode. Provided that the base field is algebraically closed and has infinite transcendence degree over its prime field, we establish the existence and uniqueness (not up to isomorphism, but in a strong sense to be specified) of a generic module for any irreducible component C , that is, of a module which displays all categorically defined generic properties of the modules parametrized by C ; the crucial point of the existence statement – a priori almost obvious – lies in the description of such a module in a format accessible to representation-theoretic techniques. Our approach to generic modules over path algebras modulo relations, by way of minimal projective resolutions, is largely constructive. It is explicit for large classes of algebras of wild type. We follow with an investigation of the properties of such generic modules in terms of quiver and relations. The sharpest specific results on all fronts are obtained for truncated path algebras, that is, for path algebras of quivers modulo ideals generated by all paths of a fixed length; this class of algebras extends the comparatively thoroughly studied hereditary case, displaying many novel features.

Palindromic Resolutions of Simple Modules

For a finite dimensional hereditary k-algebra H of Dynkin type, it is known that the trivial extension T(H) has the property that minimal projective resolutions of T(H)-modules are periodic. We examine conditions in which T(H)-modules have additional symmetry in their resolutions—namely, their resolutions are “palindromic.” A more general framework in which “palindromic” resolutions occur is also presented.

Linearity defects of modules over commutative rings

This article concerns linear parts of minimal resolutions of finitely generated modules over commutative local, or graded rings. The focus is on the linearity defect of a module, which marks the point after which the linear part of its minimal resolution is acyclic. The results established track the change in this invariant under some standard operations in commutative algebra. As one of the applications, it is proved that a local ring is Koszul if and only if it admits a Koszul module that is Cohen–Macaulay of minimal degree. An injective analogue of the linearity defect is introduced and studied. The main results express this new invariant in terms of linearity defects of free resolutions, and relate it to other ring theoretic and homological invariants of the module.

Résolution de certains modules instables et fonction de partition de Minc

One constructs minimal injective resolutions for certain unstable modules that appear to be the mod 2 cohomology of Thom spectra. The terms of the resolution are tensor products of Brown–Gitler modules and Steinberg modules introduced by S. Mitchell and S. Priddy. A combinatorial result of Andrews shows that the alternating sum of the Poincaré series of the considered modules is zero. One gives homotopical applications of this result. To cite this article: D.H.H. Nguyen et al., C. R. Acad. Sci. Paris, Ser. I 347 (2009).

The classification of special Cohen–Macaulay modules

In this paper we completely classify all the special Cohen–Macaulay (=CM) modules corresponding to the exceptional curves in the dual graph of the minimal resolutions of all two dimensional quotient singularities. In every case we exhibit the specials explicitly in a combinatorial way. Our result relies on realizing the specials as those CM modules whose first Ext group vanishes against the ring R, thus reducing the problem to combinatorics on the AR quiver; such possible AR quivers were classified by Auslander and Reiten. We also give some general homological properties of the special CM modules and their corresponding reconstruction algebras.

Minimal resolutions via algebraic discrete Morse theory

Minimal resolutions via algebraic discrete Morse theory