Composite of irreducible morphisms in the bounded derived category

We study necessary and sufficient conditions for the existence of n irreducible morphisms in the bounded derived category of an Artin algebra, with non-zero composite in the n+1-power of the radical. In the case of , the bounded derived category of an Ext-finite hereditary k-category with tilting object, such irreducible morphisms exist if and only if H is derived equivalent to a wild hereditary algebra or to a wild canonical algebra. We also characterize the cluster tilted algebras having such irreducible morphisms.     

On the relation between cluster and classical tilting

Let be a triangulated category with a cluster tilting subcategory U. The quotient category is abelian; suppose that it has finite global dimension.We show that projection from to sends cluster tilting subcategories of to support tilting subcategories of , and that, in turn, support tilting subcategories of can be lifted uniquely to weak cluster tilting subcategories of .     

Degrees of irreducible morphisms in standard components

We study the left degree of an irreducible morphism with X and Y_i indecomposable modules in a standard component of the Auslander-Reiten quiver, for 1≤i≤r. Two criteria to determine whether the left degree of these irreducible morphisms is finite or infinite are given, for standard algebras. We also study which of them has left degree two.     

Hermitian vector bundles and extension groups on arithmetic schemes. I. Geometry of numbers

We define and investigate extension groups in the context of Arakelov geometry. The “arithmetic extension groups” we introduce are extensions by groups of analytic types of the usual extension groups attached to OX-modules F and G over an arithmetic scheme X. In this paper, we focus on the first arithmetic extension group - the elements of which may be described in terms of admissible short exact sequences of hermitian vector bundles over X - and we especially consider the case when X is an “arithmetic curve”, namely the spectrum SpecOK of the ring of integers in some number field K. Then the study of arithmetic extensions over X is related to old and new problems concerning lattices and the geometry of numbers.Namely, for any two hermitian vector bundles and over X:=SpecOK, we attach a logarithmic size to any element α of , and we give an upper bound on in terms of slope invariants of and . We further illustrate this notion by relating the sizes of restrictions to points in P¹(Z) of the universal extension over to the geometry of PSL₂(Z) acting on Poincaré's upper half-plane, and by deducing some quantitative results in reduction theory from our previous upper bound on sizes. Finally, we investigate the behaviour of size by base change (i.e., under extension of the ground field K to a larger number field K^′): when the base field K is Q, we establish that the size, which cannot increase under base change, is actually invariant when the field K^′ is an abelian extension of K, or when is a direct sum of root lattices and of lattices of Voronoi's first kind.The appendices contain results concerning extensions in categories of sheaves on ringed spaces, and lattices of Voronoi's first kind which might also be of independent interest.     

Integral bilinear forms, Coxeter transformations and Coxeter polynomials of finite posets

Linear algebra technique in the study of linear representations of finite posets is developed in the paper. A concept of a quadratic wandering on a class of posets I is introduced and finite posets I are studied by means of the four integral bilinear forms (1.1), the associated Coxeter transformations, and the Coxeter polynomials (in connection with bilinear forms of Dynkin diagrams, extended Dynkin diagrams and irreducible root systems are also studied). Bilinear equivalences between some of the forms are established and equivalences with the bilinear forms of Dynkin diagrams and extended Dynkin diagrams are discussed. A homological interpretation of the bilinear forms (1.1) is given and Z-bilinear equivalences between them are discussed. By applying well-known results of Bongartz, Loupias, and Zavadskij-Shkabara, we give several characterisations of posets I, with the Euler form weakly positive (resp. with the reduced Euler form weakly positive), and posets I, with the Tits form weakly positive.     

Independence of ? in Lafforgue's theorem

Let X be a smooth curve over a finite field of characteristic p, let ?≠p be a prime number, and let be an irreducible lisse -sheaf on X whose determinant is of finite order. By a theorem of L. Lafforgue, for each prime number ?′≠p, there exists an irreducible lisse -sheaf on X which is compatible with , in the sense that at every closed point x of X, the characteristic polynomials of Frobenius at x for and are equal. We prove an “independence of ?” assertion on the fields of definition of these irreducible ?′-adic sheaves : namely, that there exists a number field F such that for any prime number ?′≠p, the -sheaf above is defined over the completion of F at one of its ?′-adic places.     

On the classification of non-normal cubic hypersurfaces

In this article we study the classification of non-normal cubic hypersurfaces over an algebraically closed field K of arbitrary characteristic. Let be an irreducible non-normal cubic hypersurface. If r≥5, then X is necessarily a cone (Remark 2.3). In view of this fact it suffices to classify irreducible non-normal cubic hypersurfaces for r≤4. We prove that there are precisely five non-normal cubic equations (resp. six non-normal cubic equations) when (resp. when is either 2 or 3), up to projective equivalence. Also we describe the normalization of X in detail.     

Toric complexes and Artin kernels

A simplicial complex L on n vertices determines a subcomplex TL of the n-torus, with fundamental group the right-angled Artin group GL. Given an epimorphism χ:GL→Z, let be the corresponding cover, with fundamental group the Artin kernel Nχ. We compute the cohomology jumping loci of the toric complex TL, as well as the homology groups of with coefficients in a field k, viewed as modules over the group algebra kZ. We give combinatorial conditions for to have trivial Z-action, allowing us to compute the truncated cohomology ring, . We also determine several Lie algebras associated to Artin kernels, under certain triviality assumptions on the monodromy Z-action, and establish the 1-formality of these (not necessarily finitely presentable) groups.     

Solution of a nonsymmetric algebraic Riccati equation from a two-dimensional transport model

For the steady-state solution of an integral-differential equation from a two-dimensional model in transport theory, we shall derive and study a nonsymmetric algebraic Riccati equation B^--XF^--F⁺X+XB⁺X=0, where , and with a nonnegative matrix P, positive diagonal matrices D^±, and nonnegative parameters f, and . We prove the existence of the minimal nonnegative solution X^∗ under the physically reasonable assumption , and study its numerical computation by fixed-point iteration, Newton’s method and doubling. We shall also study several special cases; e.g. when and P is low-ranked, then is low-ranked and can be computed using more efficient iterative processes in U and V. Numerical examples will be given to illustrate our theoretical results.     

Homology of dihedral quandles

We solve the conjecture by R. Fenn, C. Rourke and B. Sanderson that the rack homology of dihedral quandles satisfies for p odd prime [T. Ohtsuki, Problems on invariants of knots and 3-manifolds, Geom. Topol. Monogr. 4 (2002) 377-572, Conjecture 5.12]. We also show that contains Z_p for n≥3. Furthermore, we show that the torsion of is annihilated by 3. We also prove that the quandle homology contains Z_p for p odd prime. We conjecture that for n>1 quandle homology satisfies: , where f_n are “delayed” Fibonacci numbers, that is, f_n=f_n−1+f_n−3 and f(1)=f(2)=0,f(3)=1. Our paper is the first step in approaching this conjecture.     

The third homology of the special linear group of a field

We prove that for any infinite field F, the map is an isomorphism for all n≥3. When n=2 the cokernel of this map is naturally isomorphic to , where is the nth Milnor K-group of F. We deduce that the natural homomorphism from to the indecomposable K₃ of F, , is surjective for any infinite field F.     

Almost fully decomposable infinite rank lattices over orders

Let Λ be an order over a Dedekind domain R with quotient field K. An object of , the category of R-projective Λ-modules, is said to be fully decomposable if it admits a decomposition into (finitely generated) Λ-lattices. In a previous article [W. Rump, Large lattices over orders, Proc. London Math. Soc. 91 (2005) 105-128], we give a necessary and sufficient criterion for R-orders Λ in a separable K algebra A with the property that every is fully decomposable. In the present paper, we assume that is separable, but that the p-adic completion A_p is not semisimple for at least one . We show that there exists an , such that KL admits a decomposition KL=M₀⊕M₁ with finitely generated, where L∩M₁ is fully decomposable, but L itself is not fully decomposable.     

Mesh geometries of root orbits of integral quadratic forms

Integral quadratic forms q:Zⁿ→Z, with n≥1, and the sets R_q(d)={v∈Zⁿ;q(v)=d}, with d∈Z, of their integral roots are studied by means of mesh translation quivers defined by Z-bilinear morsifications b_A:Zⁿ×Zⁿ→Z of q, with Z-regular matrices A∈M_n(Z). Mesh geometries of roots of positive definite quadratic forms q:Zⁿ→Z are studied in connection with root mesh quivers of forms associated to Dynkin diagrams A_n,D_n,E₆,E₇,E₈ and the Auslander-Reiten quivers of the derived category D^b(R) of path algebras R=KQ of Dynkin quivers Q. We introduce the concepts of a Z-morsification b_A of a quadratic form q, a weighted Φ_A-mesh of vectors in Zⁿ, and a weighted Φ_A-mesh orbit translation quiver Γ(R_q,Φ_A) of vectors in Zⁿ, where R_q?R_q(1) and Φ_A:Zⁿ→Zⁿ is the Coxeter isomorphism defined by A. The existence of mesh geometries on R_q is discussed. It is shown that, under some assumptions on the morsification b_A:Zⁿ×Zⁿ→Z, the set admit a Φ_A-orbit mesh quiver , where Φ_A:Zⁿ→Zⁿ is the Coxeter isomorphism defined by A. Moreover, splits into three infinite connected components , , and , where are isomorphic to a translation quiver Z⋅Δ, with Δ an extended Dynkin quiver, and has the shape of a sand-glass tube.     

Derivative of power series attached to Γ-transform of p-adic measures

We extend the result of Anglès (2007) [1], namely for the Iwasawa power series . For the derivative , a numerical polynomial Q on Zp, and a prime π in over p, we show that if and only if i.e. for all x∈Zp. This result comes from a similar assertion for the power series attached to the Γ-transform of a p-adic measure which is related to a certain rational function in .     

Preservers of spectral radius, numerical radius, or spectral norm of the sum on nonnegative matrices

Let be the set of entrywise nonnegative n×n matrices. Denote by r(A) the spectral radius (Perron root) of . Characterization is obtained for maps such that r(f(A)+f(B))=r(A+B) for all . In particular, it is shown that such a map has the form

     

Polar syzygies in characteristic zero: The monomial case

Given a set of forms , where k is a field of characteristic zero, we focus on the first syzygy module Z of the transposed Jacobian module , whose elements are called differential syzygies of . There is a distinct submodule P⊂Z coming from the polynomial relations of through its transposed Jacobian matrix, the elements of which are called polar syzygies of . We say that is polarizable if equality P=Z holds. This paper is concerned with the situation where are monomials of degree 2, in which case one can naturally associate to them a graph with loops and translate the problem into a combinatorial one. The main result is a complete combinatorial characterization of polarizability in terms of special configurations in this graph. As a consequence, we show that polarizability implies normality of the subalgebra and that the converse holds provided the graph is free of certain degenerate configurations. One main combinatorial class of polarizability is the class of polymatroidal sets. We also prove that if the edge graph of has diameter at most 2 then is polarizable. We establish a curious connection with birationality of rational maps defined by monomial quadrics.