Journal of Pure and Applied Algebra最新文献

In the representation-theoretic study of finite dimensional associative algebras over an algebraically closed field, Brenner introduced certain partially ordered sets known as hammocks to encode factorizations of maps between indecomposable finitely generated modules. In the context of domestic string algebras, Schröer introduced a simpler version of hammocks in his doctoral thesis that are bounded discrete linear orders. In this paper, we characterize the class of order types(=order isomorphism classes) of hammock linear orders for domestic string algebras as the bounded discrete ones amongst the class $L{O}_{\mathrm{fp}}$ of finitely presented linear orders–the smallest class of linear orders containing finite linear orders as well as ω, and that is closed under isomorphisms, order reversal, finite order sums and antilexicographic products.

In fact, we provide a multi-step algorithm to compute the order type of any closed interval in the hammock, and prove the correctness of this algorithm. A major step of this algorithm is the construction of a variation, which we call the arch bridge quiver, of a finite combinatorial gadget called the bridge quiver introduced by Schröer. He utilised the graph-theoretic properties of the bridge quiver for the computation of some representation-theoretic numerical invariants of domestic string algebras. The vertices of the bridge quiver are (representatives of cyclic permutations of) bands and its arrows are certain band-free strings. There is a natural but ill-behaved partial binary operation, ∘, on a superset of the set of bridges consisting of weak bridges such that bridges are precisely the ∘-irreducibles. We equip an even larger yet finite set of weak arch bridges with another partial binary operation, ${\circ }_H$, to obtain a finite category. The binary operation ${\circ }_H$ uses isomorphisms between hammocks and explicitly relies on the description of the domestic string algebra as a bound quiver algebra. Each weak arch bridge admits a unique ${\circ }_H$-factorization into arch bridges, i.e., the ${\circ }_H$-irreducibles.

Let a group A act on the group G coprimely. Suppose that the order of the fixed point subgroup $C_G\left(A\right)$ is not divisible by an arbitrary but fixed prime p. In the present paper we determine bounds for the p-length of the group G in terms of the order of A, and investigate how some A-invariant p-subgroups are embedded in G under various additional assumptions.

We introduce and study a category of algebras strongly connected with the structure of the Gelfand-Tsetlin subalgebras of the endomorphism algebras of Bott-Samelson bimodules. We develop a series of techniques that allow us to obtain optimal presentations for the many Gelfand-Tsetlin subalgebras appearing in the context of the Diagrammatic Soergel Category.

Let $k$ be an algebraically closed field with characteristic p different from 2. We generalize the notion of a weakly triangular decomposition in [7] to the super case called a super weakly triangular decomposition. We show that the underlying even category of locally finite-dimensional left A-supermodules is an upper finite fully stratified category in the sense of [6, Definition 3.34] if the superalgebra A admits an upper finite super weakly triangular decomposition. In particular, when A is the locally unital superalgebra associated with the cyclotomic oriented Brauer-Clifford supercategory in [1], the Grothendieck group of the category of left A-supermodules admitting finite standard flags has a $g$-module structure that is isomorphic to the tensor product of an integrable lowest weight and an integrable highest weight $g$-module, where $g$ is the complex Kac-Moody Lie algebra of type $A_{2\ell }^{\left(2\right)}$ (resp., $B_{\infty }$) if $p=2\ell +1$ (resp., $p=0$).

In this paper we study marked numerical Godeaux surfaces with special bicanonical fibers. Based on the construction method of marked Godeaux surfaces in [12] we give a complete characterization for the existence of hyperelliptic bicanonical fibers and torsion fibers. Moreover, we describe how the families of Reid and Miyaoka with torsion $Z/3Z$ and $Z/5Z$ arise in our homological setting.

Let $n\in N$ and let K be a field with a henselian discrete valuation of rank n with hereditarily euclidean residue field. Let $F/K$ be a function field in one variable. It is known that every sum of squares is a sum of 3 squares. We determine the order of the group of nonzero sums of 3 squares modulo sums of 2 squares in F in terms of equivalence classes of certain discrete valuations of F of rank at most n. In the case of function fields of hyperelliptic curves of genus g, K.J. Becher and J. Van Geel showed that the order of this quotient group is bounded by $2^{n(g+1)}$. We show that this bound is optimal. Moreover, in the case where $n=1$, we show that if $F/K$ is a hyperelliptic function field such that the order of this quotient group is $2^{g+1}$, then F is nonreal.

Let $k$ be a field of characteristic $p>0$, V a finite-dimensional $k$-vector-space, and G a finite p-group acting $k$-linearly on V. Let $S=\mathrm{Sym}{V}^{⁎}$. Confirming a conjecture of Shank-Wehlau-Broer, we show that if $S^G$ is a direct summand of S, then $S^G$ is a polynomial ring, in the following cases:

• (a)

  $k=F_p$ and ${\dim }_{k}V=4$; or

• (b)

  $\left|G\right|=p^3$.

Let $G\le {\mathrm{SL}}_{n+1}\left(C\right)$ act on $R=C[X_1,\dots ,X_{n+1}]$ by change of variables. Then, the skew-group algebra $R⁎G$ is bimodule $(n+1)$-Calabi-Yau. In certain circumstances, this algebra admits a locally finite-dimensional grading of Gorenstein parameter 1, in which case it is the $(n+1)$-preprojective algebra of its n-representation infinite degree 0 piece, as defined in [10]. If the group G is abelian, the $(n+1)$-preprojective algebra is said to be of type $\tilde{A}$. For a given group G, it is not obvious whether $R⁎G$ admits such a grading making it into an $(n+1)$-preprojective algebra. We study the case when $n=2$ and G is abelian. We give an explicit classification of groups such that $R⁎G$ is 3-preprojective by constructing such gradings. This is possible as long as G is not a subgroup of ${\mathrm{SL}}_2\left(C\right)$ and not $C_2\times C_2$. For a fixed G, the algebra $R⁎G$ admits different 3-preprojective gradings, so we associate a type to a grading and classify all types. Then we show that gradings of the same type are related by a certain kind of mutation. This gives a classification of 2-representation infinite algebras of type $\tilde{A}$. The involved quivers are those arising from hexagonal dimer models on the torus, and the gradings we consider correspond to perfect matchings on the dimer, or equivalently to periodic lozenge tilings of the plane. Consequently, we classify these tilings up to flips, which correspond to the mutation we consider.

Let F be a field of characteristic 2, π an n-fold bilinear Pfister form over F and φ an arbitrary quadratic form over F. In this note, we investigate Witt index, defect, total isotropy index and higher isotropy indices of φ and $\pi \otimes \phi$ and prove relations among the indices of these two forms over certain field extensions.