Archiv der Mathematik最新文献

In this paper, we present a variety of statements that are in the spirit of the famous theorem of Pascal, often referred to as the “Mystic Hexagon”. We give explicit equations describing the conditions for (d+4) points to lie on rational normal curves. A collection of problems of Pascal type are considered for quadric surfaces in ({mathbb {P}}^3). Finally we re-prove, using computer algebra methods, a remarkable theorem of Richmond, Segre, and Brown for quadrics in ({mathbb {P}}^4) containing five general lines.

In this article, we compare the cohomology between the categories of modules of the diagram algebras and the categories of modules of its input algebras. Our main result establishes a sufficient condition for exact split pairs between these two categories, analogous to a work by Diracca and Koenig (J Pure Appl Algebra 212:471–485, 2008). To be precise, we prove the existence of the exact split pairs in A-Brauer algebras, cyclotomic Brauer algebras, and walled Brauer algebras with their respective input algebras.

We show that if C is a smooth projective curve and (mathfrak {d}) is a (mathfrak {g}^{n}_{2n}) on C, then we obtain a rational map (textrm{Sym}^{n}(C)dashrightarrow mathfrak {d}) whose fibers can be related in an interesting way to Gunning multisecants of the Kummer variety of JC. This generalizes previous work done by the first author with Codogni and Salvati Manni.

We prove the finiteness of the genus of finite-dimensional division algebras over many infinitely generated fields. More precisely, let K be a finite field extension of a field which is a purely transcendental extension of infinite transcendence degree of some subfield. We show that if ({mathcal D}) is a central division K-algebra, then (textbf{gen}({mathcal D})) consists of Brauer classes ([{mathcal D}']) such that ([{mathcal D}]) and ([{mathcal D}']) generate the same subgroup of (text {Br} (K)). In particular, the genus of any division K-algebra of exponent 2 is trivial. Note that the family of such fields is closed under finitely generated extensions. Moreover, if (text {char}(K) ne 2), we prove that the genus of a simple algebraic group of type (textrm{G}_2) over such a field K is trivial.

We study mixed multiquadratic field extensions as splitting fields for central simple algebras of exponent 2 in characteristic 2. As an application, we provide examples of nonexcellent mixed biquadratic field extensions.

Let (mathbb {k}) be a characteristic zero Dedekind domain, S be a (mathbb {k})-algebra, and (Tsubseteq S) be a full rank subalgebra. Suppose the algebra T is symmetric. It is important to know when T is a maximally symmetric subalgebra of S, i.e., no (mathbb {k})-subalgebra C satisfying (Tsubsetneq Csubseteq S) is symmetric. In this note, we establish a useful sufficient condition for this using a notion of a quasi-unit of an algebra. This condition is used to obtain an old and a new result on maximal symmetricity for generalized Schur algebras corresponding to certain Brauer tree algebras. The old result was used in our work with Evseev on RoCK blocks of symmetric groups. The new result will be used in our forthcoming work on RoCK blocks of double covers of symmetric groups.

Let ((Q, mathfrak {n} )) be a regular local ring and let (f_1, ldots , f_c in mathfrak {n} ^2) be a Q-regular sequence. Set ((A, mathfrak {m} ) = (Q/(textbf{f} ), mathfrak {n} /(textbf{f} ))). Further assume that the initial forms (f_1^*, ldots , f_c^*) form a (G(Q) = bigoplus _{n ge 0}mathfrak {n} ^i/mathfrak {n} ^{i+1})-regular sequence. Without loss of any generality, assume (operatorname {ord}_Q(f_1) ge operatorname {ord}_Q(f_2) ge cdots ge operatorname {ord}_Q(f_c)). Let M be a finitely generated A-module and let ((mathbb {F} , partial )) be a minimal free resolution of M. Then we prove that (operatorname {ord}(partial _i) le operatorname {ord}_Q(f_1) - 1) for all (i gg 0). We also construct an MCM A-module M such that (operatorname {ord}(partial _{2i+1}) = operatorname {ord}_Q(f_1) - 1) for all (i ge 0). We also give a considerably simpler proof regarding the periodicity of ideals of minors of maps in a minimal free resolution of modules over arbitrary complete intersection rings (not necessarily strict).

We show that R. Thompson’s group T is a maximal subgroup of the group V. The argument provides examples of foundational calculations which arise when expressing elements of V as products of transpositions of basic clopen sets in the Cantor space (mathfrak {C}).

We characterize the weights (w, v_1, v_2, dots , v_m ) for which the weak-type multilinear gradient inequality

holds for all (f_1, f_2, dots , f_m in C_c^{infty }({mathbb {R}}^n)) in the case (frac{1}{p} = frac{1}{p_1}+frac{1}{p_2}+ cdots + frac{1}{p_m}).

Using the decomposition of Jacobians with group action, we prove the non-existence of some Shimura subvarieties in the moduli space of ppav (A_{g}) arising from families of dihedral and quaternionic covers of the complex projective line ({{mathbb {P}}}^1).