On the geometry of lattices and finiteness of Picard groups

Let (K, O, k) be a p-modular system with k algebraically closed and O unramified, and let Λ be an O-order in a separable K-algebra. We call a Λ-lattice L rigid if Ext1Λ(L, L) = 0, in analogy with the definition of rigid modules over a finite-dimensional algebra. By partitioning the Λ-lattices of a given dimension into “varieties of lattices”, we show that there are only finitely many rigid Λ-lattices L of any given dimension. As a consequence we show that if the first Hochschild cohomology of Λ vanishes, then the Picard group and the outer automorphism group of Λ are finite. In particular the Picard groups of blocks of finite groups defined over O are always finite

The Picard group of an order and Külshammer reduction

Let (K, O, k) be a p-modular system and assume k is algebraically closed. We show that if Λ is an O-order in a separable K-algebra, then PicO(Λ) carries the structure of an algebraic group over k. As an application to the modular representation theory of finite groups, we show that a reduction theorem by Kulshammer concerned with Donovan’s conjecture remains valid over O

Blocks with a generalized quaternion defect group and three simple modules over a 2-adic ring

We show that two blocks of generalized quaternion defect with three simple modules over a sufficiently large 2-adic ring O are Morita-equivalent if and only if the corresponding blocks over the residue field of O are Morita-equivalent. As a corollary we show that any two blocks defined over O with three simple modules and the same generalized quaternion defect group are derived equivalent

On solvability of the first Hochschild cohomology of a finite-dimensional algebra

For an arbitrary finite-dimensional algebra $A$, we introduce a general approach to determining when its first Hochschild cohomology ${\rm HH}^1(A)$, considered as a Lie algebra, is solvable. If $A$ is moreover of tame or finite representation type, we are able to describe ${\rm HH}^1(A)$ as the direct sum of a solvable Lie algebra and a sum of copies of $\mathfrak{sl}_2$. We proceed to determine the exact number of such copies, and give an explicit formula for this number in terms of certain chains of Kronecker subquivers of the quiver of $A$. As a corollary, we obtain a precise answer to a question posed by Chaparro, Schroll and Solotar

A counterexample to the first Zassenhaus conjecture

Hans J. Zassenhaus conjectured that for any unit u of finite order in the integral group ring of a finite group G there exists a unit a in the rational group algebra of G such that a−1· u · a = ±g for some g ∈ G. We disprove this conjecture by first proving general results that help identify counterexamples and then providing an infinite number of examples where these results apply. Our smallest example is a metabelian group of order 27·32·5·72·192 whose integral group ring contains a unit of order 7 · 19 which, in the rational group algebra, is not conjugate to any element of the form ±g

A reduction theorem for tau -rigid modules

We prove a theorem which gives a bijection between the support τ -tilting modules over a given finite-dimensional algebra A and the support τ -tilting modules over A / I, where I is the ideal generated by the intersection of the center of A and the radical of A. This bijection is both explicit and well-behaved. We give various corollaries of this, with a particular focus on blocks of group rings of finite groups. In particular we show that there are τ -tilting-finite wild blocks with more than one simple module. We then go on to classify all support τ -tilting modules for all algebras of dihedral, semidihedral and quaternion type, as defined by Erdmann, which include all tame blocks of group rings. Note that since these algebras are symmetric, this is the same as classifying all basic two-term tilting complexes, and it turns out that a tame block has at most 32 different basic two-term tilting complexes. We do this by using the aforementioned reduction theorem, which reduces the problem to ten different algebras only depending on the ground field k, all of which happen to be string algebras. To deal with these ten algebras we give a combinatorial classification of all τ -rigid modules over (not necessarily symmetric) string algebras

Basic Orders for Defect Two Blocks of ℤpΣn

We show how basic orders for defect two blocks of symmetric groups over the ring of p-adic integers can be constructed by purely combinatorial means

The p-adic group ring of

In the present article we show that the Zp[ζpf−1]-order Zp[ζpf−1]SL2(pf) can be recognized among those orders whose reduction modulo p is isomorphic to FpfSL2(pf) using only ring-theoretic properties. In other words we show that FpfSL2(pf) lifts uniquely to a Zp[ζpf−1]-order, provided certain reasonable conditions are imposed on the lift. This proves a conjecture made by Nebe in [8] concerning the basic order of Z2[ζ2f−1]SL2(2f)

Influence of environmental factors on the growth of the corn plant under field conditions

A study, extending over a 4-year period, has been made to determine the effect of environmental factors on the response of corn plants grown under field conditions. The principal factors of the environment which were measured were available soil moisture, air temperature, evaporation and relative humidity. Factors of the environment, during each growing season, were further modified by varying the rate of planting from one to five plants per hill. Height of plants, increase in leaf area, size of stalks, dry weight of vegetative and reproductive parts, rate of photosynthesis, carbohydrate fractions of leaf samples, and nitrogen content of the ears were measured at regular intervals throughout the growing season. The relation between the plant responses and environmental factors, particularly as modified by rate of planting, was studied. The rainfall of the growing season, May, June, July and August, is a better criterion for predicting corn yield than the rainfall of the entire year. Where corn was planted one to five plants per hill there was usually less available soil moisture in the thicker rates of planting than in the thinner rates. The difference was greater in July and August during periods without rain. During the day relative humidity was 3 to 5 percent lower where plants were planted one plant per hill than where planted five plants per hill. At night the relative humidity was higher in the thinner rates of planting. The rate of evaporation from porous porcelain atmometer cups in cubic centimeters was 22 percent greater during July and August where there was one plant per hill than where there were five plants. Height of the corn plants was not appreciably influenced by rate of planting. There was considerable difference in the leaf area per plant from different rates of planting. In 1932 the maximum leaf area, per plant, was 8,900, 7,908 and 6,573 square centimeters respectively where there were one, three and five plants per hill. Loss of effective leaf area due to firing began earlier and proceeded at a more rapid rate in the thicker rates of planting. At maturity the average cross-sectional area, at the level of the ground, of stalks, where there were three plants per hill, was 60 percent as large as where there was one plant. Stalks planted, five plants per hill, were only 40 percent as large as where there was one plant. One hundred hills with one, three and five plants per hill produced a total of 141, 265 and 359 ears, respectively, in 1932. The same year 42 percent of the ears were less than 15 centimeters long- nubbins -where there were five plants per hill, while -only 12 percent of the ears were nubbins where there was one plant per hill. Rate of planting does not significantly influence the rate of food making per unit of leaf area as determined by increase in dry weight of leaf samples collected at 4 :30 a. m. and 4 p. m. Data from these experiments show that rate of food making is proportional to area and not to dry weight of leaf samples. The quantity of alcohol-soluble carbohydrate fractions separated as diastase extract, dextrins and acid hydrolyzable were determined for leaf samples collected at 4 :30 a. m. and 4 p. m. the same day from different rates of planting. There was a statistically significant difference in the non-reducing sugars and diastase extract between samples from different rates of planting and in the reducing and non-reducing sugars, and acid hydrolyzable between samples collected at 4 :30 a. m. and 4 p. m. An analysis of variance showed that there was a significant difference in the quantity of nitrogen in kernel and cob samples attributable to date of collection. There was no significant difference in the quantity of nitrogen in the kernels attributable to rate of planting, while the quantity in the cob samples was slightly significant