In this paper, we introduce the atom spectrum of an abelian category as a topological space consisting of all the equivalence classes of monoform objects. In terms of the atom spectrum, we give a classification of Serre subcategories of an arbitrary noetherian abelian category. Moreover we show that the atom spectrum of a locally noetherian Grothendieck category is homeomorphic to its Ziegler spectrum.Comment: 15 page

We introduce systematic methods to construct Grothendieck categories from colored quivers and develop a theory of the specialization orders on the atom spectra of Grothendieck categories. We show that any partially ordered set is realized as the atom spectrum of some Grothendieck category, which is an analog of Hochster's result in commutative ring theory. We also show that there exists a Grothendieck category which has empty atom spectrum but has nonempty injective spectrum.Comment: 39 page

The participants could not distinguish between the press needle and a placebo, and the data from the press needle group suggested a specific influence on patients with LBP. These findings imply that the press needle and a placebo provide an effective means of realizing a double-blind setting for clinical studies of acupuncture.

We study the elliptic algebras Q n,k (E, τ ) introduced by Feigin and Odesskii as a generalization of Sklyanin algebras. They form a family of quadratic algebras parametrized by coprime integers n > k ≥ 1, an elliptic curve E, and a point τ ∈ E. We compare several different definitions of these algebras and provide proofs of various statements about them made by Feigin and Odesskii. For example, we show that Q n,k (E, 0), and Q n,n−1 (E, τ ) for generic τ , are polynomial rings on n variables. We also show thatThis paper is the first of several we are writing about the algebras Q n,k (E, τ ). Contents 1. Introduction 1.1. Notation and conventions 1.2. The algebras Q n,k (E, τ ) 1.3. The contents of subsequent papers 1.4. The contents of this paper 1.5. Acknowledgements 2. Theta functions in one variable 2.1. The spaces Θ n (Λ) and Riemann's theta function 2.2. The standard basis for Θ n (Λ) 2.3. Θ n (Λ) as a representation of the Heisenberg group 2.4. Embedding E in P n−1 via Θ n (Λ) 2.5. Another basis for Θ n (Λ) 3. The algebras Q n,k (E, τ ) 3.1. The definition of Q n,k (E, τ ) and R n,k (E, τ ) when τ / ∈ 1 n Λ 3.2. The definition of Q n,k (E, τ ) and R n,k (E, τ ) when τ ∈ 1 n Λ 3.3. Isomorphisms and anti-isomorphisms 3.4. The Heisenberg group acts as automorphisms of Q n,k (E, τ ) 3.5. Another set of relations for Q n,k (E, τ ) 4. Twisting Q n,k (E, τ ) 4.1. Twists 4.2. The twists of Q n,k (E, τ ) induced from translations by n-torsion points 5. Q n,k (E, τ ) when τ ∈ E[n] References

Abstract. We define the extension group between an atom and an object in a locally noetherian Grothendieck category as a module over a skew field. We show that the dimension of the i-th extension group between an atom and an object coincides with the i-th Bass number of the object with respect to the atom. As an application, we give a bijection between the E-stable subcategories closed under arbitrary direct sums and direct summands and the subsets of the atom spectrum and show that such subcategories are also closed under extensions, kernels of epimorphisms, and cokernels of monomorphisms. We show some relationships to the theory of prime ideals in the case of noetherian algebras.