We introduce graded $${\mathbb {E}}_{\infty }$$ -rings and modules over them, study their properties. construct projective schemes associated to connective {N}}$$ -graded in spectral algebraic geometry. Under some finiteness conditions, we show that the $$\infty $$ -category of almost perfect quasi-coherent sheaves a scheme $$\text { {Proj}}\,(A)$$ -ring A can be described terms $${{\mathbb {Z}...

The Galois ring GR$(4^\Delta)$ is the residue $Z_4[x]/(h(x))$, where $h(x)$ a basic primitive polynomial of degree $\Delta$ over $Z_4$. For any odd larger than $1$, we construct partition GR$(4^\Delta) \backslash \{0\}$ into $6$-subsets type $\{a,b,-a-b,-a,-b,a+b\}$ and $3$-subsets $\{c,-c,2c\}$ such that invariant under multiplication by nonzero element Teichmuller set in and, if not multiple ...

The title compound, C12H14N6O, consists of three pyrazole rings bound via nitro-gen to the distal ethane carbon of meth-oxy ethane. The dihedral angles between the three pyrazole rings are 67.62 (14), 73.74 (14), and 78.92 (12)°. In the crystal, mol-ecules are linked by bifurcated C-H,H⋯N hydrogen bonds, forming double-stranded chains along [001]. The chains are linked via C-H⋯O hydrogen bonds,...

In the title compound, C(19)H(22)N(2)O(2), the morpholine ring adopts an almost perfect normal chair conformation with puckering parameters Q(T), θ and ϕ of 0.5642 (18) Å, 177.32 (17) and ϕ = 10 (4)°, respectively. The two benzene rings make a dihedral angle of 42.67 (8)° with each other. An intra-molecular O-H⋯N hydrogen bond helps to stabilize the mol-ecular conformation. Aromatic C-H⋯π inter...

Using a local monomialization result of Knaf and Kuhlmann, we prove that the valuation ring an Abhyankar function field over perfect ground prime characteristic is Frobenius split. We show splitting sufficiently well-behaved center lifts to ring. also investigate properties valuations centered on arbitrary Noetherian domains characteristic. In contrast [arXiv:1507.06009], this paper emphasizes ...

We define and study the symmetric version of differential torsion theories. We prove that the symmetric versions of some of the existing results on derivations on right modules of quotients hold for derivations on symmetric modules of quotients. In particular, we prove that the symmetric Lambek, Goldie and perfect torsion theories are differential. We also study conditions under which a derivat...