This is a continuation of our study of the foundations of D-branes from the viewpoint of Grothendieck in the region of the related Wilson’s theory-space where “branes” are still branes. In this work, we focus on D-strings and construct the moduli stack of morphisms from Azumaya prestable curves C with a fundamental module E to a fixed target Y of a given combinatorial type. Such a morphism give...

LetG be a quasisimple, connected, and simply connected algebraic group defined and split over the field k of characteristic p > 0. In this paper, we are interested in small modules for G; for us, small modules are those with dimension ≤ p. By results of Jantzen [Jan96] one knows that anyGmodule V with dimV ≤ p is semisimple. (We always understand aG-module V to be given by a morphism of algebra...

(f) Explicitly verify that the the 5 global sections above generate the invertible sheaf OP2(2)|X(−C) at every point of X. Therefore there is a unique morphism φ : X → P such that φOP4(1) equals OP2(2)|X(−C) and the pullback of the homogeneous coordinates are the 5 sections above. (g) Also verify that y0, y1, y2, y3 are identically zero on the line L. Therefore φ contracts L to the point p = [0...

Brlek et al. conjectured in 2008 that any fixed point of a primitive morphism with finite palindromic defect is either periodic or its palindromic defect is zero. Bucci and Vaslet disproved this conjecture in 2012 by a counterexample over ternary alphabet. We prove that the conjecture is valid on binary alphabet. We also describe a class of morphisms over multiliteral alphabet for which the con...

Inspired by the classical theory of modules over a monoid, we introduce the natural notion of module over a monad. The associated notion of morphism of left modules (”linear” natural transformations) captures an important property of compatibility with substitution, not only in the so-called homogeneous case but also in the heterogeneous case where ”terms” and variables therein could be of diff...

Representations of the twisted Heisenberg-Virasoro algebra at level zero. Abstract. We describe the structure of the irreducible highest weight modules for the twisted Heisenberg-Virasoro Lie algebra at level zero. We prove that such a module is either isomorphic to a Verma module or to a quotient of two Verma modules.

We introduce the notion of cumulants as applied to linear maps between associative (or commutative) algebras that are not compatible with algebraic product structure. These have a close relationship $$A_{\infty }$$ and $$C_{\infty morphisms, which classical homotopical tools for analyzing deformations algebraically maps. look at these two different perspectives understand how infinity-morphisms...

Let $T=\bigl(\begin{smallmatrix}A&0\U\&B\end{smallmatrix}\bigr)$ be a formal triangular matrix ring, where $A$ and $B$ are rings $U$ is $(B, A)$-bimodule. We prove: (1) If $U\_{A}$ ${B}U$ have finite flat dimensions, then left $T$-module $\bigl(\begin{smallmatrix}M\_1\ M\_2\end{smallmatrix}\bigr){\varphi^{M}}$ Ding projective if only $M\_1$ $M\_2/{\operatorname{im}(\varphi^{M})}$ the morphism $...

The purpose of this paper is to study certain notions metric positivity called “minimal extension property” for the lowest nonzero piece in Hodge filtration a module. Let X be complex manifold and let $$\mathcal {M}$$ polarized pure module on with strict support X. $$F_p\mathcal smallest filtration. Assume that smooth outside closed analytic subset Z $$j:X\setminus \hookrightarrow X$$ open embe...

We introduce a new concept in the area of formal logic: axioms for model morphisms. We work in the setting of specification languages that define the semantics of a theory as a category of models. While it is routine to use axioms to specify the class of models of a theory, there has so far been no analogue to systematically specify the morphisms between these models. This leads to subtle probl...