Reference request for RR-index

Let R be a noetherian domain with field of fractions F, let V be a finite-dimensional F-vector space, and let M,N⊆V be R-lattices in V (finitely generated R-submodules of V containing a basis for V over F). We define the R-index of N in M, written [M:N]R, to be the R-submodule of F generated by … Read more

An order in $\mathbb Q[G]$ which is a maximal $\mathbb Z_p$-order in $\mathbb Q_p[G]$ for finitely many primes $p$

Let $G$ be a finite group and $S$ a finite set of prime numbers. I know that every separable $\mathbb Q$-algebra $A$ contains a maximal $\mathbb Z$-order but I wonder if the following is true. Is there a $\mathbb Z$-order $\Lambda$ in $\mathbb Q[G]$ which contains $\mathbb Z[G]$ and satisfies the following two conditions 1) … Read more

Endomorphisms of free modules and extension of scalars

I asked this question on Mathematics Stackexchange, but got no answer. Let B be a commutative ring with 1, let A be a subring such that any unit of B which belongs to A is a unit of A, and let ϕ:F→F be an endomorphism of a free A-modules F such that B⊗Aϕ:B⊗AF→B⊗AF is an … Read more

What kind of module is this?

Recall that, if R is a commutative ring, then a suitably finite R-module M is projective if and only if the localization Mm is a direct sum of finitely many copies of Rm for every maximal ideal m⊆R. Consider the following alternate condition for an R-module M: each Mm is a direct sum of finitely … Read more

Holonomic sections C∞(M)C^\infty(M)-generate jet bundle

Given a vector bundle E→M with a corresponding k-th jet bundle JkE→M, denote by jk:Γ(E)→Γ(JkE) the k-th jet prolongation (k∈N∪{0}) and recall that a section σ∈Γ(JkE) is holonomic if it is in the image of jk. Is there an easy way to see why Γ(JkE) is generated as a C∞(M)-module by its holonomic sections? It’s … Read more

When does the canonical tt-structure restrict to perfect complexes?

I am interested in non-Noetherian(!) rings such that the canonical t-structure on D(R) (the derived category of left R-modules) restricts to perfect complexes i.e. to the subcategory of complexes of finitely generated projective R-modules; i.e., for a complex C of this sort the complex …0→Coker(C−1→C0)→C1→C2→… should be quasi-isomorphic to a perfect complex. This is equivalent … Read more

Row rank and column rank of matrix with entries in a commutative ring

Let R be a unital commutative ring and A∈Mn×m(R). Under which appropriate invariant “rank” of modules discussed in “Ranks of Modules” one can say that the row rank of A is equal to the column rank of A? Answer AttributionSource : Link , Question Author : Ali Taghavi , Answer Author : Community

Morphism of distinguished triangles where one of the arrows is a quasi-isomorphism

Let R be any ring and let A→B→C→[1] and A′→B′→C′→[1] be distinguished triangles of complexes of R-modules. Let f:A→A′, g:B→B′ and h:C→C′ be morphism of complexes such that (f,g,h) is a morphism of distinguished triangles. I wonder whether the following statement is true: If f:A→A′ is a quasi-isomorphism, then there is a quasi-isomorphism cone(g)∼→cone(h). Clearly, … Read more

Generalization of the second Brauer-Thrall conjecture for arbitrary Artin algebras

Let k be a field with infinite cardinality and A a finite dimensional k-Algebra. The second Brauer-Thrall conjecture states the following: There are infinitely many natural numbers n1<n2<… such that for every ni there are infinitely many indecomposable A-modules with length ni. This does not hold if the cardinality of k is finite. My question … Read more

A projective module over a domain that is not faithfully flat?

Let R be a (noncommutative) unital ring which is a domain and let N be a non-zero projective (right) module. Projectivity of course implies that N is flat, but does the fact that R is also a domain imply that N is faithfully flat? Answer AttributionSource : Link , Question Author : Tim Montegue , … Read more