Decomposition of persistence modules

Let P be an R-module with a decomposition P = M 1 ⊕⋯⊕M n , where each M i is a CS-module. In [6], a necessary and sufficient condition for P to be a CS-module for the decomposition is given, by introducing generalized relative injectivity.In this paper, we study the problem when a direct sum of uniform modules is CS for the given direct sum. In the case when each uniform module has a local endomorphism ring, this problem has been studied by several authors, e.g., Baba-Harada [2], Dung [3], Harada-Oshiro [5].

In the usual Krull-Schmidt-Azumaya theorem in abelian categories it is essential that each of the direct summands has a local ring of endomorphisms. A partial answer is given here to the case where this last condition is not satisfied by the indecomposable direct summands. It is found that those summands with local endomorphism rings are determined up to isomorphism and cardinality of occurrence.

We define a simple, explicit map sending a morphism $f:M \rightarrow N$ of pointwise finite dimensional persistence modules to a matching between the barcodes of $M$ and $N$. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of $\ker f$ and $\mathop{\mathrm{coker}} f$. As an immediate corollary, we obtain a new proof of the algebraic stability of persistence, a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a $\delta$-interleaving morphism between two persistence modules induces a $\delta$-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules, and yields a novel "single-morphism" characterization of the interleaving relation on persistence modules.

Let $(R, \mathfrak{m})$ be a Cohen–Macaulay complete local ring. We will apply an inductive argument to show that for every nonprojective locally projective maximal Cohen–Macaulay object $ \mathcal{X} $ of the morphism category of $R$ with local endomorphism ring, there exists an almost split sequence ending in $ \mathcal{X} $. Regular sequences are exploited to reduce the Krull dimension of $R$ on which the inductive argument is established. Moreover, the Auslander–Reiten translate of certain objects is described.

Krull–Schmidt categories and projective covers

Let $R$ be a local one-dimensional domain, with maximal ideal $\mathfrak M$, which is not a valuation domain. We investigate the class of the finitely generated mixed $R$-modules of Warfield type, so called since their construction goes back to R.B. Warfield. We prove that these $R$-modules have local endomorphism rings, hence they are indecomposable. We examine the torsion part $t(M)$ of a Warfield type module $M$, investigating the natural property $t(M) \subset \mathfrak M M$. This property is related to $b/a$ being integral over $R$, where $a$ and $b$ are elements of $R$ that define $M$. We also investigate $M/t(M)$ and determine its minimum number of generators.

Direct-Sum Decompositions over Local Rings

Abstract. Let R be a ring and Q be a finite quiver, and let be the number of vertices of Q. Let be the class of representations of Q by right R-modules with local endomorphism ring and R-module homomorphisms. The endomorphism ring of a representation has at most n maximal right ideals, all of which are also left ideals, and the isomorphism class of M is determined by n invariants. The main theorem of this paper states that a finite direct sum of representations in is unique up to n permutations of m elements. Let . A non-directed graph associated to M is introduced and is shown to determine the unique decomposition of M into indecomposable representations. This class of representations is shown to generalize the known classes of modules for which a theorem analogous to the case of our main theorem holds.

Local rings whose modules are almost serial

We prove a theorem on the intersection theory over a Noetherian local ring , which gives a new proof of a classical theorem of Rees about degree functions. To obtain this, we define an intersection product on schemes that are proper and birational over such rings , using the theory of rational equivalence developed by Thorup, and the Snapper–Mumford–Kleiman intersection theory for proper schemes over an Artinian local ring. Our development of this product is essentially self‐contained. As a central component of the proof of our main theorem, we extend to arbitrary Noetherian local rings a formula by Ramanujam that computes Hilbert–Samuel multiplicities. In the final section, we express mixed multiplicities in terms of intersection theory and conclude from this that they satisfy a certain multilinearity condition. Then we interpret some theorems of Rees and Sharp and of Teissier about mixed multiplicities over 2‐dimensional excellent local rings in terms of our intersection product.

Fischer (1930) stated his fundamental theorem of natural selection (ftns) as ‘the rate of increase in fitness of any organism at any time is equal to its genetic (additive) variance in fitness at that time’. The theorem related to an idealised population and he thought that it held ‘the supreme position among the biological sciences’. His proof of the theorem is, however, not understood. This has given rise to questions concerning not only the validity of the proof but also concerning the importance of the theorem.

(1981). Proof of the Fundamental Theorem of Algebra. The American Mathematical Monthly: Vol. 88, No. 4, pp. 253-256.

In this chapter we prove a localization theorem of Quillen for singular varieties, and a generalization of it due to Levine. These are then used to prove the so-called “Fundamental Theorem” (9.8), which computes K i (A[t, t −1]), and to relate the study of 0-cycles on normal surfaces to modules of finite length and finite projective dimension over the local rings at singular points. We begin with Quillen’s localization theorem, proved in Higher Algebraic K-Theory II.KeywordsExact SequenceVector BundleLocal RingFull SubcategoryExceptional DivisorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this chapter we prove a localisation theorem of Quillen for singular varieties, and a generalisation of it due to Levine. These are then used to prove the so called “fundamental theorem” (9.8) which computes Ki(A[t, t−1]), and to relate the study of 0-cycles on normal surfaces to modules of finite length and finite projective dimension over the local rings at singular points. We begin with Quillen’s localisation theorem, proved in “Higher Algebraic K-theory II”.KeywordsExact SequenceVector BundleLocal RingNatural TransformationFull SubcategoryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Introduction. The local rings, A, of the totle are complete, neotherian, local rings of dimension 0, [3, Th. 2] proved useful in the theory of cohomological dimension. By characterizing the etale sheaves (see ? 1) over a special local ring (Th. 1), we are able to further generalize these structure theorems to obtain the canonical decomposition of a sheaf over such a ring into infinitesimal and etale parts, (Th. 2). Hopefully, our structure theorem will help in the investigation of cohomological phenomena over special local rings.