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## ‣ Uma justificava da validade do teorema fundamental da álgebra para o ensino médio

Fonte: Universidade Federal do Rio Grande do Norte; BR; UFRN; Programa de Pós-Graduação em Matemática em Rede Nacional; Álgebra; Análise matemática; Ensino de matemática; Geometria e topologia; Matemática aplicada
Publicador: Universidade Federal do Rio Grande do Norte; BR; UFRN; Programa de Pós-Graduação em Matemática em Rede Nacional; Álgebra; Análise matemática; Ensino de matemática; Geometria e topologia; Matemática aplicada

Tipo: Dissertação
Formato: application/pdf

Português

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#Equação polinomial. Função polinomial. Geogebra.
Teorema fundamental da álgebra#Polynomial equation. Polynomial function. Geogebra. Fundamental. Theorem of algebra#CNPQ::CIENCIAS EXATAS E DA TERRA::MATEMATICA

Among several theorems which are taught in basic education some of them
can be proved in the classroom and others do not, because the degree of
difficulty of its formal proof.
A classic example is the Fundamental Theorem of Algebra which is not
proved, it is necessary higher-level knowledge in mathematics.
In this paper, we justify the validity of this theorem intuitively using the
software Geogebra. And, based on [2] we will present a clear formal proof of
this theorem that is addressed to school teachers and undergraduate students
in mathematics; Coordenação de Aperfeiçoamento de Pessoal de Nível Superior; Dentre os vários teoremas que são ensinados na educação básica, alguns podem ser demonstrados em sala de aula e outros não, devido o grau de dificuldade de sua prova formal. Um exemplo clássico e o Teorema Fundamental da Algébra, que não é demonstrado, pois é necessário conhecimentos em Matemática de nível superior.
Neste trabalho, justicamos intuitivamente a validade do Teorema Fundamental
da Algebra usando o software Geogebra. E, baseados em [2], apresentamos
uma clara demonstração formal desse teorema que está endereçada aos
professores do ensino básico e alunos de licenciatura em Matemática

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## ‣ Causal attributions for success or failure by passing and failing students in College Algebra

Fonte: FIU Digital Commons
Publicador: FIU Digital Commons

Tipo: Artigo de Revista Científica

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Success in mathematics has been identified as a predictor of baccalaureate degree completion. Within the coursework of college mathematics, College Algebra has been identified as a high-risk course due to its low success rates. ^ Research in the field of attribution theory and academic achievement suggests a relationship between a student's attributional style and achievement. Theorists and researchers contend that attributions influence individual reactions to success and failure. They also report that individuals use attributions to explain and justify their performance. Studies in mathematics education identify attribution theory as the theoretical orientation most suited to explain academic performance in mathematics. This study focused on the relationship among a high risk course, low success rates, and attribution by examining the difference in the attributions passing and failing students gave for their performance in College Algebra. ^ The methods for the study included a pilot administration of the Causal Dimension Scale (CDSII) which was used to conduct reliability and principal component analyses. Then, students (n = 410) self-reported their performance on an in-class test and attributed their performance along the dimensions of locus of causality...

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## ‣ An investigation into the effects of introducing algebra using a function-based approach

Fonte: University of Limerick
Publicador: University of Limerick

Tipo: info:eu-repo/semantics/masterThesis; all_ul_research; ul_published_reviewed; ul_theses_dissertations

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peer-reviewed; Ireland is currently witnessing a major overhaul of its mathematics syllabus for second level education. This syllabus is known as ‘Project Maths’ and came about as a results of concerns relating to the mathematics performance of students in Ireland in international comparative studies such as the PISA (Program for International Student Assessment) tests (Close & Oldham 2005; Cosgrove, Shiel, Sofroniou, Zastrutzki & Shortt 2005; Perkins, Moran, Cosgrove and Shiel 2010; Oldham 2002, 2006).
The author found inspiration for this research when she identified concerns in her own classroom. These concerns were two-fold; firstly the author found that first year students began secondary school with a poor attitude towards mathematics and secondly, the author found that first year students had a lot of difficulty grasping and retaining basic algebraic concepts. The author followed an action research approach to implementing an intervention in her classroom aimed at overcoming these problems. In the first phase of this research, the author carried out a comprehensive review of literature on affect pertaining to mathematics education and on the teaching and learning of algebra. As a result of this review, the author decided to use a function-based approach to teaching algebra as a means of improving students understanding of basic algebra. A collaborative peer learning environment was chosen as the main pedagogical tool for improving attitude towards mathematics. The second phase of this research saw the development and implementation of an intervention in the author’s classroom during which fourth year students tutored first year students. Quantitative and qualitative data was gathered during this phase. The third phase comprised of an analysis of data...

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## ‣ The Lie algebra perturbation lemma

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Algebraic Geometry#Mathematics - Commutative Algebra#16E45, 16W30, 17B55, 17B56, 17B65, 17B70, 18G10, 55P62

Let g be a differential graded Lie algebra and suppose given a contraction of
chain complexes of g onto a general chain complex M. We show that the data
determine an sh-Lie algebra structure on M, that is, a coalgebra perturbation
of the coalgebra differential on the cofree coaugmented differential graded
cocommutative coalgebra S' on the suspension of M, a Lie algebra twisting
cochain from the perturbed coalgebra S" to the given Lie algebra g, and an
extension of this Lie algebra twisting cochain to a contraction of chain
complexes from the Cartan-Chevalley-Eilenberg coalgebra on g onto S" which is
natural in the data. This extends a result established in a joint paper of the
author with J. Stashef [Forum math. 14 (2002), 847-868, math.AG/9906036] where
only the particular where M is the homology of g has been explored.; Comment: 20 pages; in view of a number of comments of J. Stasheff, the
exposition has been improved

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## ‣ Random Sampling in Computational Algebra: Helly Numbers and Violator Spaces

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Computer Science - Discrete Mathematics#Mathematics - Commutative Algebra#Mathematics - Algebraic Geometry#Mathematics - Combinatorics

This paper transfers a randomized algorithm originally used in geometric
optimization to computational commutative algebra. We show that Clarkson's
sampling algorithm can be applied to two problems in computational algebra:
solving large-scale polynomial systems, for which we utilize a Helly-type
result for algebraic varieties, and finding small generating sets of graded
ideals. The cornerstone of our work is showing that the theory of violator
spaces of G\"artner et al.\ applies to these polynomial ideal problems. The
resulting algorithms have expected runtime linear in the number of input
polynomials, making the method particularly interesting for handling systems
with very large numbers of polynomials, but whose rank in the vector space of
polynomials is small (e.g., when the number of variables is constant).; Comment: 14 pages; corrected Example 3.2; added some references; results
unchanged

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## ‣ Factorization in $SL_n(R)$ with elementary matrices when $R$ is the disk algebra and the Wiener algebra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 20/05/2014
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#Mathematics - Commutative Algebra#Mathematics - Complex Variables#Mathematics - Functional Analysis#Primary 46J10, Secondary 15A23, 15A54

Let $R$ be the polydisc algebra or the Wiener algebra. It is shown that the
group $SL_n(R)$ is generated by the subgroup of elementary matrices with all
diagonal entries $1$ and at most one nonzero off-diagonal entry. The result an
easy consequence of the deep result due to Ivarsson and Kutzschebauch (Ann. of
Math. 2012).; Comment: 5 pages

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## ‣ The sh-Lie algebra perturbation Lemma

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Algebraic Geometry#Mathematics - Commutative Algebra#16E45, 16W30, 17B55, 17B56, 17B65, 17B70, 18G10, 55P62

Let R be a commutative ring which contains the rationals as a subring and let
g be a chain complex. Suppose given an sh-Lie algebra structure on g, that is,
a coalgebra perturbation of the coalgebra differential on the cofree
coaugmented differential graded cocommutative coalgebra T' on the suspension of
g and write the perturbed coalgebra as T". Suppose, furthermore, given a
contraction of g onto a chain complex M. We show that the data determine an
sh-Lie algebra structure on M, that is, a coalgebra perturbation of the
coalgebra differential on the cofree coaugmented differential graded
cocommutative coalgebra S' on the suspension of M, a Lie algebra twisting
cochain from the perturbed coalgebra S" to the loop Lie algebra L on the
perturbed coalgebra T", and an extension of this Lie algebra twisting cochain
to a contraction of chain complexes from the Cartan-Chevalley-Eilenberg
coalgebra on L onto S" which is natural in the data. For the special case where
M and g are connected we also construct an explicit extension of the perturbed
retraction to an sh-Lie map. This approach includes a very general solution of
the master equation.; Comment: 20 pages

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## ‣ Algebra retracts and Stanley-Reisner rings

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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In a paper from 2002, Bruns and Gubeladze conjectured that graded algebra
retracts of polytopal algebras over a field $k$ are again polytopal algebras.
Motivated by this conjecture, we prove that graded algebra retracts of
Stanley-Reisner rings over a field $k$ are again Stanley-Reisner rings.
Extending this result further, we give partial evidence for a conjecture saying
that monomial quotients of standard graded polynomial rings over $k$ descend
along graded algebra retracts.; Comment: Incorporating several useful suggestions due to a referee. To appear
in Journal of Pure and Applied Algebra

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## ‣ Torus actions, combinatorial topology and homological algebra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 07/10/2000
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#Mathematics - Algebraic Topology#Mathematics - Commutative Algebra#Mathematics - Algebraic Geometry#Mathematics - Combinatorics#Mathematics - Geometric Topology#Mathematics - Rings and Algebras#52B70#57Q15#57R19#14M25#52B05

The paper surveys some new results and open problems connected with such
fundamental combinatorial concepts as polytopes, simplicial complexes, cubical
complexes, and subspace arrangements. Particular attention is paid to the case
of simplicial and cubical subdivisions of manifolds and, especially, spheres.
We describe important constructions which allow to study all these
combinatorial objects by means of methods of commutative and homological
algebra. The proposed approach to combinatorial problems relies on the theory
of moment-angle complexes, currently being developed by the authors. The theory
centres around the construction that assigns to each simplicial complex $K$
with $m$ vertices a $T^m$-space $\zk$ with a special bigraded cellular
decomposition. In the framework of this theory, the well-known non-singular
toric varieties arise as orbit spaces of maximally free actions of subtori on
moment-angle complexes corresponding to simplicial spheres. We express
different invariants of simplicial complexes and related
combinatorial-geometrical objects in terms of the bigraded cohomology rings of
the corresponding moment-angle complexes. Finally, we show that the new
relationships between combinatorics, geometry and topology result in solutions
to some well-known topological problems.; Comment: 87 pages...

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## ‣ Evolution algebra of a "chicken" population

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/07/2013
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We consider an evolution algebra which corresponds to a bisexual population
with a set of females partitioned into finitely many different types and the
males having only one type. We study basic properties of the algebra. This
algebra is commutative (and hence flexible), not associative and not
necessarily power associative, in general. Moreover it is not unital. A
condition is found on the structural constants of the algebra under which the
algebra is associative, alternative, power associative, nilpotent, satisfies
Jacobi and Jordan identities. In a general case, we describe the full set of
idempotent elements and the full set of absolute nilpotent elements. The set of
all operators of left (right) multiplications is described. Under some
conditions on the structural constants it is proved that the corresponding
algebra is centroidal. Moreover the classification of 2-dimensional and some
3-dimensional algebras are obtained.; Comment: 15 pages

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## ‣ Modular decomposition of the Orlik-Terao algebra of a hyperplane arrangement

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 19/11/2012
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#Mathematics - Commutative Algebra#Mathematics - Combinatorics#52C35 (Primary) 16S37, 13C40, 05B35, 13D40 (Secondary)

Let A be a collection of n linear hyperplanes in k^l, where k is an
algebraically closed field. The Orlik-Terao algebra of A is the subalgebra R(A)
of the rational functions generated by reciprocals of linear forms vanishing on
hyperplanes of A. It determines an irreducible subvariety of projective space.
We show that a flat X of A is modular if and only if R(A) is a split extension
of the Orlik-Terao algebra of the subarrangement A_X. This provides another
refinement of Stanley's modular factorization theorem and a new
characterization of modularity, similar in spirit to the modular fibration
theorem of Paris.
We deduce that if A is supersolvable, then its Orlik-Terao algebra is Koszul.
In certain cases, the algebra is also a complete intersection, and we
characterize when this happens.; Comment: 23 pages

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## ‣ Some curiosities of the algebra of bounded Dirichlet series

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/11/2015
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#Mathematics - Complex Variables#Mathematics - Commutative Algebra#Mathematics - K-Theory and Homology#Mathematics - Number Theory#Primary 11M41, Secondary 30H05, 13E99

It is shown that the algebra of bounded Dirichlet series is not a coherent
ring, and has infinite Bass stable rank. As corollaries of the latter result,
it is derived that the algebra of bounded Dirichlet series has infinite
topological stable rank and infinite Krull dimension.; Comment: 8 pages

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## ‣ The Aluffi Algebra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Commutative Algebra#Mathematics - Algebraic Geometry#13A30, 14B05, 14D06, 14H10, 13B25, 13C15, 13D02, 13F45, 13P10, 14H50

We deal with the quasi-symmetric algebra introduced by Paolo Aluffi, here
named (embedded) Aluffi algebra. The algebra is a sort of "intermediate"
algebra between the symmetric algebra and the Rees algebra of an ideal, which
serves the purpose of introducing the characteristic cycle of a hypersurface in
intersection theory. The results described in the present paper have an
algebraic flavor and naturally connect with various themes of commutative
algebra, such as standard bases \'a la Hironaka, Artin--Rees like questions,
Valabrega--Valla ideals, ideals of linear type, relation type and analytic
spread. We give estimates for the dimension of the Aluffi algebra and show
that, pretty generally, the latter is equidimensional whenever the base ring is
a hypersurface ring. There is a converse to this under certain conditions that
essentially subsume the setup in Aluffi's theory, thus suggesting that this
algebra will not handle cases other than the singular locus of a hypersurface.
The torsion and the structure of the minimal primes of the algebra are
clarified. In the case of a projective hypersurface the results are more
precise and one is naturally led to look at families of projective plane
singular curves to understand how the property of being of linear type
deforms/specializes for the singular locus of a member. It is fairly elementary
to show that the singular locus of an irreducible curve of degree at most 3 is
of linear type. This is roundly false in degree larger than 4 and the picture
looks pretty wild as we point out by means of some families. Degree 4 is the
intriguing case. Here we are able to show that the singular locus of the
generic member of a family of rational quartics...

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## ‣ The Orlik-Terao algebra and 2-formality

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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The Orlik-Solomon algebra is the cohomology ring of the complement of a
hyperplane arrangement A in C^n; it is the quotient of an exterior algebra E(V)
on |A| generators. Orlik and Terao introduced a commutative analog S(V)/I of
the Orlik-Solomon algebra to answer a question of Aomoto and showed the Hilbert
series depends only on the intersection lattice L(A). Motivated by topological
considerations, Falk and Randell introduced the property of 2-formality; we
study the relation between 2-formality and the Orlik-Terao algebra. Our main
result is a necessary and sufficient condition for 2-formality in terms of the
quadratic component I_2 of the Orlik-Terao ideal I: 2-formality is determined
by the tangent space T_p(V(I_2)) at a generic point p.; Comment: 11 pages, 2 figures. v2 typo in Example 1.4 fixed v3 Le-Mohammadi
have pointed out that one direction of Thm 2.4 (now Thm 2.5) needs a
saturation hypothesis, now corrected

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## ‣ On the invariants of the splitting algebra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 23/05/2011
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For a given monic polynomial $p(t)$ of degree $n$ over a commutative ring
$k$, the splitting algebra is the universal $k$-algebra in which $p(t)$ has $n$
roots, or, more precisely, over which $p(t)$ factors,
$p(t)=(t-\xi_1)...(t-\xi_n)$. The symmetric group $S_r$ for $1\le r\le n$ acts
on the splitting algebra by permuting the first $r$ roots $\xi_1,...,\xi_r$. We
give a natural, simple condition on the polynomial $p(t)$ that holds if and
only if there are only trivial invariants under the actions. In particular, if
the condition on $p(t)$ holds then the elements of $k$ are the only invariants
under the action of $S_n$. We show that for any $n\ge 2$ there is a polynomial
$p(t)$ of degree $n$ for which the splitting algebra contains a nontrivial
element invariant under $S_n$. The examples violate an assertion by A. D.
Barnard from 1974.

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## ‣ Algebraic structures of tropical mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/05/2013
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#Mathematics - Rings and Algebras#Mathematics - Commutative Algebra#06F20, 11C08, 12K10, 14T05, 14T99, 16Y60

Tropical mathematics often is defined over an ordered cancellative monoid
$\tM$, usually taken to be $(\RR, +)$ or $(\QQ, +)$. Although a rich theory has
arisen from this viewpoint, cf. [L1], idempotent semirings possess a restricted
algebraic structure theory, and also do not reflect certain valuation-theoretic
properties, thereby forcing researchers to rely often on combinatoric
techniques.
In this paper we describe an alternative structure, more compatible with
valuation theory, studied by the authors over the past few years, that permits
fuller use of algebraic theory especially in understanding the underlying
tropical geometry. The idempotent max-plus algebra $A$ of an ordered monoid
$\tM$ is replaced by $R: = L\times \tM$, where $L$ is a given indexing semiring
(not necessarily with 0). In this case we say $R$ layered by $L$. When $L$ is
trivial, i.e, $L=\{1\}$, $R$ is the usual bipotent max-plus algebra. When
$L=\{1,\infty\}$ we recover the "standard" supertropical structure with its
"ghost" layer. When $L = \NN $ we can describe multiple roots of polynomials
via a "layering function" $s: R \to L$. Likewise, one can define the layering
$s: R^{(n)} \to L^{(n)}$ componentwise; vectors $v_1, \dots, v_m$ are called
tropically dependent if each component of some nontrivial linear combination
$\sum \a_i v_i$ is a ghost...

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## ‣ When the orbit algebra of group is an integral domain? Proof of a conjecture of P.J. Cameron

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/04/2007
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#Mathematics - Combinatorics#Mathematics - Commutative Algebra#03 C13, 03 C52, 05 A16, 05 C30, 20 B27

P.J.Cameron introduced the orbit algebra of a permutation group and
conjectured that this algebra is an integral domain if and only if the group
has no finite orbit. We prove that this conjecture holds and in fact that the
age algebra of a relational structure $R$ is an integral domain if and only if
$R$ is age-inexhaustible. We deduce these results from a combinatorial lemma
asserting that if a product of two non-zero elements of a set algebra is zero
then there is a finite common tranversal of their supports. The proof is built
on Ramsey theorem and the integrity of a shuffle algebra.; Comment: 19 pages

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## ‣ A new discriminant algebra construction

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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A discriminant algebra operation sends a commutative ring $R$ and an
$R$-algebra $A$ of rank $n$ to an $R$-algebra $\Delta_{A/R}$ of rank $2$ with
the same discriminant bilinear form. Constructions of discriminant algebra
operations have been put forward by Rost, Deligne, and Loos. We present a
simpler and more explicit construction that does not break down into cases
based on the parity of $n$. We then prove properties of this construction, and
compute some examples explicitly.; Comment: 33 pages; the new version has been reorganized to improve readability
and contains new examples, as well as a streamlined proof of the main theorem

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## ‣ Elimination and nonlinear equations of Rees algebra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 13/11/2009
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A new approach is established to computing the image of a rational map,
whereby the use of approximation complexes is complemented with a detailed
analysis of the torsion of the symmetric algebra in certain degrees. In the
case the map is everywhere defined this analysis provides free resolutions of
graded parts of the Rees algebra of the base ideal in degrees where it does not
coincide with the corresponding symmetric algebra. A surprising fact is that
the torsion in those degrees only contributes to the first free module in the
resolution of the symmetric algebra modulo torsion. An additional point is that
this contribution -- which of course corresponds to non linear equations of the
Rees algebra -- can be described in these degrees in terms of non Koszul
syzygies via certain upgrading maps in the vein of the ones introduced earlier
by J. Herzog, the third named author and W. Vasconcelos. As a measure of the
reach of this torsion analysis we could say that, in the case of a general
everywhere defined map, half of the degrees where the torsion does not vanish
are understood.

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## ‣ Constructive Homological Algebra and Applications

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - K-Theory and Homology#Mathematics - Commutative Algebra#Mathematics - Algebraic Topology#18G40, 55T10, 55T20#G.4

This text was written and used for a MAP Summer School at the University of
Genova, August 28 to September 2, 2006. Available since then on the web site of
the second author, it has been used and referenced by several colleagues
working in Commutative Algebra and Algebraic Topology. To make safer such
references, it was suggested to place it on the Arxiv repository.
It is a relatively detailed exposition of the use of the Basic Perturbation
Lemma to make constructive Homological Algebra (standard Homological Algebra is
not constructive) and how this technology can be used in Commutative Algebra
(Koszul complexes) and Algebraic Topology (effective versions of spectral
sequences).; Comment: Version 3: Error corrected p. 111, see footnote 26

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