Commutative Algebra I
01 Jan 2012-
TL;DR: A compilation of two sets of notes at the University of Kansas was published in the Spring of 2002 by?? and the other in the spring of 2007 by Branden Stone.
Abstract: 1 A compilation of two sets of notes at the University of Kansas; one in the Spring of 2002 by ?? and the other in the Spring of 2007 by Branden Stone. These notes have been typed
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Book•
08 Oct 2012TL;DR: In this paper, the authors present a survey of the geometry of lines and cubic surfaces, including determinantal equations, theta characteristics, and the Cremona transformations.
Abstract: Preface 1. Polarity 2. Conics and quadrics 3. Plane cubics 4. Determinantal equations 5. Theta characteristics 6. Plane quartics 7. Cremona transformations 8. Del Pezzo surfaces 9. Cubic surfaces 10. Geometry of lines Bibliography Index.
663 citations
Cites background or methods from "Commutative Algebra I"
...Note that the latter property characterizes Gorenstein graded local Artinian rings, see [209], [314]....
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...By Serre’s criterion, it is a normal domain (see [209], 11....
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...This is one of the nice features of a Gorenstein Artinian algebra (see [209], 21....
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...We will use the following result from commutative algebra (see [209])....
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...and the map γ is defined by (a, b, c) 7→ at0 + bt1 + ct2 (see [209], 17....
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TL;DR: In this article, potential modularity theorems for l-adic representations of any dimension were proved for all elliptic curves with nonintegral j -invariant defined over a real field.
Abstract: We prove potential modularity theorems for l-adic representations of any dimension. From these results we deduce the Sato-Tate conjecture for all elliptic curves with nonintegral j -invariant defined over a totally real field.
459 citations
TL;DR: In this paper, the authors discuss the construction of triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky's definition of motives over fields.
Abstract: This book discusses the construction of triangulated categories of mixed motives over a noetherian scheme of finite dimension, extending Voevodsky's definition of motives over a field. In particular, it is shown that motives with rational coefficients satisfy the formalism of the six operations of Grothendieck. This is achieved by studying descent properties of motives, as well as by comparing different presentations of these categories, following and extending insights and constructions of Deligne, Beilinson, Bloch, Thomason, Gabber, Levine, Morel, Voevodsky, Ayoub, Spitzweck, R\"ondigs, {\O}stv{\ae}r, and others. In particular, the relation of motives with $K$-theory is addressed in full, and we prove the absolute purity theorem with rational coefficients, using Quillen's localization theorem in algebraic $K$-theory together with a variation on the Grothendieck-Riemann-Roch theorem. Using resolution of singularities via alterations of de Jong-Gabber, this leads to a version of Grothendieck-Verdier duality for constructible motivic sheaves with rational coefficients over rather general base schemes. We also study versions with integral coefficients, constructed via sheaves with transfers, for which we obtain partial results. Finally, we associate to any mixed Weil cohomology a system of categories of coefficients and well behaved realization functors, establishing a correspondence between mixed Weil cohomologies and suitable systems of coefficients. The results of this book have already served as ground reference in many subsequent works on motivic sheaves and their realizations, and pointers to the most recent developments of the theory are given in the introduction.
339 citations
Cites background from "Commutative Algebra I"
...Matsumura introduced in [Mat70] the weaker notion of a quasi-excellent ring A....
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TL;DR: This work establishes a general condition that must be satisfied by any degrees of freedom tuple (d1. d2....dK) achievable through linear interference alignment that implies that the total achievable DoF cannot grow linearly with K, and is in fact no more than K(M + N)/(K + 1).
Abstract: Consider a K-user flat fading MIMO interference channel where the kth transmitter (or receiver) is equipped with Mk (respectively Nk) antennas. If an exponential (in K) number of generic channel extensions are used either across time or frequency, Cadambe and Jafar [1] showed that the total achievable degrees of freedom (DoF) can be maximized via interference alignment, resulting in a total DoF that grows linearly with A even if Mk and Nk are bounded. In this work we consider the case where no channel extension is allowed, and establish a general condition that must be satisfied by any degrees of freedom tuple (d1. d2....dK) achievable through linear interference alignment. For a symmetric system with Mk = M, Nk = N, dk = d for all k, this condition implies that the total achievable DoF cannot grow linearly with K, and is in fact no more than K(M + N)/(K + 1). We also show that this bound is tight when the number of antennas at each transceiver is divisible by d, the number of data streams per user.
328 citations
Cites methods from "Commutative Algebra I"
...According to the Chevalley Theorem (see [10], Ch....
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TL;DR: In this article, the authors prove an automorphy lifting theorem for l-adic representations where they impose a new condition at l, which they call "potentential diagonalizability".
Abstract: We prove an automorphy lifting theorem for l-adic representations where we impose a new condition at l, which we call \potentential diagonalizability." This result allows for \change of weight" and seems to be substantially more exible than previous theorems along the same lines. We derive several applications. For instance we show that any irreducible, totally odd, essentially self-dual, regular, weakly compatible system of l-adic representations of the absolute Galois group of a totally real eld is potentially automorphic and hence is pure and its L-function has meromorphic continuation to the whole complex plane and satises the expected functional equation.
315 citations
References
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Book•
01 Jan 1969
TL;DR: It is shown here how the Noetherian Rings and Dedekind Domains can be transformed into rings and Modules of Fractions using the following structures:
Abstract: * Introduction * Rings and Ideals * Modules * Rings and Modules of Fractions * Primary Decomposition * Integral Dependence and Valuations * Chain Conditions * Noetherian Rings * Artin Rings * Discrete Valuation Rings and Dedekind Domains * Completions * Dimension Theory
4,168 citations
"Commutative Algebra I" refers background in this paper
...(8) Let R be a ring in which every element x satisfies x = x for some n > 1 [1] (depending on x)....
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...[1] Deduce that the sum of a nilpotent element and a unit is a unit....
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...(3) Let R be a ring and let R[x] be the ring of polynomials in an indeterminate [1] x, with coefficients in R....
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...(24) A topological space X is said to be irreducible if X 6= ∅ and if every pair [1] of non-empty open sets in X intersect, or equivalently if every non-empty open set is dense in X....
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...(22) For R a ring, consider {D(r)}r∈R (the basis of open sets for the Zariski [1] topology)....
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