Flat modules we recall here some properties of at modules as exposed in bourbaki, alg ebre commutative, ch. Print and download faithfully sheet music by journey. Prove that sgn is a homomorphism from g to the multiplicative. Resultatives under the event argument homomorphism. Hnumerically flat higgs bundles make up again a neutral tannakian category. Numerically flat vector bundles, if equipped with the zero higgs field, are hnumerically flat, so that there is a faithfully flat morphism. The aim of these pages is to expose the proofs of some of the characterizations of at and faithfully at modules given in matsumuras book commutative algebra. In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme x to a scheme y is a morphism such that the induced map on every stalk is a flat map of rings, i. Then the homomorphism allows us to view nas an amodule.
We say that an amodule m is at if for every short exact sequence 0. Then, since fi is a flat amodule and a is a flat amodule it follows. Flat modules we recall here some properties of flat. Let a be a ring assumed to be commutative throughout this section and b an aalgebra, i. Thus, we will be content with stating the theorems precisely and. Faithfully flat descent wouter zomervrucht, november 6, 20 1. More generally, if gis an abelian group written multiplicatively and n2 z is a xed integer, then the function f. Authors personal copy north dakota state university.
In this lecture we study the concept of faithfully flat descent, which is the notion that to. It is given by x e h for all x 2g where e h is the identity element of h. A morphism of schemes is faithfully flat if it is flat and epi. This section simply exists to summarize the properties of flatness that will be useful to us. If aa ring and f 1f n2agenerate the unit ideal then fspecaf i.
Higgs varieties and fundamental groups sciencedirect. Let a be a unital magma, b a monoid in c and b a a morphism of unital magmas satisfying the identities. A ring homomorphism a b is faithfully flat if for every sequence n n. Faithfully flat descent for morphism of schemes shengxuan liu november 29, 2018 theorem. For example, embed r into rx, the constants in the ring of polynomials. The following theorem shows that in addition to preserving group operation, homomorphisms must also preserve identity element and. This part is mainly devoted to the exposition of a proof of the following. A crucial ingredient in the proof is that the going down. T 6 t whenever and are distinct morphisms from an object x to an object y in c. A morphism of schemes is called faithfully flat if it. Faithful flatness we shall say that an rmodule f is faithfully flat if it. It deals with the special case where f is a faithfully flat ralgebra, and the proof relies heavily on 4,theorem 4. A module over a ring r r is called flat if its satisfies one of many equivalent conditions, the simplest to state of which is maybe.
A morphism of affine schemes spec b spec a specb \to speca, hence coming from a ring homomorphism f. Suppose that there is a homomorphism of aalgebras b. We consider a faithfully flat ring homomorphism r s such that for all special characters omitted. Because of this homomorphism between wine and winedrinking, quantification is transferred from the nominal to the verbal domain. Faithfully flat descent for morphisms of unital magmas.
Faithful flatness we shall say that an rmodule f is faithfully at if it is at if it is at and for every nonzero rmodule m, f r m 6 0. Counterexamples in algebra august 3, 2015 we use k, f, k to denote the elds, and rto denote the rings. Faithfully flat extensions of a commutative ring 257 theorem 3. Thus, we will be content with stating the theorems precisely and giving references for the proofs. A homomorphism is faithfully flat if it is faithful and flat, i. As an example, if k is a field, then kx and kx are both faithfully flat over k. Then the category of descent data for bover ais equivalent to the category of amodules. As this is best illustrated by example, we begin by studying descent. Pdf injective modules under faithfully flat ring extensions. B is local homomorphism of noetherian local rings and b is. Flat morphisms need not be injective, but they are locally injective. Denote by z the ring of rational integers, q the eld of rational numbers, r the eld of real numbers, and c the eld of complex numbers.
On a surmise of mcadam concerning quintasymptotic primes. Sheet music arranged for singer pro, and pianovocalguitar in b major transposable. Projective modules, faithful modules mathreference. Bbe a homomorphism of rings and let n be a bmodule. Sbe a local homomorphism of local rings with maximal ideals m. In this section we will show that under some conditions there exists a categorical equivalence between m o d b and s d e s. From this and the fact that b is a faithfully flat bmodule we deduce. Then b is aflat if and only if b is afaithfully flat. This is the basis for an important technique in algebraic geometry.
The predicate flat in sentence 1 is a resultative because the. A map of rings a b is called flat, if it is a homomorphism that makes b a flat amodule. This is a homomorphism which make the following diagram commute. A flat local ring homomorphism of local rings is faithfully flat. Bbe a local homomorphism of noetherian local rings. A b is called a flat homomorphism if b is an aflat. The classical theory of faithfully at descent guarantees that this functor is an equivalence of categories whenever the map f is faithfully at and quasicompact. Bis a local homomorphism of noetherian rings and nis a nitely generated bmodule, then nis at over ai the natural map m a an. Mflat if, for any amodule, m, and for any injective homomorphism.
We shall see below that the completion of a local ring r is a faithfully. Consequently, if kis a eld, every kalgebra is faithfully at. Beachy, a supplement to abstract algebraby beachy blair 21. A ring map r \to s is called faithfully flat if s is faithfully flat as an rmodule. We shall see below that the completion of a local ring r is a faithfully at ralgebra. If e is an injective rmodule, then it is a direct summand as an rmodule of the injective smodulehomr. An injective map of rings induces a dominant map on spectra, and a flat map of finite type is open, so it implies faithful flatness. Our proof begins much like the standard one, with faithfully flat splittings. The goal of this note is to prove that a flat finitetype morphism between noetherian schemes is open. Journey faithfully sheet music in b major transposable. An exact functor t from one abelian category to another is faithful if and only if tais nonzero. Moreover, when checking faithfulness, it suffices to prove v. This is similar to the definition of a flat homomorphism, as presented in the previous section. It is not apriori obvious that a homomorphism preserves identity elements or that it takes inverses to inverses.
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