By Robin Hartshorne

ISBN-10: 1441915958

ISBN-13: 9781441915955

ISBN-10: 1441915966

ISBN-13: 9781441915962

The easy challenge of deformation idea in algebraic geometry comprises observing a small deformation of 1 member of a family members of items, resembling types, or subschemes in a hard and fast house, or vector bundles on a set scheme. during this new publication, Robin Hartshorne reviews first what occurs over small infinitesimal deformations, after which steadily builds as much as extra worldwide events, utilizing tools pioneered through Kodaira and Spencer within the complicated analytic case, and tailored and increased in algebraic geometry via Grothendieck.

Topics include:

* deformations over the twin numbers;

* smoothness and the infinitesimal lifting property;

* Zariski tangent area and obstructions to deformation problems;

* pro-representable functors of Schlessinger;

* infinitesimal research of moduli areas comparable to the Hilbert scheme, Picard scheme, moduli of curves, and moduli of reliable vector bundles.

The writer comprises a variety of routines, in addition to vital examples illustrating numerous features of the speculation. this article is predicated on a graduate path taught by way of the writer on the collage of California, Berkeley.

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Deformation Theory - download pdf or read online

The fundamental challenge of deformation idea in algebraic geometry comprises observing a small deformation of 1 member of a relations of items, similar to types, or subschemes in a hard and fast area, or vector bundles on a set scheme. during this new publication, Robin Hartshorne experiences first what occurs over small infinitesimal deformations, after which steadily builds as much as extra international occasions, utilizing tools pioneered by way of Kodaira and Spencer within the advanced analytic case, and tailored and extended in algebraic geometry by means of Grothendieck.

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Extra info for Deformation Theory

Example text

Since the condition T 2 (B/k, M ) = 0 for all B-modules M depends only on B, and not on A, it follows that if B is a local complete intersection in one regular local ring, then it will be a local complete intersection in any regular local ring of which it is a quotient. Thus we can say simply that B is a local complete intersection ring without mentioning A. 2. The node of (Ex. 1) is a local complete intersection and correspondingly has T 2 = 0 for all M . The thick point of (Ex. 3) is not a local complete intersection and has T 2 (B/k, B) = 0.

The trivial extension is given by B = B ⊕ M made into a ring by the rule (b, m) · (b1 , m1 ) = (bb1 , bm1 + b1 m). 1. Let A be a ring, B an A-algebra, and M a B-module. Then equivalence classes of extensions of B by M as A-algebras are in natural oneto-one correspondence with elements of the group T 1 (B/A, M ). The trivial extension corresponds to the zero element. Proof. Let A[x] → B be a surjective map of a polynomial ring over A to B, let {ei } be a set of generators of the B-module M , and let y = {yi } be a set of indeterminates with the same index set as {ei }.

Dividing out by this ambiguity will give us a description of the set of extensions B . For the first step, we complete the above diagram by adding a top row consisting of the kernels of the vertical arrows: 0 → Q → I → I → 0. Giving B as a quotient of A[x, y] is equivalent to giving the ideal I in A[x, y]. 3) shows that the set of such diagrams is in natural one-to-one correspondence with the group HomA[x] (I, M ) = HomB (I/I 2 , M ). 5), whose proof works over any ring A in place of k, taking R = A[x], to see that the set of possible choices for f : A[x, y] → B forms a principal homogeneous space under the action of DerA (A[x], M ).

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Deformation Theory by Robin Hartshorne


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