Reports of the Midwest Category Seminar II
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61
M. Andre, D. A. Buchsbaum, E. Dubuc, R. L. Knighten, F. W. Lawvere
Reports of the Midwest Category Seminar II 1968
Edited by S. MacLane, University of Chicago
Springer-Verlag Berlin· Heidelberg' New York
All rights reserved. No part of this book may be translated or reproduced in any form without written permission from Springer Verlag. © by Springer-Verlag Berlin· Heidelberg 1968 Library of Congress Catalog Card Number 68 - 31621 Printed in Germany. Title No. 3667
Table of Contents
M. Andre, On the vanishing of the second cohomology group of a commutative algebra ••••••••••••••••••••••••••
1
David A. Buchsbaum, Homology and universality relative to
a functor.........................................
28
F. William Lawvere, Some algebraic problems in the context of functorial semantics of algebraic theories ••••• 41 R. L. Knighten, An application of categories of fractions to homotopy theory •••••••••••••••••••••••••••••••• 62 Eduardo Dubuc, Adjoint triangles •••••••••••••••••••••••••••• 69
ON THE VANISHING OF THE SECOND COHOMOLOGY GROUP OF A COMMUTATIVE ALGEBRA Michel Andrr!*
In another paper I have defined and studied cohomology groups for commutative topological algebras.
Derivations
appear in dimension 0 and extensions in dimension 1. paper I want to prove a result in dimension 2.
In this
It implies
known results on formally smooth algebras in dimension 1, on regular rings in dimension 2 and on complete intersections in dimension 3:
see [Gr]19.5.4. and [An]27.1./27.2./28.3.
All
rings of this paper are commutative with 1. We consider a commutative ring A and an ideal I.
Then the
following three properties are equivalent: i) the A/I-module 1/1 2 is projective and the A/I-algebra A/I 1/1 2 e 1 2/1 3 @ .... is symmetric. ii)
the cohomology groups
H2(A/ln, A/I, W) are zero
for all A/I-modules W. iii)
the cohomology groups
for all A/I-modules W and all
Hk(A/ln, A/I, W) are zero 2.
*This research was supported in part by the Office of Naval Research.
- 2 -
I.
Discrete cohomology.
Here is a brief review of cohomology
theory for discrete algebras.
First of all for any A-algebra
B and B-module W there are cohomology groups Hk (A, B, W), contravariant in A and in B, covariant in W.
Their basic
properties are the following. Proposition 1.1.
o
Let B be an A-algebra and W' -..
W
-+ Wit -+ 0
be a short exact sequence of B-modules.
Then there is an
exact sequence:
See [AnJ15.2. Proposition 1.2. and W be a C-module. •••• Hk
Then there is an exact sequence: Hk
(B,C,W)
Let B be an A-algebra, C be a B-algebra
(A,C,W)
->
Hk (A,B,W)
l ) •••• Hk + (B,C,W
See [AnJ18.2. Proposition 1.3. a
Let B and C be two A-algebras and W be
If the following condition holds Tor A .(B, C) = 0 1.
for all i
>
then the natural homomorphism H* (B, B @AC, W)
H* (A,
c , W)
0
- 3 is an isomorphism. See [An]19.2. Proposition 1.4. B-module.
Let B be an A-algebra and W be a
If the natural homomorphism of A into B is an epi-
morphism with kernel I, then there is a natural isomorphism:
See
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