Review of Model Categories
... Every object in the model category is fibrant. Cofibrant objects are chain complexes of projective modules. This example is of interest because the homotopy theory chain complexes turns out to be homological algebra. This indicates that model categories give us a way of doing ”homotopical algebra”. ...
... Every object in the model category is fibrant. Cofibrant objects are chain complexes of projective modules. This example is of interest because the homotopy theory chain complexes turns out to be homological algebra. This indicates that model categories give us a way of doing ”homotopical algebra”. ...
18. Fibre products of schemes The main result of this section is
... Proof. Let pi : Fi −→ X be the natural morphism and let Fij = p−1 i (Xj ). Note that Fij is isomorphic to the fibre product of Xi ∩ Xj and Y over S. Indeed if Z maps to Xij and Y over S, it maps to Xi and Y over S. But then Z maps to Fi , by the universal property of the fibre product. It is clear ...
... Proof. Let pi : Fi −→ X be the natural morphism and let Fij = p−1 i (Xj ). Note that Fij is isomorphic to the fibre product of Xi ∩ Xj and Y over S. Indeed if Z maps to Xij and Y over S, it maps to Xi and Y over S. But then Z maps to Fi , by the universal property of the fibre product. It is clear ...
Notes 1
... These simple examples already raise many questions. For instance, take the two sets Γ1 , Γ2 . Although they are both sets of three points in A2 , their equations look very different. Try to answer the questions: Can Γ1 be defined by equations of degree at most two? Did we need three equations to def ...
... These simple examples already raise many questions. For instance, take the two sets Γ1 , Γ2 . Although they are both sets of three points in A2 , their equations look very different. Try to answer the questions: Can Γ1 be defined by equations of degree at most two? Did we need three equations to def ...
Complex Spaces
... 4.1 Definition. An analytic algebra A is a C-algebra which is different from AnAlgo the zero algebra and such there exist an n and a surjective algebra homomorphism On → A. A ring R is called a local ring if it is not the zero ring and if the set of nonunits is an ideal. This ideal is then a maximal ...
... 4.1 Definition. An analytic algebra A is a C-algebra which is different from AnAlgo the zero algebra and such there exist an n and a surjective algebra homomorphism On → A. A ring R is called a local ring if it is not the zero ring and if the set of nonunits is an ideal. This ideal is then a maximal ...
EBERLEIN–ŠMULYAN THEOREM FOR ABELIAN TOPOLOGICAL
... A subset A of a topological space X is (1) relatively compact if its closure A is compact; (2) relatively countably compact if each sequence in A has a cluster point in X; (3) relatively sequentially compact if each sequence in A has a subsequence converging to a point of X; (4) countably compact, s ...
... A subset A of a topological space X is (1) relatively compact if its closure A is compact; (2) relatively countably compact if each sequence in A has a cluster point in X; (3) relatively sequentially compact if each sequence in A has a subsequence converging to a point of X; (4) countably compact, s ...
On the group of isometries of the Urysohn universal metric space
... Abstract We show that the group mentioned in the title, equipped with the topology of pointwise convergence, is a universal topological group with a countable base. Keywords: topological group, Urysohn space Classification: 22A05, 54E40 Does there exist a universal topological group with a countable ...
... Abstract We show that the group mentioned in the title, equipped with the topology of pointwise convergence, is a universal topological group with a countable base. Keywords: topological group, Urysohn space Classification: 22A05, 54E40 Does there exist a universal topological group with a countable ...
Homological algebra
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. The development of homological algebra was closely intertwined with the emergence of category theory. By and large, homological algebra is the study of homological functors and the intricate algebraic structures that they entail. One quite useful and ubiquitous concept in mathematics is that of chain complexes, which can be studied both through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other 'tangible' mathematical objects. A powerful tool for doing this is provided by spectral sequences.From its very origins, homological algebra has played an enormous role in algebraic topology. Its sphere of influence has gradually expanded and presently includes commutative algebra, algebraic geometry, algebraic number theory, representation theory, mathematical physics, operator algebras, complex analysis, and the theory of partial differential equations. K-theory is an independent discipline which draws upon methods of homological algebra, as does the noncommutative geometry of Alain Connes.