The Relation Type of Varieties

In this paper, we introduce the notion of relation type of analytic and formal algebras and prove that it is well-defined and invariant by describing this notion in terms of the Andr\'e-Quillen homology and using the Jacobi-Zariski long exact sequence of homology. In particular, the relation type is an invariant of schemes of finite type over a field, analytic varieties, and algebroid varieties.Comment: 20 pages. Comments are welcom

Adjoint algebraic entropy

The new notion of adjoint algebraic entropy of endomorphisms of Abelian groups is introduced. Various examples and basic properties are provided. It is proved that the adjoint algebraic entropy of an endomorphism equals the algebraic entropy of the adjoint endomorphism of the Pontryagin dual. As applications, we compute the adjoint algebraic entropy of the shift endomorphisms of direct sums, and we prove an Addition Theorem for the adjoint algebraic entropy of bounded Abelian groups. A dichotomy is established, stating that the adjoint algebraic entropy of any endomorphism can take only values zero or infinity. As a consequence, we obtain the following surprising discontinuity criterion for endomorphisms: every endomorphism of a compact abelian group, having finite positive algebraic entropy, is discontinuous.Comment: 27 page

Algebraic entropy of shift endomorphisms on abelian groups

For every finite-to-one map \u3bb:\u393\u2192\u393 and for every abelian group K, the generalized shift \u3c3\u3bb of the direct sum 95_\u393 K is the endomorphism defined by (x_ i)\u21a6(x_\u3bb(i)). In this paper we analyze and compute the algebraic entropy of a generalized shift, which turns out to depend on the cardinality of K, but mainly on the function \u3bb. We give many examples showing that the generalized shifts provide a very useful universal tool for producing counter-examples

G-GORENSTEIN -kompleksit

G-GORENSTEIN -kompleksit Tämän väitöskirjan tavoitteena on esittää klassiselle Gorenstein-kompleksin käsitteelle vastine Gorenstein-homologisessa algebrassa: “G-Gorenstein-kompleksi”. Väitöskirjassa tutkitaan G-Gorenstein-kompleksien rakennetta, ja selvitetään, missä määrin Gorenstein-komplekseja koskevat klassiset tulokset yleistyvät koskemaan myös näitä komplekseja. Väitöskirjassa havaitaan, että tiettyä dimensiota olevien G-Gorenstein-kompleksien kategoria on ekvivalentti modulien G-luokan kanssa. Erityisesti osoittautuu, että ne Cousin-kompleksit, joiden termit ovat Gorenstein-injektiivisiä ja joiden homologia on rajoitettu ja äärellisviritteinen, muodostavat ensin mainitun kategorian kanssa ekvivalentin kategorian.G-Gorenstein Complexes The aim of this thesis is to present in the context of Gorenstein homological algebra the notion of a “G-Gorenstein complex” as the counterpart of the classical notion of a Gorenstein complex. We investigate the structure of a G-Gorenstein complex. We will also find out in which extent classical results about Gorenstein complexes generalize to this case. We establish equivalences between the category of G-Gorenstein complexes of a fixed dimension and the G-class of modules. In particular, the first category turns out to be equivalent with a category of Cousin complexes whose terms are Gorenstein injective and homology bounded and finitely generated

G-GORENSTEIN -kompleksit

G-GORENSTEIN -kompleksit Tämän väitöskirjan tavoitteena on esittää klassiselle Gorenstein-kompleksin käsitteelle vastine Gorenstein-homologisessa algebrassa: “G-Gorenstein-kompleksi”. Väitöskirjassa tutkitaan G-Gorenstein-kompleksien rakennetta, ja selvitetään, missä määrin Gorenstein-komplekseja koskevat klassiset tulokset yleistyvät koskemaan myös näitä komplekseja. Väitöskirjassa havaitaan, että tiettyä dimensiota olevien G-Gorenstein-kompleksien kategoria on ekvivalentti modulien G-luokan kanssa. Erityisesti osoittautuu, että ne Cousin-kompleksit, joiden termit ovat Gorenstein-injektiivisiä ja joiden homologia on rajoitettu ja äärellisviritteinen, muodostavat ensin mainitun kategorian kanssa ekvivalentin kategorian.G-Gorenstein Complexes The aim of this thesis is to present in the context of Gorenstein homological algebra the notion of a “G-Gorenstein complex” as the counterpart of the classical notion of a Gorenstein complex. We investigate the structure of a G-Gorenstein complex. We will also find out in which extent classical results about Gorenstein complexes generalize to this case. We establish equivalences between the category of G-Gorenstein complexes of a fixed dimension and the G-class of modules. In particular, the first category turns out to be equivalent with a category of Cousin complexes whose terms are Gorenstein injective and homology bounded and finitely generated

G-Gorenstein Complexes

We present in the context of Gorenstein homological algebra the notion of a "G-Gorenstein complex" as the counterpart of the classical notion of a Gorenstein complex. In particular, we investigate equivalences between the category of G-Gorenstein complexes of fixed dimension and the G-class of modules