Andrei Prasolov - Academia.edu (original) (raw)

Papers by Andrei Prasolov

arXiv (Cornell University), May 16, 2011

arXiv (Cornell University), 2018

The categories pCS(X,Pro(k)) of precosheaves and CS(X,Pro(k)) of cosheaves on a small Grothendiec... more The categories pCS(X,Pro(k)) of precosheaves and CS(X,Pro(k)) of cosheaves on a small Grothendieck site X, with values in the category Pro(k) of pro-k-modules, are constructed. It is proved that pCS(X,Pro(k)) satisfies the AB4 and AB5* axioms, while CS(X,Pro(k)) satisfies AB3 and AB5*. Homology theories for cosheaves and precosheaves, based on quasi-projective resolutions, are constructed and investigated.

Topology and its Applications, Sep 1, 2013

In this paper it is investigated whether various shape homology theories satisfy the Universal Co... more In this paper it is investigated whether various shape homology theories satisfy the Universal Coefficients Formula (UCF). It is proved that pro-homology and strong homology satisfy UCF in the class FAB of finitely generated abelian groups, while they do not satisfy UCF in the class AB of all abelian groups. Two new shape homology theories (called UCF-balanced) are constructed. It is proved that balanced pro-homology satisfies UCF in the class AB, while balanced strong homology satisfies UCF only in the class FAB.

Topology and its Applications, Sep 1, 2005

A counterexample is constructed that shows that neither higher inverse limits of pro-groups nor s... more A counterexample is constructed that shows that neither higher inverse limits of pro-groups nor strong homology of topological spaces are additive. The previous counterexample by S. Mardesić and A. Prasolov depended on the Continuum Hypothesis. The approach developed in this paper is applied also to that example in order to calculate the cardinality of the corresponding strong homology groups. It appeared that the above cardinality depends essentially on a set-theoretic model: it is hypercontinuum under a weaker version of the Continuum Hypothesis, and zero under the Proper Forcing Axiom.

Topology and its Applications, 1998

The main purpose of this paper is to describe a natural four-term filtration and the associated g... more The main purpose of this paper is to describe a natural four-term filtration and the associated graded group of the strong homology group ??, (X; G) of compact Hausdorff spaces X. This result includes the Milnor exact sequence and the universal coefficient formula. It is also shown that ??, (X; G) = 0, for m < 0. The proofs depend on the fact that the derived limits limp of the homology progroups pro-H,(X; G) vanish for p 2 2. Therefore, a detailed proof of this fact is also included. 0 1998 Elsevier Science B.V.

arXiv (Cornell University), May 5, 2016

It is proved that for any Grothendieck site X, there exists a coreflection (called cosheafificati... more It is proved that for any Grothendieck site X, there exists a coreflection (called cosheafification) from the category of precosheaves on X with values in a category K, to the full subcategory of cosheaves, provided either K or K op is locally presentable. If K is cocomplete, such a coreflection is built explicitly for the (pre)cosheaves with values in the category Pro (K) of proobjects in K. In the case of precosheaves on topological spaces, it is proved that any precosheaf with values in Pro (K) is smooth, i.e. is strongly locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.

Annali Dell'universita' Di Ferrara, Oct 3, 2007

We propose a new cryptographic scheme of ElGamal type. The scheme is based on algebraic systems d... more We propose a new cryptographic scheme of ElGamal type. The scheme is based on algebraic systems defined in the paper-semialgebras (Sect. 2). The main examples are semialgebras of polynomial mappings over a finite field K , and their factor-semialgebras. Given such a semialgebra R, one chooses an invertible element a ∈ R * of finite order r , and a random integer s. One chooses also a finite dimensional K-submodule V of R. The 4-tuple (R, V, a, b) where b = a s forms the public key for the cryptosystem, while r and s form the secret key. A plain text can be viewed as a sequence of elements of the field K. That sequence is divided into blocks of length dim(V) which, in turn, correspond to uniquely determined elements X i of V. We propose three different methods (A, B, and C, see Definition 1.1) of encoding/decoding the sequence of X i. The complexity of cracking the proposed cryptosystem is based on the Discrete Logarithm Problem for polynomial mappings (see Sect. 1.1). No methods of cracking the problem, except for the "brute force" (see Sect. 1.1) with Ω(r) time, are known so far.

Topology and its Applications, Mar 1, 2012

Let D be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck... more Let D be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck site X, there exists a reflector from the category of precosheaves on X with values in D to the full subcategory of cosheaves. In the case of precosheaves on topological spaces, it is proved that any precosheaf is smooth, i.e. is locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.

arXiv (Cornell University), May 1, 2013

In this paper it is investigated whether various shape homology theories satisfy the Universal Co... more In this paper it is investigated whether various shape homology theories satisfy the Universal Coefficients Formula (UCF). It is proved that pro-homology and strong homology satisfy UCF in the class FAB of finitely generated abelian groups, while they do not satisfy UCF in the class AB of all abelian groups. Two new shape homology theories (called UCF-balanced) are constructed. It is proved that balanced pro-homology satisfies UCF in the class AB, while balanced strong homology satisfies UCF only in the class FAB.

Mathematical Notes, Jul 1, 1982

Topology and its Applications, Jun 1, 2001

We develop strong shape theory for arbitrary topological pro-spaces. When restricted to spaces, o... more We develop strong shape theory for arbitrary topological pro-spaces. When restricted to spaces, our theory is equivalent to the Lisica-Mardesić strong shape theory. We construct also extraordinary strong homology H * (_ , E) for pro-spaces with the coefficients in a spectrum. The theory is isomorphic to the Lisica-Mardesić strong homology H * (_ , G) when E = K(G) is the Eilenberg-MacLane spectrum.

Transactions of the American Mathematical Society, Jun 1, 1988

Using the continuum hypothesis (CH) we show that strong homology groups HP(X) do not satisfy Miln... more Using the continuum hypothesis (CH) we show that strong homology groups HP(X) do not satisfy Milnor's additivity axiom. Moreover, CH implies that strong homology does not have compact supports and that HP(X) need not vanish for p < 0.

arXiv: Algebraic Topology, May 1, 2013

In this paper it is investigated whether various shape homology theories satisfy the Universal Co... more In this paper it is investigated whether various shape homology theories satisfy the Universal Coefficients Formula (UCF). It is proved that pro-homology and strong homology satisfy UCF in the class FAB of finitely generated abelian groups, while they do not satisfy UCF in the class AB of all abelian groups. Two new shape homology theories (called UCF-balanced) are constructed. It is proved that balanced pro-homology satisfies UCF in the class AB, while balanced strong homology satisfies UCF only in the class FAB.

Various connections between strong homology theory and set theory are discussed. The Continuum Hy... more Various connections between strong homology theory and set theory are discussed. The Continuum Hypothesis, the Proper Forcing Axiom, and other set-theoretic axioms imply different values of higher derived limits, and strong homology groups. The topic we are going to talk about, has quite a long history. In [MP88] it was proved that strong homology neither is additive nor has compact supports, provided the Continuum Hypothesis (CH) is assumed. In that paper, an abelian pro-group A was constructed such that the groups lim ←− A, s ≥ 1, serve as the obstructions both to additivity and having compact supports. Assuming CH, it was proved that lim ←− A 6= 0. Later, in [Pra05], under a weaker assumption d = א1, it was proved that the cardinality of lim ←− A is quite large: ∣∣∣lim ←−1A∣∣∣ = אא1 1 . In [DSV89], assuming the Proper Forcing Axiom (PFA), it was proved that lim ←− A = 0. However, recently in [Ber15], it was shown that, assuming PFA, lim ←− A 6= 0, lim ←− A = 0, s 6= 0, 2. An inte...

Let D be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck... more Let D be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck site X, there exists a reflector from the category of precosheaves on X with values in D to the full subcategory of cosheaves. In the case of precosheaves on topological spaces, it is proved that any precosheaf is smooth, i.e. is locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.

It is proved that for any Grothendieck site X, there exists a coreflection (called cosheafificati... more It is proved that for any Grothendieck site X, there exists a coreflection (called cosheafification) from the category of precosheaves on X with values in a category K, to the full subcategory of cosheaves, provided either K or K^op is locally presentable. If K is cocomplete, such a coreflection is built explicitly for the (pre)cosheaves with values in the category Pro K) of pro-objects in K. In the case of precosheaves on topological spaces, it is proved that any precosheaf with values in Pro( K) is smooth, i.e. is strongly locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.

The categories pCS(X,Pro(k)) of precosheaves and CS(X,Pro(k)) of cosheaves on a small Grothendiec... more The categories pCS(X,Pro(k)) of precosheaves and CS(X,Pro(k)) of cosheaves on a small Grothendieck site X, with values in the category Pro(k) of pro-k-modules, are constructed. It is proved that pCS(X,Pro(k)) satisfies the AB4 and AB5* axioms, while CS(X,Pro(k)) satisfies AB3 and AB5*. Homology theories for cosheaves and precosheaves, based on quasi-projective resolutions, are constructed and investigated.

Selected Papers in 𝐾-Theory, 1992

Journal of Soviet Mathematics, 1986

Topology and its Applications, 1998

arXiv (Cornell University), May 16, 2011

arXiv (Cornell University), 2018

The categories pCS(X,Pro(k)) of precosheaves and CS(X,Pro(k)) of cosheaves on a small Grothendiec... more The categories pCS(X,Pro(k)) of precosheaves and CS(X,Pro(k)) of cosheaves on a small Grothendieck site X, with values in the category Pro(k) of pro-k-modules, are constructed. It is proved that pCS(X,Pro(k)) satisfies the AB4 and AB5* axioms, while CS(X,Pro(k)) satisfies AB3 and AB5*. Homology theories for cosheaves and precosheaves, based on quasi-projective resolutions, are constructed and investigated.

Topology and its Applications, Sep 1, 2013

In this paper it is investigated whether various shape homology theories satisfy the Universal Co... more In this paper it is investigated whether various shape homology theories satisfy the Universal Coefficients Formula (UCF). It is proved that pro-homology and strong homology satisfy UCF in the class FAB of finitely generated abelian groups, while they do not satisfy UCF in the class AB of all abelian groups. Two new shape homology theories (called UCF-balanced) are constructed. It is proved that balanced pro-homology satisfies UCF in the class AB, while balanced strong homology satisfies UCF only in the class FAB.

Topology and its Applications, Sep 1, 2005

A counterexample is constructed that shows that neither higher inverse limits of pro-groups nor s... more A counterexample is constructed that shows that neither higher inverse limits of pro-groups nor strong homology of topological spaces are additive. The previous counterexample by S. Mardesić and A. Prasolov depended on the Continuum Hypothesis. The approach developed in this paper is applied also to that example in order to calculate the cardinality of the corresponding strong homology groups. It appeared that the above cardinality depends essentially on a set-theoretic model: it is hypercontinuum under a weaker version of the Continuum Hypothesis, and zero under the Proper Forcing Axiom.

Topology and its Applications, 1998

The main purpose of this paper is to describe a natural four-term filtration and the associated g... more The main purpose of this paper is to describe a natural four-term filtration and the associated graded group of the strong homology group ??, (X; G) of compact Hausdorff spaces X. This result includes the Milnor exact sequence and the universal coefficient formula. It is also shown that ??, (X; G) = 0, for m < 0. The proofs depend on the fact that the derived limits limp of the homology progroups pro-H,(X; G) vanish for p 2 2. Therefore, a detailed proof of this fact is also included. 0 1998 Elsevier Science B.V.

arXiv (Cornell University), May 5, 2016

It is proved that for any Grothendieck site X, there exists a coreflection (called cosheafificati... more It is proved that for any Grothendieck site X, there exists a coreflection (called cosheafification) from the category of precosheaves on X with values in a category K, to the full subcategory of cosheaves, provided either K or K op is locally presentable. If K is cocomplete, such a coreflection is built explicitly for the (pre)cosheaves with values in the category Pro (K) of proobjects in K. In the case of precosheaves on topological spaces, it is proved that any precosheaf with values in Pro (K) is smooth, i.e. is strongly locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.

Annali Dell'universita' Di Ferrara, Oct 3, 2007

We propose a new cryptographic scheme of ElGamal type. The scheme is based on algebraic systems d... more We propose a new cryptographic scheme of ElGamal type. The scheme is based on algebraic systems defined in the paper-semialgebras (Sect. 2). The main examples are semialgebras of polynomial mappings over a finite field K , and their factor-semialgebras. Given such a semialgebra R, one chooses an invertible element a ∈ R * of finite order r , and a random integer s. One chooses also a finite dimensional K-submodule V of R. The 4-tuple (R, V, a, b) where b = a s forms the public key for the cryptosystem, while r and s form the secret key. A plain text can be viewed as a sequence of elements of the field K. That sequence is divided into blocks of length dim(V) which, in turn, correspond to uniquely determined elements X i of V. We propose three different methods (A, B, and C, see Definition 1.1) of encoding/decoding the sequence of X i. The complexity of cracking the proposed cryptosystem is based on the Discrete Logarithm Problem for polynomial mappings (see Sect. 1.1). No methods of cracking the problem, except for the "brute force" (see Sect. 1.1) with Ω(r) time, are known so far.

Topology and its Applications, Mar 1, 2012

Let D be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck... more Let D be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck site X, there exists a reflector from the category of precosheaves on X with values in D to the full subcategory of cosheaves. In the case of precosheaves on topological spaces, it is proved that any precosheaf is smooth, i.e. is locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.

arXiv (Cornell University), May 1, 2013

In this paper it is investigated whether various shape homology theories satisfy the Universal Co... more In this paper it is investigated whether various shape homology theories satisfy the Universal Coefficients Formula (UCF). It is proved that pro-homology and strong homology satisfy UCF in the class FAB of finitely generated abelian groups, while they do not satisfy UCF in the class AB of all abelian groups. Two new shape homology theories (called UCF-balanced) are constructed. It is proved that balanced pro-homology satisfies UCF in the class AB, while balanced strong homology satisfies UCF only in the class FAB.

Mathematical Notes, Jul 1, 1982

Topology and its Applications, Jun 1, 2001

We develop strong shape theory for arbitrary topological pro-spaces. When restricted to spaces, o... more We develop strong shape theory for arbitrary topological pro-spaces. When restricted to spaces, our theory is equivalent to the Lisica-Mardesić strong shape theory. We construct also extraordinary strong homology H * (_ , E) for pro-spaces with the coefficients in a spectrum. The theory is isomorphic to the Lisica-Mardesić strong homology H * (_ , G) when E = K(G) is the Eilenberg-MacLane spectrum.

Transactions of the American Mathematical Society, Jun 1, 1988

Using the continuum hypothesis (CH) we show that strong homology groups HP(X) do not satisfy Miln... more Using the continuum hypothesis (CH) we show that strong homology groups HP(X) do not satisfy Milnor's additivity axiom. Moreover, CH implies that strong homology does not have compact supports and that HP(X) need not vanish for p < 0.

arXiv: Algebraic Topology, May 1, 2013

In this paper it is investigated whether various shape homology theories satisfy the Universal Co... more In this paper it is investigated whether various shape homology theories satisfy the Universal Coefficients Formula (UCF). It is proved that pro-homology and strong homology satisfy UCF in the class FAB of finitely generated abelian groups, while they do not satisfy UCF in the class AB of all abelian groups. Two new shape homology theories (called UCF-balanced) are constructed. It is proved that balanced pro-homology satisfies UCF in the class AB, while balanced strong homology satisfies UCF only in the class FAB.

Various connections between strong homology theory and set theory are discussed. The Continuum Hy... more Various connections between strong homology theory and set theory are discussed. The Continuum Hypothesis, the Proper Forcing Axiom, and other set-theoretic axioms imply different values of higher derived limits, and strong homology groups. The topic we are going to talk about, has quite a long history. In [MP88] it was proved that strong homology neither is additive nor has compact supports, provided the Continuum Hypothesis (CH) is assumed. In that paper, an abelian pro-group A was constructed such that the groups lim ←− A, s ≥ 1, serve as the obstructions both to additivity and having compact supports. Assuming CH, it was proved that lim ←− A 6= 0. Later, in [Pra05], under a weaker assumption d = א1, it was proved that the cardinality of lim ←− A is quite large: ∣∣∣lim ←−1A∣∣∣ = אא1 1 . In [DSV89], assuming the Proper Forcing Axiom (PFA), it was proved that lim ←− A = 0. However, recently in [Ber15], it was shown that, assuming PFA, lim ←− A 6= 0, lim ←− A = 0, s 6= 0, 2. An inte...

Let D be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck... more Let D be the category of pro-sets (or abelian pro-groups). It is proved that for any Grothendieck site X, there exists a reflector from the category of precosheaves on X with values in D to the full subcategory of cosheaves. In the case of precosheaves on topological spaces, it is proved that any precosheaf is smooth, i.e. is locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.

It is proved that for any Grothendieck site X, there exists a coreflection (called cosheafificati... more It is proved that for any Grothendieck site X, there exists a coreflection (called cosheafification) from the category of precosheaves on X with values in a category K, to the full subcategory of cosheaves, provided either K or K^op is locally presentable. If K is cocomplete, such a coreflection is built explicitly for the (pre)cosheaves with values in the category Pro K) of pro-objects in K. In the case of precosheaves on topological spaces, it is proved that any precosheaf with values in Pro( K) is smooth, i.e. is strongly locally isomorphic to a cosheaf. Constant cosheaves are constructed, and there are established connections with shape theory.

The categories pCS(X,Pro(k)) of precosheaves and CS(X,Pro(k)) of cosheaves on a small Grothendiec... more The categories pCS(X,Pro(k)) of precosheaves and CS(X,Pro(k)) of cosheaves on a small Grothendieck site X, with values in the category Pro(k) of pro-k-modules, are constructed. It is proved that pCS(X,Pro(k)) satisfies the AB4 and AB5* axioms, while CS(X,Pro(k)) satisfies AB3 and AB5*. Homology theories for cosheaves and precosheaves, based on quasi-projective resolutions, are constructed and investigated.

Selected Papers in 𝐾-Theory, 1992

Journal of Soviet Mathematics, 1986

Topology and its Applications, 1998