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Gdimension over local homomorphisms. Applications to the Frobenius endomorphism
 Ill. Jour. Math
"... Abstract. We develop a theory of Gdimension for modules over local homomorphisms which encompasses the classical theory of Gdimension for finite modules over local rings. As an application, we prove that a local ring R of characteristic p is Gorenstein if and only if it possesses a nonzero finite ..."
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Abstract. We develop a theory of Gdimension for modules over local homomorphisms which encompasses the classical theory of Gdimension for finite modules over local rings. As an application, we prove that a local ring R of characteristic p is Gorenstein if and only if it possesses a nonzero finite module of finite projective dimension that has finite Gdimension when considered as an Rmodule via some power of the Frobenius endomorphism of R. We also prove results that track the behavior of Gorenstein properties of local homomorphisms under (de)composition. 1.
Homology over local homomorphisms
 Amer. J. Math
"... Abstract. The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism ϕ: R→S. Various techniques are developed to study the new invariants and to establish their basic propert ..."
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Cited by 10 (2 self)
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Abstract. The notions of Betti numbers and of Bass numbers of a finite module N over a local ring R are extended to modules that are only assumed to be finite over S, for some local homomorphism ϕ: R→S. Various techniques are developed to study the new invariants and to establish their basic properties. In several cases they are computed in closed form. Applications go in several directions. One is to identify new classes of finite Rmodules whose classical Betti numbers or Bass numbers have extremal growth. Another is to transfer ring theoretical properties between R and S in situations where S may have infinite flat dimension over R. A third is to obtain criteria for a ring equipped with a ‘contracting ’ endomorphism—such as the Frobenius endomorphism—to be regular or complete intersection; these results represent broad generalizations of Kunz’s characterization of regularity in prime characteristic.
AndreQuillen Homology Of Algebra Retracts
, 2001
"... this paper. Let K k be a surjective morphism of DG algebras, where k is a eld concentrated in degree 0, and let k ,! l be a eld extension. If K ,! KhX i k is a factorization as in 1.7, then the unique DGalgebra structure on lhX i = KhX K l is admissible. Thus, Tor de nes a functor from th ..."
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Cited by 5 (3 self)
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this paper. Let K k be a surjective morphism of DG algebras, where k is a eld concentrated in degree 0, and let k ,! l be a eld extension. If K ,! KhX i k is a factorization as in 1.7, then the unique DGalgebra structure on lhX i = KhX K l is admissible. Thus, Tor de nes a functor from the category of diagrams K k ,! l, with the obvious morphisms, to the category ofalgebras and their morphisms. 2. Factorizations of local homomorphisms Let ' : (R; m; k) ! (S; n; l) be a homomorphism of local rings, which is local in the sense that '(m) n. A regular factorization of ' is a commutative diagram # # # # # # # # ;; # # # # # # # # of local homomorphisms such that the Rmodule R is at, the ring R is regular, and the map R ! S is surjective. Regular factorizations are often easily found, for instance when ' is essentially of nite type (in particular, surjective), or when ' is the canonical embedding of R in its completion with respect to the maximal ideal. In this paper they are mostly used through the following construction of Avramov, Foxby, and B. Herzog [10]
On simplicial commutative algebras with vanishing AndréQuillen homology
 Invent. Math
, 2000
"... Abstract. In this paper, we study the AndréQuillen homology of simplicial commutative ℓalgebras, ℓ a field, having certain vanishing properties. When ℓ has nonzero characteristic, we obtain an algebraic version of a theorem of J.P. Serre and Y. Umeda that characterizes such simplicial algebras h ..."
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Cited by 4 (2 self)
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Abstract. In this paper, we study the AndréQuillen homology of simplicial commutative ℓalgebras, ℓ a field, having certain vanishing properties. When ℓ has nonzero characteristic, we obtain an algebraic version of a theorem of J.P. Serre and Y. Umeda that characterizes such simplicial algebras having bounded homotopy groups. We further discuss how this theorem fails in the rational case and, as an application, indicate how the algebraic Serre theorem can be used to resolve a conjecture of D. Quillen for algebras of finite type over Noetherian rings, which have nonzero characteristic. Overview Algebraic Serre Theorem. The following topological theorem is due to J.P. Serre [17] at the prime 2 and to Y. Umeda [19] at odd primes. Serre’s Theorem. Let X be a nilpotent space such that Hs(X; Fp) = 0 for s ≫ 0 and each Hs(X; Fp) is finite dimensional. Then the following are equivalent 1. πs(X) ⊗ Z/p = 0, s ≫ 0; 2. πs(X) ⊗ Z/p = 0, s ≥ 2. In [1, 15, 16], M. André and D. Quillen constructed the notion of a homology D∗(AR; M) for a homomorphism R → A of simplicial commutative rings, with coefficients in a simplicial Amodule M. These homology groups can be defined as π∗(L(AR) ⊗A M) where the simplicial Amodule L(AR) is called the cotangent complex of A over R. We now propose an algebraic analogue of Serre’s Theorem for simplicial augmented ℓalgebras. To accomplish this we will take simplicial homotopy π∗(−) to be the analogue of H∗(−; Fp) and H Q ∗ (−) = D∗(−ℓ; ℓ) to be the analogue of π∗(−) ⊗ Z/p. Algebraic Serre Theorem. Let A be a homotopy connected (i.e. π0A = ℓ) simplicial supplemented commutative ℓalgebra, with char ℓ ̸ = 0, such that π∗A is a finite graded ℓmodule. Then the following are equivalent 1. HQ s (A) = 0, s ≫ 0; 2. HQ s (A) = 0, s ≥ 2.
Finite Generation Of Hochschild Homology Algebras
 INVENT. MATH
, 2000
"... We prove converses of the HochschildKostantRosenberg Theorem, in particular: If a commutative algebra S is at and essentially of finite type over a noetherian ring , and the Hochschild homology HH (S j) is a finitely generated Salgebra for shuffle products, then S is smooth over . ..."
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Cited by 3 (2 self)
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We prove converses of the HochschildKostantRosenberg Theorem, in particular: If a commutative algebra S is at and essentially of finite type over a noetherian ring , and the Hochschild homology HH (S j) is a finitely generated Salgebra for shuffle products, then S is smooth over .
Jacobian Criteria For Complete Intersections. The Graded Case
, 1994
"... . Let P be a positively graded polynomial ring over a field k of characteristic zero, let I be a homogeneous ideal of P , and set R = P=I. The paper investigates the homological properties of some Rmodules canonically associated with R, among them the module\Omega Rjk of Kahler differentials ..."
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. Let P be a positively graded polynomial ring over a field k of characteristic zero, let I be a homogeneous ideal of P , and set R = P=I. The paper investigates the homological properties of some Rmodules canonically associated with R, among them the module\Omega Rjk of Kahler differentials and the conormal module I=I 2 . It is shown that a subexponential bound on the Betti numbers of any of these modules implies that I is generated by a P regular sequence. In particular, the finiteness of the projective dimension of the conormal module implies R is a complete intersection. Similarly, the finiteness of the projective dimension of the differential module implies R is a reduced complete intersection. This provides strong converses to some wellknown properties of complete intersections, and establishes special cases of conjectures of Vasconcelos. The proofs of these results make extensive use of differential graded homological algebra. The crucial step is to show that...
ON SIMPLICIAL COMMUTATIVE ALGEBRAS WITH NOETHERIAN HOMOTOPY
, 2008
"... Abstract. In this paper, we introduce a strategy for studying simplicial commutative algebras over general commutative rings R. Given such a simplicial algebra A, this strategy involves replacing A with a connected simplicial commutative k(℘)algebra A(℘), for each ℘ ∈ Spec(π0A), which we call the ..."
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Abstract. In this paper, we introduce a strategy for studying simplicial commutative algebras over general commutative rings R. Given such a simplicial algebra A, this strategy involves replacing A with a connected simplicial commutative k(℘)algebra A(℘), for each ℘ ∈ Spec(π0A), which we call the connected component of A at ℘. These components retain most of the AndréQuillen homology of A when the coefficients are k(℘)modules (k(℘) = residue field of ℘ in π0A). Thus these components should carry quite a bit of the homotopy theoretic information for A. Our aim will be to apply this strategy to those simplicial algebras which possess Noetherian homotopy. This allows us to have sophisticated techniques from commutative algebra at our disposal. One consequence of our efforts will be to resolve a more general form of a conjecture of Quillen that was posed in [13]. Overview Our focus, in this paper, is to take the view that the study of Noetherian rings and algebras through homological methods is a special case of the study of simplicial commutative algebras having Noetherian homotopy type. Our goal is to show that such
Nilpotency in the homotopy of simplicial commutative algebras
 Department of Mathematics, Purdue University, West Lafayette
, 2001
"... Abstract. In this paper, we continue a study of simplicial commutative algebras with finite AndréQuillen homology, that was begun in [19]. Here we restrict our focus to simplicial algebras having characteristic 2. Our aim is to find a generalization of the main theorem in [19]. In particular, we re ..."
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Cited by 1 (0 self)
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Abstract. In this paper, we continue a study of simplicial commutative algebras with finite AndréQuillen homology, that was begun in [19]. Here we restrict our focus to simplicial algebras having characteristic 2. Our aim is to find a generalization of the main theorem in [19]. In particular, we replace the finiteness condition on homotopy with a weaker condition expressed in terms of nilpotency for the action of the homotopy operations. Coupled with the finiteness assumption on AndréQuillen homology, this nilpotency condition provides a way to bound the height at which the homology vanishes. As a consequence, we establish a special case of an open conjecture of Quillen.
Detected typos in the paper: L.L. Avramov, Infinite free resolutions
, 1999
"... 24> = Rhx 1 ; : : : ; x e+r j @(x i ) = t i for i = 1; : : : ; e ; @(x i ) = t i @(y j ) = z j i @(x e+j ) = z j for j = 1; : : : ; ri 60 5 A 0 \Gamma! S = H 0 (AhXi) A \Gamma! S = H 0 (AhXi) 7 : A \Gamma! : AhXi \Gamma! 61 3 x = x 2 Xn + ffi< ..."
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24> = Rhx 1 ; : : : ; x e+r j @(x i ) = t i for i = 1; : : : ; e ; @(x i ) = t i @(y j ) = z j i @(x e+j ) = z j for j = 1; : : : ; ri 60 5 A 0 \Gamma! S = H 0 (AhXi) A \Gamma! S = H 0 (AhXi) 7 : A \Gamma! : AhXi \Gamma! 61 3 x = x 2 Xn + ffi<