On the Merkulov construction of A -(co)algebras
|
|
- Eugene Moody
- 5 years ago
- Views:
Transcription
1 On the Merkulov construction of A -(co)algebras Estanislao Herscovich Abstract The aim of this short note is to complete some aspects of a theorem proved by S. Merkulov in [7], Thm. 3.4, as well as to provide a complete proof of the dual result for dg coalgebras. Mathematics subject classification 2010: 16E45, 16T15. Keywords: homological algebra, dg (co)algebras, A -(co)algebras. 1 Introduction The objective of this short note is twofold: (i) omplete some aspects of a theorem of S. Merkulov which in principle produces an A -algebra from a certain dg submodule of a dg algebra, showing that the construction also gives a morphism of A -algebras and that both are strictly unitary under some further assumptions (see Theorem 3.1). These last extra components were not considered in the original statement by Merkulov, but the existence of the morphism of A -algebras appears in a particular case in [6], Prop (ii) Provide a precise statement together with a complete proof of a dual version of the previous theorem for dg coalgebras. This is done in Theorem 4.5. e are mainly interested in (ii), because we need such a result in [1] for our study of the A -coalgebra structure on the group Tor A (K, K) of a nonnegatively graded algebra A and some of their associated A -comodules (e.g. in Thm and Prop of that article). It was used in particular to compute the A -module structure of Ext A(M, k) over the Yoneda algebra of a generalized Koszul algebra A, where M is a generalized Koszul module over A. Incidentally, we find more convenient to work with the formulation in [7], for in [1] we need to deal with the slightly greater generality of (nonsymmetric) bimodules over a (noncommutative) algebra. The proof of the statements added to the result of Merkulov follows the usual philosophy of specific manipulations of equations. However, these new results, which appear in Theorem 3.1, cannot be directly deduced from [7], Thm In particular, the construction of the morphism of A -algebras added to the result by Merkulov allows to compare the dg algebra one starts with and the constructed A -algebra, which seems relevant to us, and it was indeed needed in in [6]. Finally, let us add that the proof of Theorem 4.5 is parallel to the one for dg algebras, This work was also partially supported by UBAYT BA, UBAYT BA, PIP-ONIET , MathAmSud-GR2HOPF, PIT and PIT
2 and, as in the case of dg algebras, the result obtained for dg coalgebras is slightly more general than those obtained in homological perturbation theory, since our assumptions are in general weaker than those of a SDR (cf. [3], or the nice exposition in [4], Section 6). 2 Preliminaries on basic algebraic structures In what follows, k will denote a field and K will be a noncommutative unitary k- algebra. By module (sometimes decorated by adjectives such as graded, or dg) we mean a (not necessarily symmetric) bimodule over K (correspondingly decorated), such that the induced bimodule structure over k is symmetric. For dg algebras, dg coalgebras and A -algebras, we follow the sign conventions of [5], whereas for A -coalgebras we shall use the ones given in [2], Subsection 2.1. e also recall that, if V = n Z V n is a (cohomological) graded K-module, V [m] is the graded module over K whose n-th homogeneous component V [m] n is given by V m+n, for all n, m Z, and it is called the shift of V. e are not going to consider any shift on other gradings, such as the Adams grading. All morphisms between modules will be K-linear on both sides (satisfying further requirements if the modules are decorated as before). One trivially sees that all the standard definitions of graded or dg (co)algebra, or even A -(co)algebra, eventually provided with an Adams grading, and (co)modules over them make perfect sense in the monoidal category of graded K-bimodules, correspondingly provided with an Adams grading. All unadorned tensor products would be over K. Finally, N will denote the set of (strictly) positive integers, whereas N 0 will be the set of nonnegative integers. Similarly, for N N, we denote by N N the set of positive integers greater than or equal to N. 3 On the theorem of Merkulov Let (A, µ A, d A ) be a dg algebra provided with an Adams grading and let A be a dg submodule of A respecting the Adams degree. e assume that there is a linear map Q : A A[ 1] of total degree zero, where A[ 1] denotes the shift of the cohomological degree, satisfying that the image of id A [d A, Q] lies in, where [d A, Q] = d A Q + Q d A is the graded commutator. For all n 2, construct λ n : A n A as follows. Setting formally λ 1 = Q 1, define λ n = ( 1) i+1 µ A ( (Q λ i ) (Q λ n i ) ), (3.1) for n 2. e have the following result, whose first part is [7], Thm. 3.4, whereas the rest is a slightly more general version of [6], Prop. 2.3 and Lemma 2.5. Theorem 3.1. Let (A, µ A, d A ) be a dg algebra provided with an Adams grading and let ι : (, d ) (A, d A ) be a dg submodule of A (respecting the Adams degree). Suppose there is a linear map Q : A A[ 1] of total degree zero, where A[ 1] denotes the shift of the cohomological degree, satisfying that the image of id A [d A, Q] lies in ι( ). For all n N, define m n : n as follows. Set m 1 = d = d A ι and m n = (id A [d A, Q]) λ n ι n, for n 2. Then, (, m ) is an Adams graded A - algebra. Define the collection f : A, where f n : n A is the linear map of total degree (1 n, 0) given by f n = Q λ n ι n, for n N. Then f is a morphism of Adams graded A -algebras, and it is a quasi-isomorphism if and only if ι is so. Furthermore, assume A has a unit 1 A, there is an element 1 such that ι(1 ) = 1 A, Q Q = 0 and Q ι = 0. Then, 1 is a strict unit of the Adams graded A -algebra (, m ), and f : A is a morphism of strictly unitary Adams graded A -algebras. 2
3 Proof. For the first part, the proof given in [7], based on that of the corresponding Lemmas 3.2 and 3.3, applies verbatim in this context. e also note that the sign convention of that article agrees with the one we follow here. To prove the second assertion we proceed as follows. It suffices to prove that f is a morphism of A -algebras, for the quasi-isomorphism property is immediate. e have thus to show the following reduced form of the Stasheff identities on morphisms MI(n) (see [5], Def. 4.1) ( 1) r+st f r+1+t (id r (r,s,t) I n m s id t ) = d A f n + ( 1) p 1 µ A ( (Q λ p ι p ) (Q λ n p ι (n p) ) ), (3.2) p=1 for all n N, where I n = {(r, s, t) N 0 N N 0 : r + s + t = n}. The case n = 1 is trivial since ι is a morphism of complexes. Moreover, the case n = 2 is also clear, since the left member of (3.2) gives f 1 m 2 f 2 (id m 1 + m 1 id ) = (id A [d A, Q]) µ A ι 2 + Q µ A ι 2 (id d + d id ) = µ A ι 2 d A Q µ A ι 2, (3.3) where we have used the Leibniz property for the derivation d A. The right member of (3.2) gives d A f 2 + µ A ( (Q λ 1 ι) (Q λ 1 ι) ) = d A Q µ A ι 2 + µ A ι 2, which clearly coincides with (3.3). e shall now consider n > 2. Using the same notation as in [7], by the definition of the tensors Φ n and Θ n given in Lemmas 3.2 and 3.3, respectively, we see that Q (Φ n +Θ n ) ι n = Q d A λ n ι n ( 1) r+st f r+1+t (id r m s id t ), for all n N, where In = {(r, s, t) N 0 N N 0 : r+s+t = n, r+t > 0}. Moreover, by the previously mentioned lemmas, the tensor Φ n and Θ n vanish, which implies that ( 1) r+st f r+1+t (id r (r,s,t) I n On the other hand, m s id t ) = f 1 m n + Q d A λ n ι n. f 1 m n + Q d A λ n ι n = λ n ι n d A Q λ n ι n = ( 1) i+1 µ A ( (Q λ i ι i ) (Q λ n i ι (n i) ) ) d A Q λ n ι n, where we have used the definition of m n in the first equality and equation (3.1) in the last one. It is clear that the last member of the previous chain of identities coincides with the right member of (3.2), as was to be shown. The proof of the third assertion follows the same pattern as the one given in [6], Lemma 2.5, but since we are assuming a weaker assumption on Q (called G in that article), we describe roughly how it is done. By the definition of f 2, we see 3
4 that the condition Q ι = 0 implies that f 2 (1 w) = f 2 (w 1 ) = 0. The fact ι is a morphism of dg modules and d A (1 A ) = 0 imply that m 1 (1 ) = 0. Suppose now that we have proved that, for 2 i n 1, f i (w 1,..., w i ) vanishes if there exists j {1,..., i} such that w j = 1. By (3.1) and the inductive hypothesis, we see that λ n (w 1,..., w n ) vanishes if there exists j {2,..., n 1} such that w j = 1, λ n (1, w 2,..., w n ) = f (w 2,..., w n ), and λ n (w 1,..., w, 1 ) = ( 1) n f (w 1,..., w ), for all w 1,..., w n. From the definition of f n and the assumption that Q Q = 0 we conclude that f n (w 1,..., w n ) vanishes if there exists j {1,..., n} such that w j = 1. Moreover, since the image of (id A [d A, Q]) lies in ι( ) and Q ι = 0, we see that 0 = Q (id A [d A, Q]) = Q Q d A Q = (id A [d A, Q]) Q, (3.4) where we have used in the last two equalities that Q Q vanishes. Using our previous description of λ n in terms of f = Q λ ι () for n 3 (if it does not vanish already) and (3.4), we get that m n (w 1,..., w n ) vanishes if there exists j {1,..., n} such that w j = 1. 4 The dual result e shall briefly present the dual procedure to the one introduced by S. Merkulov in [7] to produce an A -algebra structure from a particular data on a dg submodule of a dg algebra. In our case, we produce an A -coalgebra structure on a quotient dg module of a dg coalgebra. e also note that, even though the results of the article of Merkulov are stated for vector spaces, they are clearly seen to be true (by exactly the same arguments) in our more general situation of bimodules over the k-algebra K. Let (,, d ) be a dg coalgebra provided with an Adams grading and let (, d ) (, d ) be a dg module quotient of respecting the Adams degree. Denote by K the kernel of the previous quotient and assume that there is a linear map Q : [ 1] of total degree zero, where [ 1] denotes the shift of the cohomological degree whereas the Adams degree remains unchanged, satisfying that id [d, Q] vanishes on K, where [d, Q] = d Q + Q d is the graded commutator. For all n 2, define γ n : n as follows. Setting formally γ 1 = Q 1, define γ n = ( 1) i+1( (γ n i Q) (γ i Q) ), (4.1) for n 2. e shall say that Q is admissible if the family {γ n } n N 2 is locally finite, i.e. it satisfies that the induced map n 2 n factors through the canonical inclusion n 2 n n 2 n. Fact 4.1. The following identity holds. ( γp (γ n p Q) ) γ 2 + ( 1) p( ) (γ n p Q) γ p γ2 = 0 p=2 p=2 Proof. The identity just follows by replacing the two occurrences of γ p by the recurrent expression given by (4.1) and simplifying the corresponding terms. Fact 4.2. Let e be an endomorphism of of degree zero and define E n (e) = (γ s e) id t ) γ r+1+t, 4
5 where I n = {(r, s, t) N 0 N 2 N 0 : r + s + t = n and r + t > 0}. Then E n(e) = + r=1 + t=1 r (r,s,t) În i=r+1 + t ( 1) i (γ n te id t )(γ iq γ t i+1q) ( 1) r(n r)+r i (id r r+t r (r,s,t) În ( 1) rs+r i( ( (id r γ n re)(γ iq γ 1+r iq) ( 1) rs+r+s i(s+1)( γ iq ( (id (r i) γ se id (i r 1) )γ ) ) iq γ r+1+t iq γ se id t )γ r+1+t iq )), (4.2) where În = {(r, s, t) N N 2 N : r + s + t = n}, and we have omitted the composition symbol to economize space. Proof. The statement follows from the following chain of identities, which uses the definition (4.1), E n(e) = ( 1) t (γ n te id t t=1 + (r,s,t) În = t=1 + r=1 + )γ 1+t + r=1 γ se id t )γ r+1+t t ( 1) i (γ n te id t )(γ iq γ t i+1q) r (r,s,t) În i=r+1 + ( 1) r(n r)+r i (id r r+t r (r,s,t) În ( 1) rs+r i( ( (id r ( 1) r(n r) (id r γ n re)γ r+1 γ n re)(γ iq γ 1+r iq) ( 1) rs+r+s i(s+1)( γ iq ( (id (r i) γ se id (i r 1) )γ ) ) iq γ r+1+t iq γ se id t )γ r+1+t iq )), where we have omitted the composition symbol to reduce space. Lemma 4.3. For n N 3, define Γ n = γ s id t ) γ r+1+t, where I n = {(r, s, t) N 0 N 2 N 0 : r + s + t = n and r + t > 0}. Then Γ n 0, for all n 3. Proof. First note that Γ 3 = (id id ), so the coassociativity of implies that Γ 3 vanishes. Let us now consider n > 3. e note that Γ n = E n (id ), so it can be written as indicated in equation (4.2). Moreover, the terms corresponding to i = 1 in the first sum and to i = r in the second sum in that latter expression of Γ n cancel due to Fact 4.1. Using the definition of Γ m for 3 m n 1 in the remaining terms of the new expression of Γ n we get that n 3 Γ n = ( 1) n i( (γ i Q) (Γ n i Q) ) n 3 ( (Γn j Q) (γ j Q) ). j=1 The lemma thus follows from an inductive argument. 5
6 Lemma 4.4. For n N 2, define H n = γ n d + ( 1) (id r r=0 d id (n r 1) ) γ n ( 1) rs+t( id r (γ s [d, Q]) id t ) γr+1+t, where I n = {(r, s, t) N 0 N 2 N 0 : r + s + t = n and r + t > 0}. Then H n 0, for all n 2. Proof. Note that H 2 = d (id d + d id ), so it vanishes by the Leibniz identity for. Assume now that n > 2. Note that H n = γ n d ( 1) n (id r r=0 Using the definition (4.1), we can write ( H n = (γi Q d ) (γ n i Q) ) ( 1) n i( (γ i Q) (γ n i Q d ) ) i 1 + ( 1) i( ( (id r i=0 r=0 i=0 n i 1 r=0 ( (γ i Q) ( (id r E n (d Q) E n (Q d ). d id (n r 1) ) γ n E n ([d, Q]). d id (i r 1) ) (γ i Q) ) ) (γ n i Q) d id (n i r 1) ) (γ n i Q) )) (4.3) Fact 4.2 expresses E n (Q d ) as a linear combination of sums satisfying that the terms corresponding to i = 1 of its first sum and the terms corresponding to i = r of its second sum cancel the first two sums in (4.3). ombining the remaining terms of (4.3) and using the definition of H m, for 3 m n 1, we get that H n = ( 1) n i( (γ i Q) (H n i Q) ) ( (Hn j Q) (γ j Q) ). j=1 The lemma thus follows from an inductive argument. Theorem 4.5. Let (,, d ) be a dg coalgebra provided with an Adams grading and let ρ : (, d ) (, d ) be a quotient dg module of (respecting the Adams degree) with kernel K. Suppose there is an admissible linear map Q : [ 1] of total degree zero, where [ 1] denotes the shift of the cohomological degree, satisfying that id [d, Q] vanishes on K. For all n N, define n : n as follows. Set 1 = d and n to be the unique map satisfying that n ρ = ρ n γ n (id [d, Q]), for n 2. Then, (, ) is an Adams graded A -coalgebra. Define the collection f :, where f n : n is the linear map of homological degree n 1 and zero Adams degree given by f n = ρ n γ n Q, for n N. Then f is a morphism of Adams graded A -coalgebras. 6
7 Furthermore, assume has a counit ɛ, there a linear map ɛ : K such that ɛ ρ = ɛ, Q Q = 0 and ρ Q = 0. Then, ɛ is a strict counit of the Adams graded A -coalgebra (, ), and f : is a morphism of strictly counitary Adams graded A -coalgebras. Proof. The first part follows from the fact that the Stasheff identity SI(n) for n 3 for the operations is by definition given by ρ n (Γ n + H n ) (id [d, Q]), so it vanishes. The two first Stasheff identities SI(1) and SI(2) are trivial, for is a quotient of. To prove the second assertion we proceed as follows. e have thus to prove the following reduced form of the Stasheff identities on morphisms MI(n) (see [2], eq. (2.2)) (r,s,t) I n s id t ) f r+1+t = f n d + ( 1) n p 1( (ρ p γ p Q) (ρ (n p) γ n p Q) ), (4.4) p=1 for all n N, where I n = {(r, s, t) N 0 N N 0 : r + s + t = n}. The case n = 1 is trivial since ρ is a morphism of complexes. Moreover, the case n = 2 is also clear, since the left member of (4.4) gives 2 f 1 (id id ) f 2 = ρ 2 (id A [d, Q]) + ρ 2 (id d + d id ) Q = ρ 2 ρ 2 Q d, (4.5) where we have used the Leibniz property for the coderivation d. The right member of (4.4) gives f 2 d + ( (ρ γ 1 Q) (ρ γ 1 Q) ) = ρ 2 Q d + ρ 2, which clearly coincides with (4.5). e shall now consider n > 2. By the definition of the tensors Γ n and H n given in Lemmas 4.3 and 4.4, respectively, we see that ρ n (Γ n +H n ) Q = ρ n γ n d Q+ s id t ) f r+1+t, for all n N, where we recall that In = {(r, s, t) N 0 N N 0 : r+s+t = n, r+t > 0}. Moreover, by the previously mentioned lemmas, the tensor Γ n and H n vanish, which implies that (r,s,t) I n On the other hand, s id t ) f r+1+t = n f 1 ρ n γ n d Q. n f 1 ρ n γ n d Q = ρ n γ n ρ n γ n Q d = ( 1) n i 1( (ρ i γ i Q) (ρ (n i) γ n i Q) ) ρ n γ n Q d, where we have used the definition of n in the first equality, and equation (4.1) in the last one. It is clear that the last member of the previous chain of identities coincides with the right member of (4.4), as was to be shown. 7
8 The proof of the third assertion is parallel to the one given to Theorem 3.1. Using the definition of f 2, we see that the condition ρ Q = 0 implies that (ɛ id ) f 2 = (id ɛ ) f 2 = 0. The fact ρ is a morphism of dg modules and ɛ d = 0 imply that ɛ 1 = 0. Suppose now that we have proved that, for 2 i n 1, (id j ɛ id (i j 1) ) f i vanishes for all j {0,..., i 1}. By (4.1) and the inductive hypothesis, we see that (id j ɛ id (n j 1) ) ρ n γ n vanishes for all j {1,..., n 2}, (ɛ id () ) ρ n γ n = ( 1) f, and (id () ɛ ) ρ n γ n = f. By the definition of f n and the assumption that Q Q = 0 we conclude that (id j ɛ id (n j) ) f n vanishes for all j {0,..., n 1}. Moreover, since the image of (id [d, Q]) vanishes on the kernel of ρ and ρ Q = 0, we see that 0 = (id [d, Q]) Q = Q Q d Q = Q (id [d, Q]), (4.6) where we have used in the last two equalities that Q Q vanishes. Using our previous description of (id j ɛ id (n j 1) ) ρ n γ n in terms of f = ρ () γ Q for n 3 (if it does not vanish already) and (4.6), we get that (id j ɛ id (n j) ) n vanishes for all j {0,..., n 1}. The structure of A -coalgebra on given by the previous theorem is called a Merkulov model on, or simply a model. As in the case of dg algebras, note that the result stated in the first two paragraphs of the previous theorem is slightly more general than those obtained in homological perturbation theory, since the conditions of a SDR are not necessarily satisfied (cf. [3], or [4], Section 6). References [1] Estanislao Herscovich, Applications of one-point extensions to compute the A -(co)module structure of several Ext (resp., Tor) groups, available at ~eherscov/articles/applications-of-one-point-extensions.pdf. Preprint. 1 [2], Using torsion theory to compute the algebraic structure of Hochschild (co)homology, available at Using-torsion-theory. Preprint. 2, 7 [3] V. K. A. M. Gugenheim, On a perturbation theory for the homology of the loop-space, J. Pure Appl. Algebra 25 (1982), no. 2, , 8 [4] Johannes Huebschmann, On the construction of A -structures, Georgian Math. J. 17 (2010), no. 1, , 8 [5] D. M. Lu, J. H. Palmieri, Q. S. u, and J. J. Zhang, A -algebras for ring theorists, Proceedings of the International onference on Algebra, 2004, pp , 3 [6] D.-M. Lu, J. H. Palmieri, Q.-S. u, and J. J. Zhang, A-infinity structure on Ext-algebras, J. Pure Appl. Algebra 213 (2009), no. 11, , 2, 3 [7] S. A. Merkulov, Strong homotopy algebras of a Kähler manifold, Internat. Math. Res. Notices 3 (1999), , 2, 3, 4 Estanislao HERSOVIH Institut Joseph Fourier, Université Grenoble Alpes, Grenoble, FRANE, Estanislao.Herscovich@ujf-grenoble.fr, Departamento de Matemática, FEyN, UBA, Buenos Aires, ARGENTINA, eherscov@dm.uba.ar, and ONIET (ARGENTINA). 8
Applications of one-point extensions to compute the A -(co)module structure of several Ext (resp., Tor) groups
Applications of one-point extensions to compute the A -(co)module structure of several Ext (resp., Tor) groups Estanislao Herscovich Abstract Let A be a nonnegatively graded connected algebra over a noncommutative
More informationA-INFINITY STRUCTURE ON EXT-ALGEBRAS
A-INFINITY STRUCTURE ON EXT-ALGEBRAS D.-M. LU, J. H. PALMIERI, Q.-S. WU AND J. J. ZHANG Abstract. Let A be a connected graded algebra and let E denote its Extalgebra L i Exti A (k A, k A ). There is a
More informationA -algebras. Matt Booth. October 2017 revised March 2018
A -algebras Matt Booth October 2017 revised March 2018 1 Definitions Work over a field k of characteristic zero. All complexes are cochain complexes; i.e. the differential has degree 1. A differential
More informationSECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS
SECTION 5: EILENBERG ZILBER EQUIVALENCES AND THE KÜNNETH THEOREMS In this section we will prove the Künneth theorem which in principle allows us to calculate the (co)homology of product spaces as soon
More informationKOSZUL DUALITY AND CODERIVED CATEGORIES (AFTER K. LEFÈVRE)
KOSZUL DUALITY AND CODERIVED CATEGORIES (AFTER K. LEFÈVRE) BERNHARD KELLER Abstract. This is a brief report on a part of Chapter 2 of K. Lefèvre s thesis [5]. We sketch a framework for Koszul duality [1]
More information(communicated by James Stasheff)
Homology, Homotopy and Applications, vol.5(1), 2003, pp.407 421 EXTENSIONS OF HOMOGENEOUS COORDINATE RINGS TO A -ALGEBRAS A. POLISHCHUK (communicated by James Stasheff) Abstract We study A -structures
More informationDe Rham Cohomology. Smooth singular cochains. (Hatcher, 2.1)
II. De Rham Cohomology There is an obvious similarity between the condition d o q 1 d q = 0 for the differentials in a singular chain complex and the condition d[q] o d[q 1] = 0 which is satisfied by the
More informationKOSZUL DUALITY COMPLEXES FOR THE COHOMOLOGY OF ITERATED LOOP SPACES OF SPHERES. Benoit Fresse
KOSZUL DUALITY COMPLEXES FOR THE COHOMOLOGY OF ITERATED LOOP SPACES OF SPHERES by Benoit Fresse Abstract. The goal of this article is to make explicit a structured complex computing the cohomology of a
More informationHomological Algebra and Differential Linear Logic
Homological Algebra and Differential Linear Logic Richard Blute University of Ottawa Ongoing discussions with Robin Cockett, Geoff Cruttwell, Keith O Neill, Christine Tasson, Trevor Wares February 24,
More informationGeneralized Alexander duality and applications. Osaka Journal of Mathematics. 38(2) P.469-P.485
Title Generalized Alexander duality and applications Author(s) Romer, Tim Citation Osaka Journal of Mathematics. 38(2) P.469-P.485 Issue Date 2001-06 Text Version publisher URL https://doi.org/10.18910/4757
More informationIterated Bar Complexes of E-infinity Algebras and Homology Theories
Iterated Bar Complexes of E-infinity Algebras and Homology Theories BENOIT FRESSE We proved in a previous article that the bar complex of an E -algebra inherits a natural E -algebra structure. As a consequence,
More informationRational Homotopy Theory Seminar Week 11: Obstruction theory for rational homotopy equivalences J.D. Quigley
Rational Homotopy Theory Seminar Week 11: Obstruction theory for rational homotopy equivalences J.D. Quigley Reference. Halperin-Stasheff Obstructions to homotopy equivalences Question. When can a given
More informationHOMOLOGICAL PROPERTIES OF MODULES OVER DING-CHEN RINGS
J. Korean Math. Soc. 49 (2012), No. 1, pp. 31 47 http://dx.doi.org/10.4134/jkms.2012.49.1.031 HOMOLOGICAL POPETIES OF MODULES OVE DING-CHEN INGS Gang Yang Abstract. The so-called Ding-Chen ring is an n-fc
More informationarxiv: v1 [math.ag] 2 Oct 2009
THE GENERALIZED BURNSIDE THEOREM IN NONCOMMUTATIVE DEFORMATION THEORY arxiv:09100340v1 [mathag] 2 Oct 2009 EIVIND ERIKSEN Abstract Let A be an associative algebra over a field k, and let M be a finite
More informationFormal power series rings, inverse limits, and I-adic completions of rings
Formal power series rings, inverse limits, and I-adic completions of rings Formal semigroup rings and formal power series rings We next want to explore the notion of a (formal) power series ring in finitely
More informationON THE VANISHING OF HOMOLOGY WITH MODULES OF FINITE LENGTH
ON THE VANISHING OF HOMOLOGY WITH MODULES OF FINITE LENGTH PETTER ANDREAS BERGH Abstract We study the vanishing of homology and cohomology of a module of finite complete intersection dimension over a local
More informationA Primer on Homological Algebra
A Primer on Homological Algebra Henry Y Chan July 12, 213 1 Modules For people who have taken the algebra sequence, you can pretty much skip the first section Before telling you what a module is, you probably
More informationHochschild (co)homology of Koszul dual pairs
Hochschild (co)homology of Koszul dual pairs Estanislao Herscovich * Abstract In this article we prove that the Tamarkin-Tsygan calculus of an Adams connected augmented dg algebra and of its Koszul dual
More informationCohomology operations and the Steenrod algebra
Cohomology operations and the Steenrod algebra John H. Palmieri Department of Mathematics University of Washington WCATSS, 27 August 2011 Cohomology operations cohomology operations = NatTransf(H n ( ;
More informationAN ABSTRACT CHARACTERIZATION OF NONCOMMUTATIVE PROJECTIVE LINES
AN ABSTRACT CHARACTERIZATION OF NONCOMMUTATIVE PROJECTIVE LINES A. NYMAN Abstract. Let k be a field. We describe necessary and sufficient conditions for a k-linear abelian category to be a noncommutative
More informationMATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 23
MATH 101B: ALGEBRA II PART A: HOMOLOGICAL ALGEBRA 23 6.4. Homotopy uniqueness of projective resolutions. Here I proved that the projective resolution of any R-module (or any object of an abelian category
More informationp,q H (X), H (Y ) ), where the index p has the same meaning as the
There are two Eilenberg-Moore spectral sequences that we shall consider, one for homology and the other for cohomology. In contrast with the situation for the Serre spectral sequence, for the Eilenberg-Moore
More informationTCC Homological Algebra: Assignment #3 (Solutions)
TCC Homological Algebra: Assignment #3 (Solutions) David Loeffler, d.a.loeffler@warwick.ac.uk 30th November 2016 This is the third of 4 problem sheets. Solutions should be submitted to me (via any appropriate
More informationEILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY
EILENBERG-ZILBER VIA ACYCLIC MODELS, AND PRODUCTS IN HOMOLOGY AND COHOMOLOGY CHRIS KOTTKE 1. The Eilenberg-Zilber Theorem 1.1. Tensor products of chain complexes. Let C and D be chain complexes. We define
More informationHomological Methods in Commutative Algebra
Homological Methods in Commutative Algebra Olivier Haution Ludwig-Maximilians-Universität München Sommersemester 2017 1 Contents Chapter 1. Associated primes 3 1. Support of a module 3 2. Associated primes
More informationA TALE OF TWO FUNCTORS. Marc Culler. 1. Hom and Tensor
A TALE OF TWO FUNCTORS Marc Culler 1. Hom and Tensor It was the best of times, it was the worst of times, it was the age of covariance, it was the age of contravariance, it was the epoch of homology, it
More informationWIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES
WIDE SUBCATEGORIES OF d-cluster TILTING SUBCATEGORIES MARTIN HERSCHEND, PETER JØRGENSEN, AND LAERTIS VASO Abstract. A subcategory of an abelian category is wide if it is closed under sums, summands, kernels,
More informationarxiv:math/ v3 [math.at] 11 Jul 2004
arxiv:math/0401160v3 [math.at] 11 Jul 2004 HOMOLOGICAL PERTURBATIONS, EQUIVARIANT COHOMOLOGY, AND KOSZUL DUALITY Johannes Huebschmann Université des Sciences et Technologies de Lille UFR de Mathématiques
More informationSome Remarks on D-Koszul Algebras
International Journal of Algebra, Vol. 4, 2010, no. 24, 1177-1183 Some Remarks on D-Koszul Algebras Chen Pei-Sen Yiwu Industrial and Commercial College Yiwu, Zhejiang, 322000, P.R. China peisenchen@126.com
More informationA Leibniz Algebra Structure on the Second Tensor Power
Journal of Lie Theory Volume 12 (2002) 583 596 c 2002 Heldermann Verlag A Leibniz Algebra Structure on the Second Tensor Power R. Kurdiani and T. Pirashvili Communicated by K.-H. Neeb Abstract. For any
More informationDerivations and differentials
Derivations and differentials Johan Commelin April 24, 2012 In the following text all rings are commutative with 1, unless otherwise specified. 1 Modules of derivations Let A be a ring, α : A B an A algebra,
More informationARCHIVUM MATHEMATICUM (BRNO) Tomus 53 (2017), Jakub Kopřiva. Dedicated to the memory of Martin Doubek
ARCHIUM MATHEMATICUM BRNO Tomus 53 2017, 267 312 ON THE HOMOTOPY TRANSFER OF A STRUCTURES Jakub Kopřiva Dedicated to the memory of Martin Doubek Abstract. The present article is devoted to the study of
More informationarxiv:math/ v1 [math.at] 6 Oct 2004
arxiv:math/0410162v1 [math.at] 6 Oct 2004 EQUIVARIANT UNIVERSAL COEFFICIENT AND KÜNNETH SPECTRAL SEQUENCES L. GAUNCE LEWIS, JR. AND MICHAEL A. MANDELL Abstract. We construct hyper-homology spectral sequences
More informationGORENSTEIN DIMENSIONS OF UNBOUNDED COMPLEXES AND CHANGE OF BASE (WITH AN APPENDIX BY DRISS BENNIS)
GORENSTEIN DIMENSIONS OF UNBOUNDED COMPLEXES AND CHANGE OF BASE (WITH AN APPENDIX BY DRISS BENNIS) LARS WINTHER CHRISTENSEN, FATIH KÖKSAL, AND LI LIANG Abstract. For a commutative ring R and a faithfully
More informationConfluence Algebras and Acyclicity of the Koszul Complex
Confluence Algebras and Acyclicity of the Koszul Complex Cyrille Chenavier To cite this version: Cyrille Chenavier. Confluence Algebras and Acyclicity of the Koszul Complex. Algebras and Representation
More informationSERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN
SERRE FINITENESS AND SERRE VANISHING FOR NON-COMMUTATIVE P 1 -BUNDLES ADAM NYMAN Abstract. Suppose X is a smooth projective scheme of finite type over a field K, E is a locally free O X -bimodule of rank
More informationRealization problems in algebraic topology
Realization problems in algebraic topology Martin Frankland Universität Osnabrück Adam Mickiewicz University in Poznań Geometry and Topology Seminar June 2, 2017 Martin Frankland (Osnabrück) Realization
More informationTHE DERIVED CATEGORY OF A GRADED GORENSTEIN RING
THE DERIVED CATEGORY OF A GRADED GORENSTEIN RING JESSE BURKE AND GREG STEVENSON Abstract. We give an exposition and generalization of Orlov s theorem on graded Gorenstein rings. We show the theorem holds
More informationLecture 17: Invertible Topological Quantum Field Theories
Lecture 17: Invertible Topological Quantum Field Theories In this lecture we introduce the notion of an invertible TQFT. These arise in both topological and non-topological quantum field theory as anomaly
More informationON sfp-injective AND sfp-flat MODULES
Gulf Journal of Mathematics Vol 5, Issue 3 (2017) 79-90 ON sfp-injective AND sfp-flat MODULES C. SELVARAJ 1 AND P. PRABAKARAN 2 Abstract. Let R be a ring. A left R-module M is said to be sfp-injective
More informationAN INTRODUCTION TO A -ALGEBRAS
AN INTRODUCTION TO A -ALGEBRAS Nathan Menzies Supervisor: Dr. Daniel Chan School of Mathematics, The University of New South Wales. November 2007 Submitted in partial fulfillment of the requirements of
More informationLINKAGE CLASSES OF GRADE 3 PERFECT IDEALS
LINKAGE CLASSES OF GRADE 3 PERFECT IDEALS LARS WINTHER CHRISTENSEN, OANA VELICHE, AND JERZY WEYMAN Abstract. While every grade 2 perfect ideal in a regular local ring is linked to a complete intersection
More informationNotes on p-divisible Groups
Notes on p-divisible Groups March 24, 2006 This is a note for the talk in STAGE in MIT. The content is basically following the paper [T]. 1 Preliminaries and Notations Notation 1.1. Let R be a complete
More informationOn Hochschild Cohomology, Koszul Duality and DG Categories
On Hochschild Cohomology, Koszul Duality and DG Categories Dhyan Aranha Born 10th December 1989 in Lubbock, Texas, U.S.A. April 26, 2015 Master s Thesis Mathematics Advisor: Prof. Dr. Catharina Stroppel
More informationA NEW PROOF OF SERRE S HOMOLOGICAL CHARACTERIZATION OF REGULAR LOCAL RINGS
A NEW PROOF OF SERRE S HOMOLOGICAL CHARACTERIZATION OF REGULAR LOCAL RINGS RAVI JAGADEESAN AND AARON LANDESMAN Abstract. We give a new proof of Serre s result that a Noetherian local ring is regular if
More informationNoetherian property of infinite EI categories
Noetherian property of infinite EI categories Wee Liang Gan and Liping Li Abstract. It is known that finitely generated FI-modules over a field of characteristic 0 are Noetherian. We generalize this result
More informationON THE COBAR CONSTRUCTION OF A BIALGEBRA. 1. Introduction. Homology, Homotopy and Applications, vol.7(2), 2005, pp T.
Homology, Homotopy and Applications, vol.7(2), 2005, pp.109 122 ON THE COBAR CONSTRUCTION OF A BIALGEBRA T. KADEISHVILI (communicated by Tom Lada) Abstract We show that the cobar construction of a DG-bialgebra
More informationarxiv: v1 [math.co] 25 Jun 2014
THE NON-PURE VERSION OF THE SIMPLEX AND THE BOUNDARY OF THE SIMPLEX NICOLÁS A. CAPITELLI arxiv:1406.6434v1 [math.co] 25 Jun 2014 Abstract. We introduce the non-pure versions of simplicial balls and spheres
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationNoncommutative motives and their applications
MSRI 2013 The classical theory of pure motives (Grothendieck) V k category of smooth projective varieties over a field k; morphisms of varieties (Pure) Motives over k: linearization and idempotent completion
More informationProperties of Triangular Matrix and Gorenstein Differential Graded Algebras
Properties of Triangular Matrix and Gorenstein Differential Graded Algebras Daniel Maycock Thesis submitted for the degree of Doctor of Philosophy chool of Mathematics & tatistics Newcastle University
More informationMath 210B. Artin Rees and completions
Math 210B. Artin Rees and completions 1. Definitions and an example Let A be a ring, I an ideal, and M an A-module. In class we defined the I-adic completion of M to be M = lim M/I n M. We will soon show
More informationLECTURE 3: RELATIVE SINGULAR HOMOLOGY
LECTURE 3: RELATIVE SINGULAR HOMOLOGY In this lecture we want to cover some basic concepts from homological algebra. These prove to be very helpful in our discussion of singular homology. The following
More informationSTABLE MODULE THEORY WITH KERNELS
Math. J. Okayama Univ. 43(21), 31 41 STABLE MODULE THEORY WITH KERNELS Kiriko KATO 1. Introduction Auslander and Bridger introduced the notion of projective stabilization mod R of a category of finite
More informationA generalized Koszul theory and its applications in representation theory
A generalized Koszul theory and its applications in representation theory A DISSERTATION SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Liping Li IN PARTIAL FULFILLMENT
More informationIntroduction to Chiral Algebras
Introduction to Chiral Algebras Nick Rozenblyum Our goal will be to prove the fact that the algebra End(V ac) is commutative. The proof itself will be very easy - a version of the Eckmann Hilton argument
More informationA Version of the Grothendieck Conjecture for p-adic Local Fields
A Version of the Grothendieck Conjecture for p-adic Local Fields by Shinichi MOCHIZUKI* Section 0: Introduction The purpose of this paper is to prove an absolute version of the Grothendieck Conjecture
More informationINTRO TO TENSOR PRODUCTS MATH 250B
INTRO TO TENSOR PRODUCTS MATH 250B ADAM TOPAZ 1. Definition of the Tensor Product Throughout this note, A will denote a commutative ring. Let M, N be two A-modules. For a third A-module Z, consider the
More informationStasheffs A -algebras and Homotopy Gerstenhaber Algebras. Tornike Kadeishvili A. Razmadze Mathematical Institute of Tbilisi State University
1 Stasheffs A -algebras and Homotopy Gerstenhaber Algebras Tornike Kadeishvili A. Razmadze Mathematical Institute of Tbilisi State University 1 Twisting Elements 2 A dg algebra (A, d : A A +1, : A A A
More informationDEFORMATIONS OF ALGEBRAS IN NONCOMMUTATIVE ALGEBRAIC GEOMETRY EXERCISE SHEET 1
DEFORMATIONS OF ALGEBRAS IN NONCOMMUTATIVE ALGEBRAIC GEOMETRY EXERCISE SHEET 1 TRAVIS SCHEDLER Note: it is possible that the numbers referring to the notes here (e.g., Exercise 1.9, etc.,) could change
More informationLECTURE X: KOSZUL DUALITY
LECTURE X: KOSZUL DUALITY Fix a prime number p and an integer n > 0, and let S vn denote the -category of v n -periodic spaces. Last semester, we proved the following theorem of Heuts: Theorem 1. The Bousfield-Kuhn
More informationRelative Left Derived Functors of Tensor Product Functors. Junfu Wang and Zhaoyong Huang
Relative Left Derived Functors of Tensor Product Functors Junfu Wang and Zhaoyong Huang Department of Mathematics, Nanjing University, Nanjing 210093, Jiangsu Province, China Abstract We introduce and
More informationMath 121 Homework 4: Notes on Selected Problems
Math 121 Homework 4: Notes on Selected Problems 11.2.9. If W is a subspace of the vector space V stable under the linear transformation (i.e., (W ) W ), show that induces linear transformations W on W
More informationMath 210B. The bar resolution
Math 210B. The bar resolution 1. Motivation Let G be a group. In class we saw that the functorial identification of M G with Hom Z[G] (Z, M) for G-modules M (where Z is viewed as a G-module with trivial
More informationDuality, Residues, Fundamental class
Duality, Residues, Fundamental class Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu May 22, 2011 Joseph Lipman (Purdue University) Duality, Residues, Fundamental class
More informationMODEL-CATEGORIES OF COALGEBRAS OVER OPERADS
Theory and Applications of Categories, Vol. 25, No. 8, 2011, pp. 189 246. MODEL-CATEGORIES OF COALGEBRAS OVER OPERADS JUSTIN R. SMITH Abstract. This paper constructs model structures on the categories
More informationarxiv: v1 [math.kt] 2 Feb 2019
Cyclic A -algebras and double Poisson algebras David Fernández and Estanislao Herscovich arxiv:1902.00787v1 [math.kt] 2 Feb 2019 Abstract In this article we prove that there exists an explicit bijection
More informationarxiv: v4 [math.rt] 14 Jun 2016
TWO HOMOLOGICAL PROOFS OF THE NOETHERIANITY OF FI G LIPING LI arxiv:163.4552v4 [math.rt] 14 Jun 216 Abstract. We give two homological proofs of the Noetherianity of the category F I, a fundamental result
More informationHopf algebroids and the structure of MU (MU)
Hopf algebroids and the structure of MU (MU) Vitaly Lorman July 1, 2012 Note: These are my notes on sections B.3 and B.4 of Doug Ravenel s Orange Book (Nilpotence and Periodicity in Stable Homotopy Theory).
More informationAlgebraic Geometry Spring 2009
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More information1 Hochschild Cohomology and A : Jeff Hicks
1 Hochschild Cohomology and A : Jeff Hicks Here s the general strategy of what we would like to do. ˆ From the previous two talks, we have some hope of understanding the triangulated envelope of the Fukaya
More informationLECTURE IV: PERFECT PRISMS AND PERFECTOID RINGS
LECTURE IV: PERFECT PRISMS AND PERFECTOID RINGS In this lecture, we study the commutative algebra properties of perfect prisms. These turn out to be equivalent to perfectoid rings, and most of the lecture
More informationPART II.1. IND-COHERENT SHEAVES ON SCHEMES
PART II.1. IND-COHERENT SHEAVES ON SCHEMES Contents Introduction 1 1. Ind-coherent sheaves on a scheme 2 1.1. Definition of the category 2 1.2. t-structure 3 2. The direct image functor 4 2.1. Direct image
More informationMODULES OVER OPERADS AND FUNCTORS. Benoit Fresse
MODULES OVER OPERADS AND FUNCTORS Benoit Fresse Benoit Fresse UMR 8524 de l Université des Sciences et Technologies de Lille et du CNRS, Cité Scientifique Bâtiment M2, F-59655 Villeneuve d Ascq Cédex (France).
More informationDeformation theory of algebraic structures
Deformation theory of algebraic structures Second Congrès Canada-France June 2008 (Bruno Vallette) Deformation theory of algebraic structures June 2008 1 / 22 Data: An algebraic structure P A vector space
More informationREPRESENTATION THEORY IN HOMOTOPY AND THE EHP SEQUENCES FOR (p 1)-CELL COMPLEXES
REPRESENTATION THEORY IN HOMOTOPY AND THE EHP SEQUENCES FOR (p 1)-CELL COMPLEXES J. WU Abstract. For spaces localized at 2, the classical EHP fibrations [1, 13] Ω 2 S 2n+1 P S n E ΩS n+1 H ΩS 2n+1 play
More informationFINITE REGULARITY AND KOSZUL ALGEBRAS
FINITE REGULARITY AND KOSZUL ALGEBRAS LUCHEZAR L. AVRAMOV AND IRENA PEEVA ABSTRACT: We determine the positively graded commutative algebras over which the residue field modulo the homogeneous maximal ideal
More informationNotes on the definitions of group cohomology and homology.
Notes on the definitions of group cohomology and homology. Kevin Buzzard February 9, 2012 VERY sloppy notes on homology and cohomology. Needs work in several places. Last updated 3/12/07. 1 Derived functors.
More informationCATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS. Contents. 1. The ring K(R) and the group Pic(R)
CATEGORICAL GROTHENDIECK RINGS AND PICARD GROUPS J. P. MAY Contents 1. The ring K(R) and the group Pic(R) 1 2. Symmetric monoidal categories, K(C), and Pic(C) 2 3. The unit endomorphism ring R(C ) 5 4.
More informationTEST GROUPS FOR WHITEHEAD GROUPS
TEST GROUPS FOR WHITEHEAD GROUPS PAUL C. EKLOF, LÁSZLÓ FUCHS, AND SAHARON SHELAH Abstract. We consider the question of when the dual of a Whitehead group is a test group for Whitehead groups. This turns
More informationDEFORMATION THEORY MICHAEL KEMENY
DEFORMATION THEORY MICHAEL KEMENY 1. Lecture 1: Deformations of Algebras We wish to first illustrate the theory with a technically simple case, namely deformations of algebras. We follow On the Deformation
More informationCOHEN-MACAULAY RINGS SELECTED EXERCISES. 1. Problem 1.1.9
COHEN-MACAULAY RINGS SELECTED EXERCISES KELLER VANDEBOGERT 1. Problem 1.1.9 Proceed by induction, and suppose x R is a U and N-regular element for the base case. Suppose now that xm = 0 for some m M. We
More informationHungry, Hungry Homology
September 27, 2017 Motiving Problem: Algebra Problem (Preliminary Version) Given two groups A, C, does there exist a group E so that A E and E /A = C? If such an group exists, we call E an extension of
More informationDecompositions of Modules and Comodules
Decompositions of Modules and Comodules Robert Wisbauer University of Düsseldorf, Germany Abstract It is well-known that any semiperfect A ring has a decomposition as a direct sum (product) of indecomposable
More informationKathryn Hess. Conference on Algebraic Topology, Group Theory and Representation Theory Isle of Skye 9 June 2009
Institute of Geometry, lgebra and Topology Ecole Polytechnique Fédérale de Lausanne Conference on lgebraic Topology, Group Theory and Representation Theory Isle of Skye 9 June 2009 Outline 1 2 3 4 of rings:
More informationMath Homotopy Theory Hurewicz theorem
Math 527 - Homotopy Theory Hurewicz theorem Martin Frankland March 25, 2013 1 Background material Proposition 1.1. For all n 1, we have π n (S n ) = Z, generated by the class of the identity map id: S
More informationOn a theorem of Ziv Ran
INSTITUTUL DE MATEMATICA SIMION STOILOW AL ACADEMIEI ROMANE PREPRINT SERIES OF THE INSTITUTE OF MATHEMATICS OF THE ROMANIAN ACADEMY ISSN 0250 3638 On a theorem of Ziv Ran by Cristian Anghel and Nicolae
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime
More informationSURGERY EQUIVALENCE AND FINITE TYPE INVARIANTS FOR HOMOLOGY 3-SPHERES L. FUNAR Abstract. One considers two equivalence relations on 3-manifolds relate
SURGERY EQUIVALENCE AND FINITE TYPE INVARIANTS FOR HOMOLOGY 3-SPHERES L. FUNAR Abstract. One considers two equivalence relations on 3-manifolds related to nite type invariants. The rst one requires to
More informationAN ELEMENTARY GUIDE TO THE ADAMS-NOVIKOV EXT. The Adams-Novikov spectral sequence for the Brown-Peterson spectrum
AN ELEMENTARY GUIDE TO THE ADAMS-NOVIKOV EXT MICHA L ADAMASZEK The Adams-Novikov spectral sequence for the Brown-Peterson spectrum E s,t = Ext s,t BP BP (BP, BP ) = π S s t(s 0 ) (p) has been one of the
More informationKOSZUL DUALITY FOR STRATIFIED ALGEBRAS II. STANDARDLY STRATIFIED ALGEBRAS
KOSZUL DUALITY FOR STRATIFIED ALGEBRAS II. STANDARDLY STRATIFIED ALGEBRAS VOLODYMYR MAZORCHUK Abstract. We give a complete picture of the interaction between the Koszul and Ringel dualities for graded
More informationCOMPARISON MORPHISMS BETWEEN TWO PROJECTIVE RESOLUTIONS OF MONOMIAL ALGEBRAS
REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 59, No. 1, 2018, Pages 1 31 Published online: July 7, 2017 COMPARISON MORPHISMS BETWEEN TWO PROJECTIVE RESOLUTIONS OF MONOMIAL ALGEBRAS Abstract. We construct
More informationsset(x, Y ) n = sset(x [n], Y ).
1. Symmetric monoidal categories and enriched categories In practice, categories come in nature with more structure than just sets of morphisms. This extra structure is central to all of category theory,
More informationA note on standard equivalences
Bull. London Math. Soc. 48 (2016) 797 801 C 2016 London Mathematical Society doi:10.1112/blms/bdw038 A note on standard equivalences Xiao-Wu Chen Abstract We prove that any derived equivalence between
More informationNONCOMMUTATIVE GRADED GORENSTEIN ISOLATED SINGULARITIES
NONCOMMUTATIVE GRADED GORENSTEIN ISOLATED SINGULARITIES KENTA UEYAMA Abstract. Gorenstein isolated singularities play an essential role in representation theory of Cohen-Macaulay modules. In this article,
More informationQuillen cohomology and Hochschild cohomology
Quillen cohomology and Hochschild cohomology Haynes Miller June, 2003 1 Introduction In their initial work ([?], [?], [?]), Michel André and Daniel Quillen described a cohomology theory applicable in very
More informationarxiv: v3 [math.rt] 11 Apr 2016
A LONG EXACT SEQUENCE FOR HOMOLOGY OF FI-MODULES arxiv:1602.08873v3 [math.rt] 11 Apr 2016 WEE LIANG GAN Abstract. We construct a long exact sequence involving the homology of an FI-module. Using the long
More informationCohomology and Base Change
Cohomology and Base Change Let A and B be abelian categories and T : A B and additive functor. We say T is half-exact if whenever 0 M M M 0 is an exact sequence of A-modules, the sequence T (M ) T (M)
More informationHochschild cohomology
Hochschild cohomology Seminar talk complementing the lecture Homological algebra and applications by Prof. Dr. Christoph Schweigert in winter term 2011. by Steffen Thaysen Inhaltsverzeichnis 9. Juni 2011
More informationOMEGA-CATEGORIES AND CHAIN COMPLEXES. 1. Introduction. Homology, Homotopy and Applications, vol.6(1), 2004, pp RICHARD STEINER
Homology, Homotopy and Applications, vol.6(1), 2004, pp.175 200 OMEGA-CATEGORIES AND CHAIN COMPLEXES RICHARD STEINER (communicated by Ronald Brown) Abstract There are several ways to construct omega-categories
More information