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A theory of cohomological support for pairs of DG modules over a Koszul complex is investigated. These specialize to the support varieties of Avramov and Buchweitz defined over a complete intersection ring, as well as support varieties over an exterior algebra. The main objects of study are certain DG modules over a polynomial ring; these determine the aforementioned cohomological supports and are shown to encode (co)homological information about pairs of DG modules over a Koszul complex. The perspective in this article leads to new proofs of well-known results for pairs of complexes over a complete intersection. Furthermore, these cohomological supports are used to define a support theory for pairs of objects in the derived category of an arbitrary commutative noetherian local ring. Finally, we calculate several examples and provide an application by answering a question of D. Jorgensen in the negative., 40 pages. V2: New introduction and minor changes. To appear in Math. Zeit

In this paper we are concerned with the following question: if the tensor product of finitely generated modules $M$ and $N$ over a local complete intersection domain is maximal Cohen-Macaulay, then must $M$ or $N$ be a maximal Cohen-Macaulay? Celebrated results of Auslander, Lichtenbaum, and Huneke and Wiegand, yield affirmative answers to the question when the ring considered has codimension zero or one, but the question is very much open for complete intersection domains that have codimension at least two, even open for those that are one-dimensional, or isolated singularities. Our argument exploits Tor-rigidity and proves the following, which seems to give a new perspective to the aforementioned question: if $R$ is a complete intersection ring which is an isolated singularity such that dim($R$) > codim($R$), and the tensor product $M\otimes_R N$ is maximal Cohen-Macaulay, then $M$ is maximal Cohen-Macaulay if and only if $N$ is maximal Cohen-Macaulay., This is a pre-print of an article published in Annali di Matematica. The final authenticated version is available online at: https://doi.org/10.1007/s10231-020-01019-9

We study homological properties of a locally complete intersection ring by importing facts from homological algebra over exterior algebras. One application is showing that the thick subcategories of the bounded derived category of a locally complete intersection ring are self-dual under Grothendieck duality. This was proved by Stevenson when the ring is a quotient of a regular ring modulo a regular sequence; we offer two independent proofs in the more general setting. Second, we use these techniques to supply new proofs that complete intersections possess symmetry of complexity., 13 pages, comments welcome

Let ( A , m ) be a local complete intersection ring of dimension d and let I be an m -primary ideal. Let M be a maximal Cohen–Macaulay A -module. For i = 0 , 1 , ⋯ , d , let e i I ( M ) denote the i th Hilbert-coefficient of M with respect to I . We prove that for i = 0 , 1 , 2 , the function j ↦ e i I ( Syz j A ( M ) ) is of quasi-polynomial type with period 2. Let G I ( M ) be the associated graded module of M with respect to I . If G I ( A ) is Cohen–Macaulay and dim ⁡ A ≤ 2 we also prove that the functions j ↦ depth G I ( Syz 2 j + i A ( M ) ) are eventually constant for i = 0 , 1 . Let ξ I ( M ) = lim l → ∞ ⁡ depth G I l ( M ) . Finally we prove that if dim ⁡ A = 2 and G I ( A ) is Cohen–Macaulay then the functions j ↦ ξ I ( Syz 2 j + i A ( M ) ) are eventually constant for i = 0 , 1 .

We introduce a new algebra associated with a hyperplane arrangement $\mathcal{A}$, called the Solomon–Terao algebra $ST(\mathcal{A}, \eta)$, where $\eta$ is a homogeneous polynomial. It is shown by Solomon and Terao that $ST(\mathcal{A}, \eta)$ is Artinian when $\eta$ is generic. This algebra can be considered as a generalization of coinvariant algebras in the setting of hyperplane arrangements. The class of Solomon–Terao algebras contains cohomology rings of regular nilpotent Hessenberg varieties. We show that $ST(\mathcal{A}, \eta)$ is a complete intersection if and only if $\mathcal{A}$ is free. We also give a factorization formula of the Hilbert polynomials of $ST(\mathcal{A}, \eta)$ when $\mathcal{A}$ is free, and pose several related questions, problems and conjectures.

For decades, differential ideals have played an important role in algebra. In this paper, if A is a Noetherian local ring with positive residual characteristic, we characterize when a maximally differential ideal P ⊂ A is an integrally closed ideal of finite projective dimension. Our main argument yields, in a characteristic-free setting, that if P has finite projective dimension then P must be a complete intersection. This generalizes a well-known result which assumes A to be regular. We derive further homological criteria and, along the way, we conjecture (in positive residual characteristic) that if P is maximally differential with respect to the entire derivation module, then A must be a complete intersection ring if P is integrally closed and has finite complete intersection dimension.

It is proved that a module $M$ over a Noetherian local ring $R$ of prime characteristic and positive dimension has finite flat dimension if Tor$_i^R({}^e R, M)=0$ for dim $R$ consecutive positive values of $i$ and infinitely many $e$. Here ${}^e R$ denotes the ring $R$ viewed as an $R$-module via the $e$th iteration of the Frobenius endomorphism. In the case $R$ is Cohen-Macualay, it suffices that the Tor vanishing above holds for a single $e\geq \log_p e(R)$, where $e(R)$ is the multiplicity of the ring. This improves a result of D. Dailey, S. Iyengar, and the second author, as well as generalizing a theorem due to C. Miller from finitely generated modules to arbitrary modules. We also show that if $R$ is a complete intersection ring then the vanishing of Tor$_i^R({}^e R, M)$ for single positive values of $i$ and $e$ is sufficient to imply $M$ has finite flat dimension. This extends a result of L. Avramov and C. Miller., Comment: This version corrects an error in the proof of Theorem 3.2 of the original manuscript

Let $A$ be a local complete intersection ring. Let $M,N$ be two finitely generated $A$-modules and $I$ an ideal of $A$. We prove that \[ \bigcup_{i\geqslant 0}\bigcup_{n \geqslant 0}\mathrm{Ass}_A\left(\mathrm{Ext}_A^i(M,N/I^n N)\right) \] is a finite set. Moreover, we prove that there exist $i_0,n_0\geqslant 0$ such that for all $i\geqslant i_0$ and $n \geqslant n_0$, we have \[ \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2i}(M,N/I^nN)\right) = \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2 i_0}(M,N/I^{n_0}N)\right), \] \[ \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2i+1}(M,N/I^nN)\right) = \mathrm{Ass}_A\left(\mathrm{Ext}_A^{2 i_0 + 1}(M,N/I^{n_0}N)\right). \] We also prove the analogous results for complete intersection rings which arise in algebraic geometry. Further, we prove that the complexity $\mathrm{cx}_A(M,N/I^nN)$ is constant for all sufficiently large $n$., Comment: 17 pages, final version

Let $(A,\mathfrak{m})$ be a local complete intersection ring and let $I$ be an ideal in $A$. Let $M, N$ be finitely generated $A$-modules. Then for $l = 0,1$, the values $depth \ Ext^{2i+l}_A(M, N/I^nN)$ become independent of $i, n$ for $i,n \gg 0$. We also show that if $\mathfrak{p}$ is a prime ideal in $A$ then the $j^{th}$ Bass numbers $\mu_j\big(\mathfrak{p},\ Ext^{2i+l}_A(M,N/{I^nN})\big)$ has polynomial growth in $(n,i)$ with rational coefficients for all sufficiently large $(n,i)$.

We introduce higher-order support varieties for pairs of modules over a commutative local complete intersection ring and give a complete description of which varieties occur as such support varieties. In the context of a group algebra of a finite elementary abelian group, we also prove a higher-order Avrunin–Scott-type theorem, linking higher-order support varieties and higher-order rank varieties for pairs of modules. © 2017. This is the authors' accepted and refereed manuscript to the article. The final authenticated version is available online at: https://doi.org/10.1142/S021949881850158X

We observe that there is an equivalence between the singularity category of an affine complete intersection and the homotopy category of matrix factorizations over a related scheme. This relies in part on a theorem of Orlov. Using this equivalence, we give a geometric construction of the ring of cohomology operators, and a generalization of the theory of support varieties, which we call stable support sets. We settle a question of Avramov about which stable support sets can arise for a given complete intersection ring. We also use the equivalence to construct a projective resolution of a module over a complete intersection ring from a matrix factorization, generalizing the well-known result in the hypersurface case., 41 pages

Let $(R,m)$ be a commutative Noetherian local ring. It is known that R is Cohen-Macaulay if there exists either a nonzero Cohen-Macaulay R-module of finite projective dimension or a nonzero finitely generated R-module of finite injective dimension. In this article, we will prove the complete intersection analogues of these facts. Also, by using complete intersection homological dimensions, we will characterize local rings which are either regular, complete intersection or Gorenstein.

Let R be a complete intersection ring and let M and N be R-modules. It is shown that the vanishing of Ext^i_R(M,N) for a certain number of consecutive values of i starting at n forces the complete intersection dimension of M to be at most n-1. We also estimate the complete intersection dimension of the dual of M, in terms of vanishing of the cohomology modules, Ext^i_R(M,N)., 11 pages

In this paper, we answer a question of Dwyer, Greenlees, and Iyengar by proving a local ring $R$ is a complete intersection if and only if every complex of $R$-modules with finitely generated homology is proxy small. Moreover, we establish that a commutative noetherian ring $R$ is locally a complete intersection if and only if every complex of $R$-modules with finitely generated homology is virtually small., Comment: 14 pages

We show that every thick subcategory of the singularity category of a complete intersection ring is self dual. We also prove the analogous statement for thick subcategories of the bounded derived category and give applications to the symmetry of vanishing of cohomology. These results are also proved for certain complete intersection schemes., 12 pages, comments welcome - minor corrections and added a few details

We give an application of the New Intersection Theorem and prove the following: let $R$ be a local complete intersection ring of codimension $c$ and let $M$ and $N$ be nonzero finitely generated $R$-modules. Assume $n$ is a nonnegative integer and that the tensor product $M\tensor_{R}N$ is an $(n+c)$th syzygy of some finitely generated $R$-module. If Tor$^{R}_{>0}(M,N)=0$, then both $M$ and $N$ are $n$th syzygies of some finitely generated $R$-modules., some typos are fixed; to appear in Mathematische Zeitschrift

Let g be the nil radical of the Borel subalgebra of one of the classical simple Lie algebras over a field F of characteristic p ⩾ 0 . For p > 0 we find an explicit realization of the center Z ( g ) of the enveloping algebra U ( g ) by generators and relations. This constructive approach yields an explicit isomorphism between Z ( g ) and the polynomial invariants algebra S ( g ) g . While realizing Z ( g ) , we also prove that Z ( g ) is a complete intersection ring. Moreover, it leads to an explicit realization of Z ( g ) and S ( g ) g for p = 0 as well. This extends a result of Dixmier in type A n .

This work concerns finite free complexes over commutative noetherian rings, in particular over group algebras of elementary abelian groups. The main contribution is the construction of complexes such that the total rank of their underlying free modules, or the total length of their homology, is less than predicted by various conjectures in the theory of transformation groups and in local algebra., Comment: 13 pages. Version 2 is a significant revision of the first one. This paper will appear in Acta Mathematica

We investigate three cases regarding asymptotic associate primes. First, assume $ (A,\mathfrak{m}) $ is an excellent Cohen-Macaulay (CM) non-regular local ring, and $ M = \operatorname{Syz}^A_1(L) $ for some maximal CM $ A $-module $ L $ which is free on the punctured spectrum. Let $ I $ be a normal ideal. In this case, we examine when $ \mathfrak{m} \notin \operatorname{Ass}(M/I^nM) $ for all $ n \gg 0 $. We give sufficient evidence to show that this occurs rarely. Next, assume that $ (A,\mathfrak{m}) $ is excellent Gorenstein non-regular isolated singularity, and $ M $ is a CM $ A $-module with $\operatorname{projdim}_A(M) = \infty $ and $ \dim(M) = \dim(A) -1 $. Let $ I $ be a normal ideal with analytic spread $ l(I) < \dim(A) $. In this case, we investigate when $\mathfrak{m} \notin \operatorname{Ass} \operatorname{Tor}^A_1(M, A/I^n)$ for all $n \gg 0$. We give sufficient evidence to show that this also occurs rarely. Finally, suppose $ A $ is a local complete intersection ring. For finitely generated $ A $-modules $ M $ and $ N $, we show that if $ \operatorname{Tor}_i^A(M, N) \neq 0 $ for some $ i > \dim(A) $, then there exists a non-empty finite subset $ \mathcal{A} $ of $ \operatorname{Spec}(A) $ such that for every $ \mathfrak{p} \in \mathcal{A} $, at least one of the following holds true: (i) $ \mathfrak{p} \in \operatorname{Ass}\left( \operatorname{Tor}_{2i}^A(M, N) \right) $ for all $ i \gg 0 $; (ii) $ \mathfrak{p} \in \operatorname{Ass}\left( \operatorname{Tor}_{2i+1}^A(M, N) \right) $ for all $ i \gg 0 $. We also analyze the asymptotic behaviour of $\operatorname{Tor}^A_i(M, A/I^n)$ for $i,n \gg 0$ in the case when $I$ is principal or $I$ has a principal reduction generated by a regular element., 34 pages

A quotient ring of a regular local ring of embedding dimension~3 has a minimal free resolution that carries a differential graded algebra structure. This structure has been used to classify such rings. For some classes very few examples have been observed. One such class is called \lt B>{B}\lt /B>, named for Anne Brown, who discovered and gave the first constructions of rings in this class. Her constructions were the only known rings in that class until recently. In this paper, we show how to obtain Artinian rings in the class \textbf {B} by expanding the socle of a complete intersection.

We study H. Dao’s invariant \({\eta_c^R}\) of pairs of modules defined over a complete intersection ring R of codimension c having an isolated singularity. Our main result is that \({\eta_c^R}\) vanishes for all pairs of modules when R is a graded complete intersection ring of codimension c > 1 having an isolated singularity. A consequence of this result is that all pairs of modules over such a ring are c-Tor-rigid.

We study (slope-)stability properties of syzygy bundles on a projective space P^N given by ideal generators of a homogeneous primary ideal. In particular we give a combinatorial criterion for a monomial ideal to have a semistable syzygy bundle. Restriction theorems for semistable bundles yield the same stability results on the generic complete intersection curve. From this we deduce a numerical formula for the tight closure of an ideal generated by monomials or by generic homogeneous elements in a generic two-dimensional complete intersection ring., This paper contains an appendix by Georg Hein: Semistability of the general syzygy bundle. The new version is quite new

Support and rank varieties of modules over a group algebra of an elementary abelian [math] -group have been well studied. In particular, Avrunin and Scott showed that in this setting, the rank and support varieties are equivalent. Avramov and Buchweitz proved an analogous result for pairs of modules over arbitrary commutative local complete intersection rings. In this paper we study support and rank varieties in the triangulated category of totally acyclic chain complexes over a complete intersection ring and show that these varieties are also equivalent.

Bleher and Chinburg recently used modular representation theory to produce an example of a linear representation of a finite group whose universal deformation ring is not a complete intersection ring. We prove this by using only elementary cohomological obstruction calculus. To cite this article: J. Byszewski, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Recently Avramov and Miller proved that over a local complete intersection ring (R, m. k) in characteristic p > 0, a finitely generated module M has finite projective dimension if for some i > 0 and for some n > 0, Tor R i (M,f n R ) = 0 - f n being the frobenius map repeated n times. They used the notion of complexity and several related theorems. Here we offer a very simple proof of the above theorem without using complexity at all.

Quillen’s geometric approach to the cohomology of finite groups [30] has revolutionized modular representation theory. The ideology and techniques involved have been generalized and extended to representations of various Hopf algebras, culminating in the recent work of Friedlander and Suslin [17] on finite group schemes. In this paper we develop geometric methods for the study of finite modules over a local complete intersection R. If R is such a ring and m is its maximal ideal, then the m-adic completion R has the form Q/(f), where f is a regular sequence and Q is a regular local ring that can be taken to be a ring of formal power series over a field or a discrete valuation ring. The least number of equations needed to cut out R from a regular local ring equals the codimension of R, where codim R = νR(m)− dim R and νR(M) denotes the minimal number of generators of an R-module M. In [5] a cone, that is, a homogeneous algebraic set VR(M) is attached to each finite R-module M and used to study its minimal free resolution. Here we prove

Let A be a field and C a noetherian A-algebra such that C q is a quotient ring of an equicharacteristic local regular ring for each prime ideal q of C We prove that if C is geometrically reduced over A and Ω C/A is flat, then C is regular. We apply this result to prove that a reduced ring homomorphism u :A→C is regular when Ω C/A is a flat C-module and C is a complete noetherian local ring or a Nagata local ring. We also study the property of C being locally a complete intersection ring when the flat dimension of Ω C/A finite.

Several cases of the monomial conjecture are proved. An equivalent form of the direct summand conjecture is discussed. Let (A, m, k) be a noetherian local ring of dimension n, m its maximal ideal and k = A/rn. The Monomial Conjecture (henceforth MC) of Hochster asserts that, given any system of parameters (henceforth s.o.p.) xI, .. ., xn of A, (XIX2 ... Xn) ? Xv***)Xn)bt>O Hochster proved the conjecture in the equicharacteristic case [HI], [H2]. He also established the fact that in the mixed and the positive characteristic cases the Direct Summand Conjecture (henceforth DSC) and hence MC is equivalent to the Canonical Element Conjecture [H12] (henceforth CEC). Thus MC occupies a central position in the study of several homological conjectures. In [I1l] Hochster pointed out that, given any s.o.p. xl,...,xn of A, xt, ... , xt satisfies MC for t sufficiently large. Next Goto [G] proved MC for Buchsbaum rings and Koh proved DSC for degree p extensions [K]. Several special cases of CEC were proved in [DI] when depthA = dimA 1. In [D2], the following result was established: If Ji AnnH;-j(A) and J JiJ2 ... Jr where r dimA depthA > 0, then Xl,... ,xn satisfies MC if J ? (xl,... , xn). This in turn implies that, given any s.o.p. xl, . . , xn in a complete local normal domain A, XI, X2, X3) ... , xt satisfies MC for t > 0. We will have several more applications of this result in Section 2. We recall that there is no loss of generality in assuming A to be a complete local normal domain. In our most recent work we established the validity of CEC over i) A/xA when A is a complete local normal domain and x E mJ1 [D3], ii) any almost complete intersection domain A or any almost complete intersection ring A over which p (= the mixed characteristic) is a non-zero-divisor [D4], iii) rings of the form R/Q, where R is a complete Gorenstein ring such that the complete local normal domain A is a homomorphic image of R, dimR = dimA, and Q is the canonical module of A [D4], and iv) complete local normal domains A for which Q is S3. Thus MC holds in all the above cases. The main results of this paper are arranged in the following way. In Section 1, first we reduce the study of MC to almost complete intersection rings. Recall that, due to the results mentioned earlier, the problem boils down to almost complete intersection rings of depth 0. Next we prove the following: Received by the editors July 20, 1996. 1991 Mathematics Subject Classification. Primary 13D02; Secondary 13H1.0. This work was partially supported by an NSA grant and an NSF grant. (?)1998 American Mathematical Society

A new homological invariant is introduced for a finite module over a commutative noetherian ring: its CI-dimension. In the local case, sharp quantitative and structural data are obtained for modules of finite CI-dimension, providing the first class of modules of (possibly) infinite projective dimension with a rich structure theory of free resolutions.

Let $R$ be a local complete intersection ring and let $M$ and $N$ be nonzero finitely generated $R$-modules. We employ Auslander's transpose in the study of the vanishing of Tor and obtain useful bounds for the depth of the tensor product $M\otimes_{R}N$. An application of our main argument shows that, if $M$ is locally free on the the punctured spectrum of $R$, then either $\depth(M\otimes_{R}N)\geq \depth(M)+\depth(N)-\depth(R)$, or $\depth(M\otimes_{R}N)\leq \cod(R)$. Along the way we generalize an important theorem of D. A. Jorgensen and determine the number of consecutive vanishing of $\Tor_i^R(M,N)$ required to ensure the vanishing of all higher $\Tor_i^R(M,N)$., Grant information included. To appear in Journal of Pure and Applied Algebra

Avramov and Buchweitz proved that for finitely generated modules $M$ and $N$ over a complete intersection local ring $R$, $\Ext^i_R(M,N)=0$ for all $i\gg 0$ implies $\Ext^i_R(N,M)=0$ for all $i\gg 0$. In this note we give some generalizations of this result. Indeed we prove the above mentioned result when (1) $M$ is finitely generated and $N$ is arbitrary, (2) $M$ is arbitrary and $N$ has finite length and (3) $M$ is complete and $N$ is finitely generated.

Let S=K[x1,…,xn] be a polynomial ring over a field kand let / be a monomial ideal of S. The main result of this paper is an explicit minimal resolution of kover R= S/Iwhen / is a monomial almost complete intersection ideal of S. We also compute an upper bound on the multigraded resolution of k over a generalization of monomial almost complete intersection ring.

A new proof of the classification for tensor ideal thick subcategories of the bounded derived category, and the stable category, of modular representations of a finite group is obtained. The arguments apply more generally to yield a classification of thick subcategories of the bounded derived category of an artinian complete intersection ring. One of the salient features of this work is that it takes no recourse to infinite constructions, unlike the previous proofs of these results., 18 pages

A new construction of rings is introduced, studied, and applied. Given surjective homomorphisms $R\to T\gets S$ of local rings, and ideals in $R$ and $S$ that are isomorphic to some $T$-module $V$, the \emph{connected sum} $R#_TS$ is defined to be the local ring obtained by factoring out the diagonal image of $V$ in the fiber product $R\times_TS$. When $T$ is Cohen-Macaulay of dimension $d$ and $V$ is a canonical module of $T$, it is proved that if $R$ and $S$ are Gorenstein of dimension $d$, then so is $R#_TS$. This result is used to study how closely an artinian ring can be approximated by Gorenstein rings mapping onto it. It is proved that when $T$ is a field the cohomology algebra $\Ext^*_{R#_kS}(k,k)$ is an amalgam of the algebras $\Ext^*_{R}(k,k)$ and $\Ext^*_{S}(k,k)$ over isomorphic polynomial subalgebras generated by one element of degree 2. This is used to show that when $T$ is regular, the ring $R#_TS$ almost never is complete intersection., This version includes a new theorem (Theorem 1.8), in which results due to D'Anna and Shapiro are completed and strengthened. Other changes to the text are minor. To appear in Crelle's J

A cohomological support, Supp_A(M), is defined for finitely generated modules M over an left noetherian ring R, with respect to a ring A of central cohomology operations on the derived category of R-modules. It is proved that if the A-module Ext^R(M,M) is noetherian and Ext_i^R(M,R)=0 for i>>0, then every closed subset of Supp_A(M) is the support of some finitely generated R-module. This theorem specializes to known realizability results for varieties of modules over group algebras, over local complete intersections, and over finite dimensional algebras over a field. The theorem is also used to produce large families of finitely generated modules of finite projective dimension over commutative local noetherian rings., To appear in the Illinois Journal of Mathematics, the issue honoring Phillip Griffith. Revised version has 18 pages. A word (the first one) has been added to the title and the material has been reorganized into seven sections, in place of the original six. There are, however, no changes of any substance

This text is based on the notes for a series of five lectures to the Barcelona Summer School in Commutative Algebra at the Centre de Recerca Matematica, Institut d’Estudis Catalans, July 15–26, 1996.

Let R be a regular local ring, and let A = R/(x), where x is any nonunit of R. We prove that every minimal free resolution of a finitely generated A -module becomes periodic of period 1 or 2 after at most dim A steps, and we examine generalizations and extensions of this for complete intersections. Our theorems follow from the properties of certain universally defined endomorphisms of complexes over such rings. Let A be a commutative ring, and let x G A be a nonzero divisor. How does homological algebra over A/(x) = B differ from that over AI In this paper we will study a certain natural endomorphism / of complexes of free A / (x)-modu\es which seems to reflect some of the difference. For example, the (homotopic) triviality of t is an obstruction (closely related to the usual one in Ext2,) to the lifting of a complex of free 5-modules to a complex of free /I-modules. More generally, if x,, . . . , xn is an A -sequence, we study « natural endomorphisms /,,..., tn of complexes of free A/(xx, . . . , x")-modules, and try to use them to explain the way in which free resolutions over A/(xx, . . . , x") differ from free resolutions over A (the construction and elementary properties of these endomorphisms is given in §1). In this paper, we will study the case in which A is a regular local ring and B = A/(xx, . . . , x") is not regular. (It would also be very interesting to understand the case in which both A and A/(x) = B were regular-with, say, A of mixed characteristic and B ramified or of characteristic p.) In this case, the homological algebra over A is dominated, roughly speaking, by the fact that minimal /I-free resolutions are finite; we seek to understand the eventual behavior of minimal 5-free resolutions in terms of the tt. For example, if « = 1, so that B = A/(x), we prove that / is eventually an isomorphism, so that every minimal 5-free resolution becomes periodic of period 2 after at most 1 + dim B steps (§6). We also show that the 5-modules with periodic resolutions are the maximal Cohen-Macaulay modules without free direct summands. Since the periodic part of a periodic resolution over A/(x) (or more generally, over A/(xx, . . . , xn), if x,, . . . , x" is an A -sequence) is easy to describe explicitly (§5), this yields information on maximal Cohen- Macaulay modules.

A simple criterion is given for determining “almost completely” whether the positively graded complete intersection ring R = K [ X 1 , … , X n + t ] / ( G i , … , G t ) R = K[{X_1},\, \ldots ,\,{X_{n + t}}]/({G_i},\, \ldots ,\,{G_t}) , of dimension n n , has an F F -pure type singularity at m = ( X 1 , … , X n + t ) m = ({X_1},\, \ldots ,\,{X_{n + t}}) . Specifically, if deg ⁡ ( X i ) = α i > 0 \operatorname {deg} ({X_i}) = {\alpha _i} > 0 for 1 ≤ i ≤ n + t 1 \leq i \leq n + t and deg ⁡ ( G i ) = d i > 0 \operatorname {deg} ({G_i}) = {d_i} > 0 for i ≤ i ≤ t i \leq i \leq t , then there exists an integer δ \delta determined by the singular locus of R R such that: (1) R R has F F -pure type if Σ i = 1 t d i − Σ i = 1 n + t α i > δ \Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} > \delta . (2) R R does not have F F -pure type if Σ i = 1 t d i − Σ i = 1 n + t α i > 0 \Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} > 0 . The characterization given by this theorem is particularly effective if the singularity of R R at m m is isolated. In that case, δ = 0 \delta = 0 so that only the condition Σ i = 1 t d i − Σ i = 1 n + t α i = 0 \Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} = 0 is not solved by the above result. In particular, it follows from work of Kei-ichi Watanabe that if R R has an isolated rational singularity, then R R has F F -pure type. The converse is also “almost true” with the only exception being the case where Σ i = 1 t d i − Σ i = 1 n + t α i = 0 \Sigma _{i = 1}^t{d_i} - \Sigma _{i = 1}^{n + t}{\alpha _i} = 0 . In proving this criterion, a weak but more stable form of F F -purity, called F F -contractedness, is defined and explored. R R is F F -contracted (in characteristic p > 0 p > 0 ) if every system of parameters for m m is contracted with respect to the Frobenius map F : R → R F:\,R \to R . Just as for F F -purity, the notion of F F -contracted type is defined in characteristic 0 by reduction to characteristic p p . The two notions of F F -pure (type) and F F -contracted (type) coincide when R R is Gorenstein; whence, in particular, when R R is a complete intersection ring.

Let R be a noetherian commutative local ring, and M,N be finitely generated R-modules. Then a generalized form of Serre's Vanishing conjecture can be stated as follows: if (1) length (M⊗RN) (2) pd(M),pd(N) (3) dim M+ dim N dim R , then χ(M,N)≔ ∑ i=0 ∞ (−1) i length ( Tor i R (M,N))=0. It is known that Serre's Vanishing conjecture holds for a complete intersection ring R, but is not known for a Gorenstein ring R. We can make a similar conjecture replacing Tor by Ext, namely, if M and N satisfy the above three conditions, then ξ(M,N)≔ ∑ i=0 ∞ (−1) i length ( Ext R i (M,N))=0. In this paper, we will prove that, over a Gorenstein ring R, the Tor-version of the Vanishing conjecture, the Ext-version of the Vanishing conjecture, and the commutativity of the intersection multiplicity defined in Mori and Smith (J. Pure Appl. Algebra 157 (2001) 279) are all equivalent. Further, we will prove that a certain Ext-version of the Vanishing conjecture holds for a large class of noncommutative projective schemes, typically including all commutative projective schemes over a field, by extending Bezout's Theorem.

Let $\MS_g$ be the moduli space of stable holomorphic vector bundles of rank 2 and fixed determinant of odd degree over a smooth complex projective curve of genus $g$. This paper proves various properties of the rational cohomology ring $H^*(\MS_g)$. It is shown that the first relation in genus $g$ between the standard generators satisfies a recurrence relation in $g$ and that the invariant subring for the mapping class group is a complete intersection ring. (These two results have been obtained independently by Zagier, Baranovsky and Siebert & Tian.) A Gr��bner basis is found for the ideal of invariant relations. A structural formula for $H^*(\MS_g)$ (originally conjectured by Mumford) is verified and a natural monomial basis is given., 14 pages, Plain TeX with amssym, no figures. Hard copies available

Let k be an effective infinite perfect field, k [ x 1 ,…, x n ] the polynomial ring in n variables and F ∈ k [ x 1 ,…, x n ] M × M a square polynomial matrix verifying F 2 = F . Suppose that the entries of F are polynomials given by a straight-line program of size L and their total degrees are bounded by an integer D . We show that there exists a well parallelizable algorithm which computes bases of the kernel and the image of F in time ( nL ) O(1) ( MD ) O( n ) . By means of this result we obtain a single exponential algorithm to compute a basis of a complete intersection ring in Noether position. More precisely, let f 1 ,…, f n − r ∈ k [ x 1 ,…, x n ] be a regular sequence of polynomials given by a slp of size l, whose degrees are bounded by d . Let R ≔ k [ x 1 ,…, x r ] and S ≔ k [ x 1 ,…, x n ]/( f 1 ,…, f n − r ) such that S is integral over R ; we show that there exists an algorithm running in time O( n )l d O( n 2 ) which computes a basis of S over R . Also, as a consequence of our techniques, we show a single exponential well parallelizable algorithm which decides the freeness of a finite k [ x 1 ,…, x n ]-module given by a presentation matrix, and in the affirmative case it computes a basis.