Vladimir baranovsky abstract the multiplicative structure of the cohomology ring of the moduli space of stable rank 2 bundles on a smooth projective curve is computed. Introduction our starting point is the following 1957 result from hk. When the topological space is connected, the zeroth graded component of the cohomology ring is the base ring itself. Every vector bundle over a variety x gives a projective bundle by taking the projective spaces of the fibers, but not all projective bundles arise in this way. This definition is illuminative because many of the results about spheres, particularly. As a graded abelian group, it has, in its graded component, the cohomology group. As another example, one may consider the commutative ring cx of con tinuous maps x c.
The hochschild cohomology ring of a global quotient orbifold cris negron and travis schedler with appendices by pieter belmans, pavel etingof and the authors abstract. We start with the real projective spaces rpn, which we think of as ob. When xis projective, in addition to being symplectic, we conjecture that there is furthermore an algebra isomorphism hh xg. Consider the cw structure on the real projective space. Introduction let c be a complete smooth curve of genus g. One helpful way of thinking of integral cohomology maybe the following. These are notes intended for the authors algebraic topology ii lectures at the university of oslo in the fall term of 2011. Pdf on the spectrum of the equivariant cohomology ring.
A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. However, we will see that it yields more information than homology precisely because certain kinds of. In this survey, we discuss whether the complex projective space can be characterized by its integral cohomology ring among compact complex manifolds. It is thus easy to compute by hand even by picture. Graded rings and varieties in weighted projective space miles reid jan 2002 abstract this chapter is a rst introduction to weighted projective spaces. On hochschild cohomology ring of the integral group ring of the quaternion group hayami, takao, tsukuba journal of mathematics, 2005. Allen hatchers book algebraic topology 2, drawing on chapter 3 on cohomology and chapter 4 on homotopy theory. Weighted projective space is a toric delignemumford stack this is spelled out in so one can compute the orbifold cohomology ring using.
Borel construction, configuration space, integral cohomology ring. Introduction to equivariant cohomology in algebraic geometry dave anderson april 30, 2011 abstract introduced by borel in the late 1950s, equivariant cohomology encodes information about how the topology of a space interacts with a group action. In algebraic geometry, an affine variety or, more generally, an affine algebraic set is defined as the subset of an affine space that is the set of the common zeros of a set of socalled polynomial functions over the affine space. This is an associative ring with identity, and for homogeneous elements of degrees respectively. Introduction to equivariant cohomology in algebraic geometry. That is, we will use the koszulcech connection discussed two days ago. We introduce the cech cohomology and use it to calculate cohomology. The group cohomology of a discrete group g is the singular cohomology of a suitable space having g as its fundamental group, namely the corresponding eilenbergmaclane space. Nov 22, 2010 now, let us compute the cohomology of projective space over a ring.
For each group gand representation mof gthere are abelian groups hng,m and hng,m where n 0,1,2,3. For example, the grassmannian gr1, v is the space of lines through the origin in v, so it is the same as the projective space of one dimension lower than v when v is a real or complex vector space, grassmannians are compact smooth manifolds. For the existence of the mappings it is enough to have a set of mappings satisfying certain additional properties, and a mapping, that is, a multiplication in the coefficient group see. The cohomology ring of the moduli space of stable vector bundles with odd determinant.
A gentle introduction to homology, cohomology, and sheaf. An introduction to the cohomology of groups peter j. In this survey, we discuss whether the complex projective space can be char acterized by its integral cohomology ring among compact complex manifolds. In particular, we will consider the quasicoherent sheaf. Some remarks on the uniqueness of the complex projective spaces. Cohomology of projective space let us calculate the. In cohomology, you have the dual scenario, ie you attach an integer to every simplex in the topological space, and make identifications upto coboundaries. Brown department of mathematics, cornell university ithaca, ny 14853, usa email. Cohomology groups need not determine cohomology ring. Looking at the decomposition, we see that each of those classes is in fact the fundamental class of a projective subspace. For example, in 16 the equivariant cohomology ring h. Let xbe a projective variety over a noetherian ring and let o x1 be a very ample line bundle on x.
In mathematics, specifically algebraic topology, the cohomology ring of a topological space x is a ring formed from the cohomology groups of x together with the cup product serving as the ring multiplication. Mackey functor rings used in our discussion of the ring structure of the cohomology of complex projective spaces. The invariant is cohomology and the topological space family is complex projective space get more specific information about complex projective space get more computations of cohomology. Later in this course we will see a shorter proof of this theorem using poincar e duality. It is possible to have two topological spaces and in fact, we can choose both and to be compact connected orientable manifolds such that for every, we have an isomorphism of cohomology group. We first notice in this article that if a compact kahler manifold has the same integral cohomology ring. Pdf on the equivariant cohomology ring of hilbert scheme. A projective space rp1 is homeomorphic to the circle s1. Cohomology of projective space let us calculate the cohomology of projective space. Singular cohomology is a powerful invariant in topology, associating a gradedcommutative ring to any topological space. X y determines a homomorphism from the cohomology ring of y to that of x. The cohomology ring away from 2 of configuration spaces on.
What is the difference between homology and cohomology. For defining a polynomial function over the affine space. If is a commutative ring, the cohomology ring is gradedcommutative. We describe the integral equivariant cohomology ring of a weighted projective space in terms of piecewise polynomials, and thence by generators and relations. Homology, cohomology, and sheaf cohomology university of. It comes from the fact that if x is projective, one gets a ka. We study the 0th local cohomology module h0 mrf of the jacobian ring rf of a singular reduced complex projective hypersurface x, by relating it to the sheaf of logarithmic vector. In homology, you look at sums of simplices in the topological space, upto boundaries.
The cohomology of projective space climbing mount bourbaki. On the spectrum of the equivariant cohomology ring article pdf available in canadian journal of mathematics 622 april 2010 with 88 reads how we measure reads. We deduce that the ring is a perfect invariant, and prove a chern class formula for weighted projective bundles. Statement cohomology groups need not determine cohomology ring.
The resulting computation is almost completely geometric. In mathematics, the grassmannian grk, v is a space which parameterizes all kdimensional linear subspaces of the ndimensional vector space v. Free algebraic topology books download ebooks online textbooks. As an application of general techniques of cohomology we prove the grothendieck and serre vanishing theorems. Thus, the group cohomology of can be thought of as the singular cohomology of the circle s 1, and similarly for and. From the point of view of derived categories, the ith hochschild cohomology group hhia,a of an algebraaovera. We investigate the structure of the cohomology ring of the quot schemes quotr,n, i. In his paper, hurewicz proved that these spaces are. Analogously, the product defines a ring structure on the group, where is the dimensional singular cohomology group of a topological space with coefficients in. We do not assume kalgebraically closed since the most interesting case of this theorem is the case k q. Free algebraic topology books download ebooks online. The local cohomology of the jacobian ring edoardosernesi received.
Cohomology is more abstract because it usually deals with functions on a space. This is a basic note in algebraic topology, it introduce the notion of fundamental groups, covering spaces, methods for computing fundamental groups using seifert van kampen theorem and some applications such as the brouwers fixed point theorem, borsuk ulam theorem, fundamental theorem of algebra. Hence the cohomology ring of a connected space with coefficients in is a connected graded algebra. The significance of the cohomology ring is that two spaces x and y may have isomorphic.
Let us calculate the cohomology of projective space. February5,2014 communicatedbygavrilfarkas abstract. Let xi yi be the union of the real projective plane and a onesphere. We compute the r cohomology ring of the configuration space conf r p m, k of k ordered points in the mdimensional real projective space r p m. Inductively, is defined as a discrete twopoint space, and for any natural number, is defined as the suspension of. Chenruan orbifold cohomology ring of weighted projective space, which is obtained from the quantum cohomology ring by setting q 0, has been studied by a number of authors. The cohomology ring of the moduli space of stable vector. The cohomology ring turns out to be equipped with the structure of a graded ring. Rpn and all coe ecients for the cohomology groups are z2z coe cients. Pdf a gkm description of the equivariant cohomology ring of. Thus, the calculation of the cohomology ring of the grassmannian becomes a central task in studying vector bundles. Xy be a quasicompact and quasi separated morphism of algebraic spaces over s. The method is based on the fact that the orbit configuration space of k ordered points in the mdimensional sphere with respect to the antipodal action is a 2 kfold covering of conf r p m, k.
An implication of the classifying space nature of the grassmannian is that characteristic classes are in one to one correspondence with cohomology classes of the grassmannian. On the equivariant cohomology ring of hilbert scheme of points in the projective plane. The starting point for the topological aspect of the theory. It is clear from the computations in the proof of lemma 30. For a general reduced weighted projective space, we give a formula to compute the 3point function which is the key in the definition of chenruan cohomology ring. The equivariant cohomology ring of weighted projective space. Contents 1 introduction 2 2 motivic stable homotopy theory 4 3 the bott inverted in. Graded rings and varieties in weighted projective space.
The cohomology ring of a topological space with coefficients in a ring is a graded ring defined as follows. The atiyahhirzebruch spectral sequence allows one to compute the a cohomology groups of the space xin terms of the a cohomology groups of a point and the ordinary cohomology groups of the space x. However the equivariant cohomology is often easier to understand as a consequence of the localization theorem 3. The rational cohomology ring of the moduli space of. This article describes the value and the process used to compute it of some homotopy invariants for a topological space or family of topological spaces. Note that is quasicompact and separated, so we can compute the cech cohomology by the above machinery. The unit sphere is defined as the subset of euclidean space comprising those points whose distance from the origin is inductive definition. C of an equivariantly formal space x was described in terms of the. Following the standard notation, we will say that bg is an eilenbergmaclane space kg,1. A projective frame is a set of points in a projective space that allows defining coordinates. The purpose of this note is to survey the structure of the cohomology ring of a simply connected topological space x having only nontrivial cohomology groups hinix, zp zj, for t 0, 1, 2, k. We study the cup product on the hochschild cohomology of the stack quotient xg of a smooth quasi projective variety x by a nite group g. On the symmetric squares of complex and quaternionic projective.
This motivates the definition of an abelian semigroup of isomorphism classes of vector bundles over compact bases. The invariant is cohomology group and the topological space family is real projective space get more specific information about real projective space get more computations of cohomology group. The chenruan cohomology of weighted projective spaces. Transformation groups on cohomology protective spacesi. A gkm description of the equivariant cohomology ring of a homogeneous space article pdf available in journal of algebraic combinatorics 231.