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Graded vector space

A graded vector space is a vector space V over a field k that admits a decomposition V = \bigoplus_{n \in \mathbb{Z}} V_n as a direct sum of subspaces V_n, where each V_n consists of the homogeneous elements of degree n.^[1] This structure equips the vector space with an additional grading, allowing operations and morphisms to respect the degrees.^[2] More generally, the grading index set can be any set G, yielding a G-graded vector space V = \bigoplus_{g \in G} V_g, with the category of such spaces forming a functor category from the discrete category on G to the category of vector spaces over k.^[3] Key properties include the assignment of degrees to basis elements, degree-preserving linear maps as morphisms, and compatibility with algebraic operations such as the tensor product, where (V \otimes W)_n = \bigoplus_{i+j=n} V_i \otimes W_j.^[4] The dual of a graded vector space V has components V_n^* = (V_{-n})^*, and shifts V redefine degrees by adding n.^[3] Graded vector spaces form the foundational structure for numerous areas in mathematics, including homological algebra—where chain complexes are \mathbb{Z}-graded with differentials of degree 1—and supergeometry, where \mathbb{Z}/2\mathbb{Z}-gradings distinguish even and odd components in supervector spaces.^[5]^[4] They underpin graded algebras, such as symmetric and exterior algebras, and play a central role in the study of Lie superalgebras, operads, and graded manifolds in Poisson geometry.^[4] Special cases like \mathbb{N}-gradings appear in polynomial rings, while finite group gradings lead to fusion categories with braided structures.^[3]

Definitions and Basic Concepts

Definition

A graded vector space over a field k is a vector space V equipped with a family of subspaces (V_i)_{i \in I} such that V = \bigoplus_{i \in I} V_i, where I is an abelian monoid (for example, the integers \mathbb{Z}).^[6]^[7]^[3] The direct sum decomposition implies that every element of V is uniquely expressible as a finite sum \sum v_i with v_i \in V_i for each i \in I, and the subspaces V_i satisfy V_i \cap V_j = \{0\} for i \neq j.^[6]^[7] An element v \in V_i is called homogeneous of degree i, and general elements are finite sums of such homogeneous components.^[7]^[6] The subspaces V_i are the homogeneous components (or graded pieces) of V.^[6] If each V_i is finite-dimensional and only finitely many are nonzero, then \dim V = \sum_{i \in I} \dim V_i.^[6] This constitutes an internal grading, where the V_i are subspaces of the single vector space V, in contrast to an external grading that assembles V from separate vector spaces via their direct sum.^[8] A common special case is the integer grading, where I = \mathbb{Z}.^[6]

Examples

A fundamental example of a graded vector space is the polynomial ring k over a field k, which is \mathbb{Z}_{\geq 0}-graded by assigning to each monomial x^n the degree n, so that the n-th graded component is the one-dimensional subspace spanned by x^n. Another standard example is the exterior algebra \Lambda(V) of a finite-dimensional vector space V over a field, which is \mathbb{Z}_{\geq 0}-graded with the n-th component \Lambda^n(V) consisting of the alternating n-forms on V, and the grading induced by the degree of the forms.^[9] In algebraic topology, chain complexes provide examples of \mathbb{Z}-graded vector spaces; for instance, the space of n-chains C_n in the singular chain complex of a topological space forms the n-th graded piece, with the total chain space being the direct sum \bigoplus_n C_n.^[10] Free graded modules over a graded ring, such as those appearing in minimal free resolutions of modules in commutative algebra, illustrate graded vector spaces where the grading is compatible with the module structure, often used to study syzygies and projective dimensions.^[11] In superalgebra, superspaces are \mathbb{Z}/2\mathbb{Z}-graded vector spaces, decomposed into even and odd parts V = V_0 \oplus V_1, where elements in V_0 have even parity and those in V_1 have odd parity, forming the foundation for Lie superalgebras and related structures.

Types of Gradings

Integer grading

A ℤ-graded vector space, often simply called an integer-graded vector space, is a vector space V equipped with a decomposition V = \bigoplus_{n \in \mathbb{Z}} V_n, where each V_n is a subspace (possibly zero-dimensional) and the index set \mathbb{Z} allows for both positive and negative degrees. This structure enables the assignment of an integer degree to each homogeneous component, facilitating applications where directionality in grading is essential, such as in contexts involving inverses or reversals. In such a space, vector addition and scalar multiplication preserve degrees: if u \in V_m and v \in V_n, then u + v \in V_m if m = n (and otherwise the sum decomposes accordingly), while \lambda u \in V_m for any scalar \lambda. If V is finite-dimensional, the total dimension is \dim V = \sum_{n \in \mathbb{Z}} \dim V_n < \infty, providing a measure of the space's overall size while respecting the grading. A key operation on ℤ-graded vector spaces is the graded shift, or suspension, denoted \Sigma V, defined by (\Sigma V)_n = V_{n-1} for all n \in \mathbb{Z}. This shifts all degrees upward by 1, with the underlying vector space isomorphism given by the identity map on elements, but reassigning degrees accordingly. The inverse operation, desuspension \Sigma^{-1} V, satisfies (\Sigma^{-1} V)_n = V_{n+1}, shifting degrees downward by 1. In the context of homological algebra, these shifts extend to chain complexes of vector spaces, where the differential on \Sigma C is the negative of that on C, preserving the graded structure while adjusting homological positions. Integer gradings are prevalent in algebraic topology, where homology and cohomology groups are naturally ℤ-graded by homological degree, capturing topological invariants across all integers (though often concentrated in non-negative degrees for simplicial complexes).^[12] For instance, the homology H_n(X) of a space X forms a ℤ-graded vector space over the coefficients field.^[12]

General grading

In general, a grading on a vector space V over a field k is specified by an arbitrary index set I, which may be a set without additional structure, such as a monoid or partially ordered set (poset), resulting in a direct sum decomposition V = \bigoplus_{i \in I} V_i, where each V_i is a subspace of V. This generalizes the notion of grading by allowing the components V_i to be indexed without requiring a total order on I, focusing solely on the decomposition into homogeneous subspaces. Unlike the integer-graded case, which assumes a linear ordering on \mathbb{Z}, this setup does not inherently impose shifts or ordered functors but emphasizes the partitioning of V. When the index set I is equipped with a partial order \leq, the grading becomes partially ordered, enabling the definition of subgradings and associated filtrations. For a subset J \subseteq I, the filtration is given by F_J V = \bigoplus_{\substack{i \leq j \\ j \in J}} V_i, which captures the "lower" components relative to the order. This construction induces a filtration on V, where the associated graded space recovers the original grading components, analogous to the \mathbb{[Z](/page/Z)}-case where \mathrm{gr}(F_n V) = V_n. Such partially ordered gradings are particularly useful in settings where the index set reflects a lattice or poset structure, allowing for more flexible decompositions in algebraic contexts.^[13] A prominent example of general grading arises in multi-graded spaces, where I = \mathbb{Z}^k or \mathbb{N}^m for k, m \geq 1, decomposing V into components labeled by multi-indices, such as bigraded spaces with bidegrees (p, q). In commutative algebra, polynomial rings provide a concrete illustration: the ring k[x_1, \dots, x_n] admits a multi-grading by \mathbb{N}^n, where the homogeneous component of multi-degree (a_1, \dots, a_n) consists of monomials x_1^{a_1} \cdots x_n^{a_n} scaled by coefficients in k.^[14] This multi-grading induces filtrations via the partial order on \mathbb{N}^n, facilitating the study of ideals and modules in algebraic geometry and representation theory.^[14]

Graded Morphisms

Homomorphisms

In the category of graded vector spaces, a homomorphism between two graded vector spaces V = \bigoplus_{i \in I} V_i and W = \bigoplus_{j \in J} W_j over a field k is a k-linear map f: V \to W that preserves the grading, meaning f(V_i) \subseteq W_i for each i \in I. Such maps are also called degree-zero homomorphisms, as they map homogeneous elements of degree i to homogeneous elements of the same degree i. This structure ensures that homomorphisms respect the direct sum decomposition inherent to the grading.^[15]^[16] Any graded homomorphism f: V \to W decomposes componentwise as f = \bigoplus_{i \in I} f_i, where each f_i: V_i \to W_i is a linear map between the corresponding homogeneous components. This decomposition follows directly from the grading preservation, allowing f to act independently on each graded piece while maintaining overall linearity. The space of all such homomorphisms, denoted \mathrm{Hom}_0(V, W), is the degree-zero component of the full homomorphism space \mathrm{Hom}(V, W) = \bigoplus_k \mathrm{Hom}^k(V, W), which is itself a graded vector space graded by degree.^[15]^[16] More generally, one considers homogeneous linear maps of degree k, which are linear maps f: V \to W satisfying f(V_i) \subseteq W_{i+k} for each i, where the grading sets allow such shifts (e.g., when I = J = \mathbb{Z}). The full space of linear maps between graded vector spaces decomposes as \mathrm{Hom}(V, W) = \bigoplus_k \mathrm{Hom}^k(V, W), where \mathrm{Hom}^k(V, W) consists of the homogeneous maps of degree k. Graded homomorphisms correspond precisely to the degree-zero component \mathrm{Hom}^0(V, W). This graded structure on the homomorphism space is crucial for applications in algebra and topology.^[15] For a graded homomorphism f: V \to W, both the kernel and image are graded subspaces. Specifically, \ker f = \bigoplus_{i \in I} \ker f_i, where each \ker f_i \subseteq V_i, making \ker f a graded sub-space of V. Similarly, \mathrm{im} f = \bigoplus_{i \in I} \mathrm{im} f_i \subseteq \bigoplus_{i \in I} W_i, inheriting the grading from W. These properties ensure that kernels and images respect the categorical structure of graded vector spaces. In the special case of \mathbb{Z}-graded vector spaces, graded homomorphisms of degree zero are closely related to chain maps in homological algebra. When V and W are equipped with differentials forming chain complexes, a graded homomorphism f that additionally commutes with the differentials—that is, df = fd—becomes a chain map, preserving the homological structure. This connection underpins much of algebraic topology and derived categories.^[15]

Isomorphisms

A graded isomorphism between two G-graded vector spaces V and W is defined as a bijective graded homomorphism f: V \to W such that its inverse f^{-1}: W \to V is also a graded homomorphism, meaning f(V_g) \subseteq W_g and f^{-1}(W_g) \subseteq V_g for all g \in G.^[17] Such an isomorphism is characterized by the existence of linear isomorphisms \phi_g: V_g \to W_g for each grading component g \in G, which collectively define f componentwise.^[3] This componentwise bijectivity implies that the graded dimensions are preserved, i.e., \dim V_g = \dim W_g for every g \in G.^[18] The category of G-graded vector spaces, often denoted \mathbf{GrVect} or \mathbf{Vect}^G, has graded vector spaces as objects and graded homomorphisms (degree-zero linear maps) as morphisms, with isomorphisms being the standard invertible morphisms in this category.^[3] Graded homomorphisms form the broader class of morphisms, of which isomorphisms are the bijective special case. Two G-graded vector spaces are isomorphic if and only if \dim V_g = \dim W_g for every g \in G, determining their isomorphism class by the graded dimension function.^[18] In the multi-graded setting, such as \mathbb{Z}^n-graded vector spaces, a graded isomorphism preserves the multi-degrees, equivalently consisting of componentwise linear isomorphisms V_{\mathbf{i}} \to W_{\mathbf{i}} for each multi-index \mathbf{i} \in \mathbb{Z}^n.^[3]

Operations and Constructions

Direct sums and products

The direct sum of a family of graded vector spaces \{V^j\}_{j \in J}, where each V^j = \bigoplus_i (V^j)_i, is the graded vector space \bigoplus_{j \in J} V^j defined by

\left( \bigoplus_{j \in J} V^j \right)_i = \bigoplus_{j \in J} (V^j)_i

for each degree i.^[19] This construction preserves the grading componentwise, making the direct sum itself a graded vector space.^[20] For infinite index sets J, the direct sum in each degree consists of elements with finite support, meaning only finitely many components (V^j)_i are non-zero for any fixed i; this ensures the result remains a vector space under the usual algebraic operations.^[21] The direct product of the family is the graded vector space \prod_{j \in J} V^j with

\left( \prod_{j \in J} V^j \right)_i = \prod_{j \in J} (V^j)_i

for each i, allowing arbitrary support across the index set.^[20] This structure is particularly useful in contexts involving duals, where the dual of a direct sum corresponds to the direct product of the duals, and in completions of graded spaces.^[20] Both the direct sum and direct product are graded operations, inheriting associativity and commutativity from the underlying category of vector spaces (up to canonical isomorphism). The unit for both is the zero graded vector space, with all components trivial. In the \mathbb{Z}-graded case, these operations commute with the suspension functor \Sigma, satisfying \Sigma\left( \bigoplus_{j \in J} V^j \right) \cong \bigoplus_{j \in J} \Sigma V^j.^[19]

Tensor products

In the category of \mathbb{Z}-graded vector spaces over a field k, the tensor product V \otimes W of two \mathbb{Z}-graded vector spaces V = \bigoplus_{i \in \mathbb{Z}} V_i and W = \bigoplus_{j \in \mathbb{Z}} W_j is the \mathbb{Z}-graded vector space whose nth homogeneous component is (V \otimes W)_n = \bigoplus_{i+j=n} V_i \otimes_k W_j, where each V_i \otimes_k W_j is the ordinary tensor product of vector spaces.^[22] This construction extends to gradings indexed by an abelian monoid I, with (V \otimes W)_n = \bigoplus_{i+j=n \in I} V_i \otimes_k W_j for n \in I.^[22] The graded tensor product satisfies a universal property with respect to graded bilinear maps: for any \mathbb{Z}-graded vector space Z, there is a bijection between k-bilinear maps V \times W \to Z that are graded (i.e., map V_i \times W_j into Z_{i+j}) and graded k-linear maps V \otimes W \to Z.^[22] As a consequence, the graded dual complexifies appropriately, yielding an isomorphism of graded vector spaces (V \otimes W)^* \cong \Hom_{\gr}(V, W^*), where \Hom_{\gr} denotes the space of graded linear maps and W^* is the graded dual of W with (W^*)_n = \Hom_k(W_{-n}, k).^[22] The graded tensor product is associative up to canonical isomorphisms of graded vector spaces: for \mathbb{Z}-graded vector spaces V, W, U, there exist graded isomorphisms (V \otimes W) \otimes U \cong V \otimes (W \otimes U) natural in all variables, induced by the associativity of the underlying ungraded tensor products componentwise.^[23] For \mathbb{Z}/2\mathbb{Z}-graded vector spaces (super vector spaces) V = V_0 \oplus V_1 and W = W_0 \oplus W_1, the super tensor product V \otimes W has homogeneous components (V \otimes W)_0 = (V_0 \otimes W_0) \oplus (V_1 \otimes W_1) and (V \otimes W)_1 = (V_0 \otimes W_1) \oplus (V_1 \otimes W_0).^[24] It is equipped with a symmetry (braiding) c_{V,W}: V \otimes W \to W \otimes V defined on homogeneous elements v \in V_p, w \in W_q (with parity p,q \in \{0,1\}) by c_{V,W}(v \otimes w) = (-1)^{pq} w \otimes v, ensuring the category of super vector spaces is symmetric monoidal with this sign convention for odd-odd interchanges.^[24] A prominent application arises in the exterior algebra \Lambda(V) of a \mathbb{Z}-graded vector space V, which is the quotient of the tensor algebra T(V) = \bigoplus_{n \geq 0} (V^{\otimes n}) (graded by total degree) by the two-sided ideal generated by homogeneous elements of the form v \otimes w + (-1)^{|v||w|} w \otimes v for v,w \in V homogeneous; this yields a graded-commutative algebra where the wedge product \wedge realizes alternating tensors compatible with the grading.^[23]

Invariants and Series

Hilbert–Poincaré series

The Hilbert–Poincaré series of a graded vector space provides a generating function that encodes the dimensions of its homogeneous components, serving as an important invariant in algebraic and combinatorial contexts.^[25] For a \mathbb{Z}_{\geq 0}-graded vector space V = \bigoplus_{n=0}^\infty V_n over a field k, with each V_n finite-dimensional, the Hilbert–Poincaré series is defined as the formal power series

P_V(t) = \sum_{n=0}^\infty \dim_k(V_n) t^n \in k[].

^[26] For a general \mathbb{Z}-graded vector space V = \bigoplus_{n \in \mathbb{Z}} V_n, the series extends to a Laurent series

P_V(t) = \sum_{n \in \mathbb{Z}} \dim_k(V_n) t^n.

In the multi-graded case, for a \mathbb{Z}^k-graded vector space V = \bigoplus_{(n_1, \dots, n_k) \in \mathbb{Z}^k} V_{n_1 \dots n_k}, the Hilbert–Poincaré series is the multivariable series

P_V(t_1, \dots, t_k) = \sum_{(n_1, \dots, n_k) \in \mathbb{Z}^k} \dim_k(V_{n_1 \dots n_k}) t_1^{n_1} \cdots t_k^{n_k}.

^[25] This series exhibits additivity under direct sums: if V and W are graded vector spaces, then P_{V \oplus W}(t) = P_V(t) + P_W(t), since the dimensions add componentwise.^[27] It is multiplicative under tensor products: for graded vector spaces V and W, P_{V \otimes W}(t) = P_V(t) P_W(t), reflecting the graded structure of the tensor product.^[25] In many algebraic settings, such as finitely generated graded modules over polynomial rings, the Hilbert–Poincaré series is a rational function.^[27]

Euler characteristic

The Euler characteristic of a finite-dimensional \mathbb{Z}-graded vector space V = \bigoplus_{n \in \mathbb{Z}} V_n over a field is defined as the alternating sum

\chi(V) = \sum_{n \in \mathbb{Z}} (-1)^n \dim V_n.

This integer invariant captures a parity-based summary of the grading structure, vanishing if the total dimension in even degrees equals that in odd degrees. It arises naturally as the evaluation of the Hilbert–Poincaré series P_V(t) = \sum_{n \in \mathbb{Z}} \dim V_n \, t^n at t = -1, so \chi(V) = P_V(-1).^[12]^[28] In the setting of a bounded chain complex C_\bullet of graded vector spaces with finite-dimensional total homology, the Euler characteristic is similarly \chi(C) = \sum_n (-1)^n \dim C_n. This equals the Euler characteristic of the homology graded vector space H_\bullet(C), \sum_n (-1)^n \dim H_n(C), because the alternating sum is preserved under the passage to homology via the rank-nullity theorem applied to the boundary maps in each degree. The invariance holds more generally for quasi-isomorphisms between complexes of finite type.^[12] A key topological interpretation appears in singular homology, where for a topological space X with finite-dimensional rational homology groups, the Euler characteristic is \chi(X) = \sum_{n \geq 0} (-1)^n \dim H_n(X; \mathbb{Q}), treating H_\bullet(X; \mathbb{Q}) as a \mathbb{Z}_{\geq 0}-graded vector space. This coincides with the alternating sum over the ranks of the cellular or simplicial chain groups when applicable.^[12] The Euler characteristic exhibits additivity over direct sums of finite-dimensional graded vector spaces: \chi(V \oplus W) = \chi(V) + \chi(W), as dimensions add componentwise. It is multiplicative over tensor products: \chi(V \otimes_k W) = \chi(V) \chi(W), reflecting the multiplicative property of the underlying Hilbert–Poincaré series P_{V \otimes W}(t) = P_V(t) P_W(t). These properties extend to chain complexes via the corresponding operations.^[28]

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